Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.44.0-wmf.6 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk 12 1654 2690315 2690212 2024-12-04T23:07:11Z MathXplore 2888076 Reverted edits by [[Special:Contributions/38.158.204.171|38.158.204.171]] ([[User_talk:38.158.204.171|talk]]) to last version by [[User:Tule-hog|Tule-hog]] using [[Wikiversity:Rollback|rollback]] 2669079 wikitext text/x-wiki [[w:Hyperlink|Hyperlinks]] allow users to move between pages. For some basic information about using hyperlinks, see [[Help:Editing#Links, URLs]]. There are three general types of hyperlinks recognized by [[MediaWiki]], each with associated [[w:Cascading Style Sheets|CSS formatting]] to distinguish them. This page (below) is like a user manual page about links. If you are new to making links, you might prefer to explore [[Making links]], a short tutorial about how to make links on [[Wikiversity]] pages. == Wikilinks == A '''wikilink''' or '''internal link''' links a page to another page within the same project. To link to a page, you need to put the name of the page within double square brackets <nowiki>[[like this]]</nowiki>. So, for example, if you add <nowiki>[[Technology in the classroom]]</nowiki> to a page in edit mode, it will create a link to [[Technology in the classroom]]. You can also name this link anything by adding a "|" (not the letter l, but the vertical line character) and other descriptive text after this character - eg. adding <nowiki>[[Technology in the classroom|a page about teaching with technology]]</nowiki> will display as: [[Technology in the classroom|a page about teaching with technology]]. '''Further detail:''' *<nowiki>[[a]]</nowiki> gives [[a]]. *<nowiki>[[a|b]]</nowiki> gives [[a|b]] (link to a, labelled b). *<nowiki>[[a]]b</nowiki> gives [[a]]b, just like <nowiki>[[a|ab]]</nowiki> does: [[a|ab]]. *<nowiki>[[a|b]]c</nowiki> gives [[a|b]]c, just like <nowiki>[[a|bc]]</nowiki> does: [[a|bc]]. *<nowiki>a[[b]]</nowiki> gives a[[b]]. *<nowiki>[[a]]<</nowiki>nowiki>b</nowiki> gives [[a]]<nowiki>b</nowiki>. *<nowiki>[[a]]''b''</nowiki> gives [[a]]''b''. *<nowiki>''[[a]]''b</nowiki> gives ''[[a]]''b. *<nowiki>[[a|b]]c<</nowiki>nowiki>d</nowiki> gives [[a|b]]c<nowiki>d</nowiki>. *<nowiki>[[a]][[b]]</nowiki> gives [[a]][[b]] (two links, but looking equal to the single link [[ab]]), even if the links are underlined ([[a]] [[b]] and [[a b]] look the same only if links are not underlined). Links with parameters (the link name) are said to be "piped" because of the pipe symbol used ( | ). MediaWiki automatically checks if the target of a wikilink exists ("existence detection"). If the page doesn't exist, the link leads to the editing screen, and it is assigned the class "new". Such wikilinks are nicknamed "red links" because they are colored red in the default stylesheet on a default installation of MediaWiki. Red links are useful in determining the current status of the page (created or not created), create incoming links to a future page, facilitates and incites page creation. Note that the image, category, and interlanguage syntax are the same as the wikilink syntax. Attempting to link normally will place the image on the page, add the page to the category and create an interlanguage link at the edge of the page. This can be prevented by prefixing a colon, which escapes the specific syntax. For example, <code><nowiki>[[:Category:Help]]</nowiki></code>, <code><nowiki>[[:fr:Help:Link]]</nowiki></code>, and <code><nowiki>[[:Image:Mediawiki.png]]</nowiki></code>. The existence of an internal link from a page to an existing or non-existing page is recorded in the [[pagelinks table]]. ===Stub feature=== A wikilink to an existing page will be in class 'stub' if the page is in the main namespace, it is not a redirect, '''''and''''' the number of bytes of the wikitext is less than the "threshold for stub display" set in the [[Help:Preferences|user's preferences]]. This allows users to immediately identify links to very short pages that probably need to be expanded. Alternately, a user may set a very high threshold to achieve any of the following: * Identify links to very large pages. However, the criterion is the size of the wikitext; possible inclusion of templates and images can make the rendered page large, even if the amount of wikitext is small. * Determine at a glance whether a link leads to the main namespace or not. However, this does not take into account ''redirects'' to the main namespace (even if the redirect itself is in the main namespace). * Identify links to redirects, for clean-up work such as bypassing redirects. However, section linking to a "stub" does not work. Although this is normally a minor issue, this may cause problems with users who set a very high threshold. == Interwiki links == An '''[[Help:Interwiki linking|interwiki link]]''' links a page to a page on another website. Unlike the name suggests, the target site need not be a wiki, but it has to be on the [[m:Interwiki map|interwiki map]] specified for the source wiki. These links have the associated CSS class "extiw". These are in the same form as wikilinks above, but take a prefix which specifies the target site. For example, on Wikimedia projects and many other wikis <code><nowiki>[[wikipedia:Main Page]]</nowiki></code> links to Wikipedia's main page. The prefix can be hidden using the same piped syntax as wikilinks. === Interwiki links to the same project === Although interwiki links can be used to point to a wiki from itself, this is not generally recommended. MediaWiki does not detect whether or not the target page of an interwiki list exists, so there is no special formatting and the link is always to the view page. Further, MediaWiki does not check if the page is linking to itself. A [[Help:Self link|self wikilink]] is bolded (like [[{{NAMESPACE}}:{{PAGENAME}}]]), whereas a self interwiki link is normal ([[m:{{NAMESPACE}}:{{PAGENAME}}]]). Pros: * A copy of the wikitext on a sister project may still point to the same page. Sometimes two prefixes are needed for that purpose, e.g. [[w:de:a]]. == External links == {{for|the policy proposal on which links are appropriate|Wikiversity:External links}} External links use absolute URLs that link directly to any webpage. External links are in the form <code><nowiki>[http://www.example.org link name]</nowiki></code> (resulting in [http://www.example.org link name]), with the link name separated from the URL by a space. Links without link names will be numbered: <code><nowiki>[http://www.example.org]</nowiki></code> becomes [http://www.example.org]. Links with no square brackets will be displayed in their entirety: http://www.example.org. See [[m:Help:URL|URLs in external links]] for more detailed information. [[Special:Linksearch]] finds all pages linking to a given site. === External links to the same project === External links are often used to use special URL parameters in links. This allows links directly to the edit history of a page, to a page in edit view, a diff of two versions, et cetera. They can also be used to create a [[Help:Navigational image|navigational image]]. However, the use of external links to link to a normal page on the same project is not recommended. These links benefit from none of the features of a wikilink, and may break the web of links when the content is exported to another domain. ===Arrow icon=== Monobook skin produces an arrow icon after every external link. This can be suppressed with class="plainlinks": *<nowiki><</nowiki>span class="plainlinks">http<nowiki>://a</</nowiki>span> gives <span class="plainlinks">http://a</span> *http<nowiki>://a</nowiki> gives http://a == Section linking == === Headings === Links in the form <code><nowiki>[[#heading_name]]</nowiki></code> will link to the corresponding section header on the same page. === Anchors === {{shortcut|WV:ANCHOR}} Links in the form <code><nowiki>[[#anchor_name]]</nowiki></code> will link to any anchor named "anchor_name" on the page. This may be either a heading named "anchor_name", or an arbitrary position. <code><nowiki>[[#top]]</nowiki></code> is a reserved name that links to the [[#top|top of a page]]. It is possible to create an arbitrary anchor name using the HTML code <code><nowiki><div id="anchor_name"></div></nowiki></code>. Anchor links can also be appended to any type of link; for more information, see [[w:Help:Section#Section_linking]]. === Problems with page name conversion === Note that if the page name is automatically converted (for example, from "/wiki/main Page" to "/wiki/Main Page"), the section link will still work but will disappear from the address bar. As a consequence, this will make it more difficult to bookmark the section itself. This is not applicable for wikilinks, because the conversions have already taken place on Preview or Save of the referring page. For example, consider [[w:Wikipedia:how_to_edit_a_page#Links_and_URLs|http://en.wikipedia.org/wiki/Wikipedia:how_to_edit_a_page#Links_and_URLs]]. In this case, the anchor part of the address will disappear because the "how" will be converted to "How". === Redirects with section links === A [[Help:Redirect|redirect]] to a page section does not go to the section. However, one can add the section anyway as a clarification, and it will work if the redirect is manually clicked from the redirect page. However, links with a section to a redirect will lead to the section on the redirect's page. ==Subpage feature== MediaWiki has a subpage feature, although activation depends on project and namespace. If activated, the following applies (if not, "A/b" is just a page with that name). A [[w:tree structure|tree structure]] of pages is established by using forward slashes in pagenames: A/b is a child of A, hence A is a parent of A/b; also A/b/c is a child of A/b; A/a, A/b, and A/c are siblings. At the top of the subpage body links to all ancestor pages are shown automatically, without any corresponding wikitext. The links show up even if the child page does not exist, but the sequence of ancestors stops before any non-existing ancestor page (e.g., if the grandparent page does not exist, the parent page is not shown either). Like most letters of a page name, the first letter after the slash is case-sensitive; "/subpage" and "/Subpage" are different pages. [[Help:What links here|What links here]] and [[Help:Related changes|Related changes]] ignore these automatic links. === Relative links === Relative links still work if all pages of a tree are renamed according a name change of the root, including making it a child of a new root. Inside a subpage hierarchy the following relative links can be used: * <nowiki>[[../]]</nowiki> links to the parent of the current subpage, e.g., on A/b it links to A, on A/b/c it links to A/b. * <nowiki>[[../s]]</nowiki> links to a sibling of the current subpage, e.g., on A/b, it links to A/s. *<nowiki>[[/s]]</nowiki> links to a subpage, e.g. on A it is the same as <nowiki>[[A/s]]</nowiki>. See also [[w:Wikipedia:Subpages]], and the example pages [[m:Link/a/b]] and [[m:Help:Link/a/b]]. For the latter the subpage feature does not work because of the namespace. === User space === Subpages of a user page (<code><nowiki>[[User:Username/Subpage]]</nowiki></code>) are considered to be in that user's "user space". Rules are often relaxed in a user's own subpages, whereas they are typically tightened for a user editing another user's subpages. ===Unintended subpage structure=== Any slash in a pagename causes a subpage structure, e.g. [[Subpage demo Season 2006/2007]] is a subpage of "Subpage demo Season 2006". As long as the latter does not exist, this has no effect on the former, However, a page with a slash in its name cannot be the root page of a subpage structure. For example, [[Subpage demo Season 2006/2007 /soccer]] does not show its parent, because its unintended grandparent does not exist. A dummy grandparent page can fix this. === Subpage activation === Wikipedia has this feature activated in all talk namespaces and the user and project namespace. The Meta-Wiki also has it in the main namespace. The default is set in [[mw:Help:Configuration settings|DefaultSettings.php]][http://cvs.sourceforge.net/viewcvs.py/wikipedia/phpwiki/newcodebase/DefaultSettings.php?rev=1.21&view=markup]. As of revision 1.21, the following namespaces have it activated by default: Special, Main talk, User and User_talk, Meta_talk, and Image_talk. Settings per project are changed in [[LocalSettings.php]][http://cvs.sourceforge.net/viewcvs.py/wikipedia/phpwiki/newcodebase/LocalSettings.php?rev=1.25&view=markup]. == Miscellaneous == === Linking to a page with images === It is possible to use images as links to other pages. For more information, see [[Help:Navigational images|use an image as a link to a page]]. === "Hover box" on links === On some browsers, holding the cursor over link will show a hover box containing the text of the link's HTML title attribute. MediaWiki sets this to the target page name (without the possible section indication) if it's a wikilink, the page name with prefix if it's an interwiki link, and the URL if it's an external link. This can be switched off in the [[Help:Preferences|user preferences]]. The browser may also show similar info, but with the possible section indication, in the address bar. For these effects a piped link is useful even if it not followed; for example, for displaying the meaning of an acronym (e.g. [[neutral point of view|NPOV]]) or any other remark. It is possible to produce a hover box without a link, see {{tim|H:title}}. === Disallowed characters === In internal and interwiki link style, a plus sign in a page name is not allowed, the HTML and hence the rendered page just shows the wikitext, e.g. [[a+b]]. In external link style a plus sign in the URL is retained. It is often equivalent with a space. See also below. In accordance with the rules explained in [[Help:Page name]], conversions are automatically made to [[Help:Special characters|non-literal characters]] in wiki and interwiki links. For example, "<code><nowiki>[[Help:Page%20name]]</nowiki></code>" becomes "[[Help:Page name]]". However, the opposite is true for external links; literal characters are converted into non-literal characters. For example, most browsers will convert ".../wiki/!" to ".../wiki/%21". A code like %70 in a redirect disables it, although the link works from the redirect page. For a redirect that works, the redirect page shows the canonical form of the target, unlike its preview page, which renders the link in the usual way. === Special pipe syntax === Using an empty pipe syntax on wiki and interwiki links will hide interwiki prefixes and parentheses. For example, <nowiki>[[w:Mercury (planet)|]]</nowiki> becomes [[w:Mercury (planet)|Mercury]]. This pipe syntax should only be used where the unqualified reference is not ambiguous, such as in an article about the solar system. See [[Help:Piped link]]. === Additional effects of links === *[[Help:Related changes|Related changes]] *[[Help:What links here|Backlink]]s *[[Help:Preferences#Date_format|Date format]] * Using a space after the pipe syntax (<nowiki>[[Main page| ]]</nowiki>) produces (perhaps depending on the browser) a space only, not a link ("[[Main page| ]]"), but it is treated as a link for the "what links here" feature. ===Links from a page=== With [[query]] the links from page ''pagename'', sorted by namespace, and for each namespace alphabetically, are given by <nowiki>{{SERVER}}{{SCRIPTPATH}}</nowiki>/query.php?what=links&titles=''pagename'' , e.g. {{SERVER}}{{SCRIPTPATH}}/query.php?what=links&titles={{FULLPAGENAME}} . ==See also== *[[Help:Calculation]] *[[Help:Interwiki linking]] *[[Help:Piped link]] *[[Help:Self link]] *[[Help:Template#Restrictions]] (and the next section) *[[Help:URL]] *[[Help:What links here]] *[[m:Parser testing/replaceInternalLinks]] *[[m:Parser testing/replaceExternalLinks]] *[[m:Links table]] *[[m:Brokenlinks table]] *[[w:Hyperlink|Hyperlink]] *[[w:Template:linkless|Linkless]] *[[Making links]] *[[w:Red Link|Red Link]] *[[w:Wikipedia:Canonicalization|Wikipedia:Canonicalization]] [[Category:Help]] [[es:Ayuda:Enlace]] 704binq9suq44154o7ogeesmxblnhm3 4 1786 2690285 2690160 2024-12-04T16:58:21Z Atcovi 276019 /* Vandalism */ Reply 2690285 wikitext text/x-wiki {{Shortcut|WV:NOTICE|WV:AN}} {{/Header}} == [[Special:Contribs/84.126.165.13]] == Vandalism. – [[User:Tryvix1509|Tryvix1509]] ([[User talk:Tryvix1509|discuss]] • [[Special:Contributions/Tryvix1509|contribs]]) 14:42, 11 January 2023 (UTC) :CC {{ping|Antandrus}} since you were deleted a vandalism page. – [[User:Tryvix1509|Tryvix1509]] ([[User talk:Tryvix1509|discuss]] • [[Special:Contributions/Tryvix1509|contribs]]) 15:23, 11 January 2023 (UTC) :I'm not able to block, only protect and delete - I can protect it from re-creation though. [[User:Antandrus|Antandrus]] ([[User talk:Antandrus|discuss]] • [[Special:Contributions/Antandrus|contribs]]) 15:25, 11 January 2023 (UTC) ::Blocked and hidden. Thank you, &mdash;Hasley&nbsp;[[user talk:Hasley|<span style="color: #0645AD; vertical-align: super; font-size: smaller;">talk</span>]] 16:10, 11 January 2023 (UTC) :::See also [[Special:Contributions/Goggdoggdetroyer]]. This is a long-term abuser who will edit from several IPs. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:54, 12 January 2023 (UTC) ::::[[Special:Contributions/201.146.32.131]] —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:55, 12 January 2023 (UTC) :::::Please make sure to revdel all of these as well. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:56, 12 January 2023 (UTC) ::::::[[Special:Contributions/189.129.121.186]] —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:17, 12 January 2023 (UTC) :::::::[[Special:Contributions/189.223.219.134]] —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:20, 12 January 2023 (UTC) ::::::::{{done}}. &mdash;Hasley&nbsp;[[user talk:Hasley|<span style="color: #0645AD; vertical-align: super; font-size: smaller;">talk</span>]] 15:16, 12 January 2023 (UTC) == [[Special:Contribs/41.214.18.231]] == Long-term abuse. [[User:Tryvix1509|Tryvix1509]] ([[User talk:Tryvix1509|discuss]] • [[Special:Contributions/Tryvix1509|contribs]]) 08:05, 24 January 2023 (UTC) :{{done}} by GV. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:33, 24 January 2023 (UTC) == [[Special:Contribs/ApprenticeFan]] == Long-term abuse. – [[Special:Contributions/2405:4802:1AB:E140:91F9:4320:8977:E239|2405:4802:1AB:E140:91F9:4320:8977:E239]] ([[User talk:2405:4802:1AB:E140:91F9:4320:8977:E239|discuss]]) 14:59, 28 January 2023 (UTC) :Globally locked. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 15:17, 28 January 2023 (UTC) == [[Special:Contribs/41.202.78.57]] == Long-term abuse: Globally+WMF banned: [[:w:en:WP:LTA/GRP]]. – [[Special:Contributions/2405:4802:36C:3B0:1C90:9B74:603F:E6F|2405:4802:36C:3B0:1C90:9B74:603F:E6F]] ([[User talk:2405:4802:36C:3B0:1C90:9B74:603F:E6F|discuss]]) 13:02, 30 January 2023 (UTC) :I will take no action on this because I am confused: One IP user is requesting that action be taken against another IP user. Who is [[User:2405:4802:36C:3B0:1C90:9B74:603F:E6F]]?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:43, 31 January 2023 (UTC) == [[Special:Contributions/2A02:1210:5493:6F00:8D60:CF27:AC3E:C328]] == Block and revdel, please. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:29, 13 February 2023 (UTC) :{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:59, 13 February 2023 (UTC) == [[Special:Contributions/KEMONO PANTSU X]] == Vandalism, please block. [[User:Blua lago|Blua lago]] ([[User talk:Blua lago|discuss]] • [[Special:Contributions/Blua lago|contribs]]) 08:29, 29 March 2023 (UTC) : {{done}} The user has been locked. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:15, 29 March 2023 (UTC) ==[[Special:Contributions/2600:1003:B03E:C660:0:19:5833:C401]]== Revdel, please. --[[User:Jan.Kamenicek|Jan Kameníček]] ([[User talk:Jan.Kamenicek|discuss]] • [[Special:Contributions/Jan.Kamenicek|contribs]]) 19:12, 23 April 2023 (UTC) :@[[User:Jan.Kamenicek|Jan.Kamenicek]]: {{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 16:13, 24 April 2023 (UTC) == GRP socks == * {{user|170.83.119.121}} * {{user|181.115.60.123}} Long-term abuse: [[w:WP:LTA/GRP]]. [[User:Leonidlednev|Leonidlednev]] ([[User talk:Leonidlednev|discuss]] • [[Special:Contributions/Leonidlednev|contribs]]) 02:13, 4 May 2023 (UTC) :{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 03:26, 4 May 2023 (UTC) ::Revdel request: [[Special:Contributions/154.246.203.175]] —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:44, 7 May 2023 (UTC) == [[special:contribs/189.223.144.31|189.223.144.31]] == * {{user|189.223.144.31}} * Report reason: Open proxy (used by [[w:WP:LTA/GRP|GRP]]), left an abusive edit summary on my talk page. [[User:Leonidlednev|Leonidlednev]] ([[User talk:Leonidlednev|discuss]] • [[Special:Contributions/Leonidlednev|contribs]]) 01:50, 14 May 2023 (UTC) :{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 02:09, 14 May 2023 (UTC) == [[special:contribs/2601:206:8580:35D0:714E:77DC:EAE3:DFE3|2601:206:8580:35D0:714E:77DC:EAE3:DFE3]] == * {{user|2601:206:8580:35D0:714E:77DC:EAE3:DFE3}} * Report reason: Edit warring on [[Types of Dinosaur]] (see [[special:diff/2521104|1]], [[special:diff/2521102|2]], [[special:diff/2521100|3]], [[special:diff/2521098|4]]) [[User:Leonidlednev|Leonidlednev]] ([[User talk:Leonidlednev|discuss]] • [[Special:Contributions/Leonidlednev|contribs]]) 23:10, 16 May 2023 (UTC) :Now blocked, thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:56, 17 May 2023 (UTC) == [[special:contribs/I Join john|I Join john]] == * {{user|I Join john}} * Report reason: spam [[User:Leonidlednev|Leonidlednev]] ([[User talk:Leonidlednev|discuss]] • [[Special:Contributions/Leonidlednev|contribs]]) 02:58, 10 June 2023 (UTC) :Late notice, but for recording purposes: blocked by Dave Braunschweig on June 17. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:52, 3 October 2023 (UTC) == New special page to fight spam == {{int:please-translate}} <div lang="en" dir="ltr" class="mw-content-ltr"> Hello, We are replacing most of the functionalities of [[MediaWiki:Spam-blacklist]] with a new special page called [[Special:BlockedExternalDomains]]. In this special page, admins can simply add a domain and notes on the block (usually reasoning and/or link to a discussion) and the added domain would automatically be blocked to be linked in Wikis anymore (including its subdomains). Content of this list is stored in [[MediaWiki:BlockedExternalDomains.json]]. You can see [[:w:fa:Special:BlockedExternalDomains]] as an example. Check [[phab:T337431|the phabricator ticket]] for more information. This would make fighting spam easier and safer without needing to know regex or accidentally breaking wikis while also addressing the need to have some notes next to each domain on why it’s blocked. It would also make the list of blocked domains searchable and would make editing Wikis in general faster by optimizing matching links added against the blocked list in every edit (see [[phab:T337431#8936498]] for some measurements). If you want to migrate your entries in [[MediaWiki:Spam-blacklist]], there is a python script in [[phab:P49299]] that would produce contents of [[MediaWiki:Spam-blacklist]] and [[MediaWiki:BlockedExternalDomains.json]] for you automatically migrating off simple regex cases. Note that this new feature doesn’t support regex (for complex cases) nor URL paths matching. Also it doesn’t support bypass by spam whitelist. For those, please either keep using [[MediaWiki:Spam-blacklist]] or switch to an abuse filter if possible. And adding a link to the list might take up to five minutes to be fully in effect (due to server-side caching, this is already the case with the old system) and admins and bots automatically bypass the blocked list. Let me know if you have any questions or encounter any issues. Happy editing. [[User:Ladsgroup|Amir]] ([[User talk:Ladsgroup|talk]]) 09:41, 19 June 2023 (UTC) </div> <!-- Message sent by User:Ladsgroup@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Ladsgroup/target_ANs&oldid=25167735 --> ==Vandal: [[User:Ceoalphonso]]== They were spamming on Wikipedia, later claimed that their account was hacked. They were blocked for having a compromised account. They have started editing Wikiversity. High risk of future vandalism. They should be blocked for having a compromised account, in spite of the fact that this is dubious, because of AGF. [[User:Janhrach|Janhrach]] ([[User talk:Janhrach|discuss]] • [[Special:Contributions/Janhrach|contribs]]) 07:45, 30 September 2023 (UTC) :This user hasn't edited since February. Unless their editing picks back up and becomes problematic, there's no action necessary at this time. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:47, 30 September 2023 (UTC) ::{{ping|Atcovi}} They have not been active here since February, but they have made several edits across other wikis in the meantime. Their edits include highly visible vandalism, such as spamming in TemplateData. I would highly recommend a block. [[User:Janhrach|Janhrach]] ([[User talk:Janhrach|discuss]] • [[Special:Contributions/Janhrach|contribs]]) 14:52, 30 September 2023 (UTC) :::Since I still personally disagree, I'll leave this another custodian to take whatever action they deem reasonable. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:06, 30 September 2023 (UTC) == [[Special:Contributions/2600:1003:B05E:6EC6:0:33:95BF:E701]] == * {{user|2600:1003:B05E:6EC6:0:33:95BF:E701}} Cross-wiki vandalism. --[[User:SHB2000|SHB2000]] ([[User talk:SHB2000|discuss]] • [[Special:Contributions/SHB2000|contribs]]) 01:21, 2 June 2024 (UTC) :{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:22, 2 June 2024 (UTC) ::Thank you, Koavf! --[[User:SHB2000|SHB2000]] ([[User talk:SHB2000|discuss]] • [[Special:Contributions/SHB2000|contribs]]) 01:26, 2 June 2024 (UTC) == Maintenance vs administration == Bumping a [[User_talk:Tule-hog#Wikiversity:Administration|redirect for discussion]] concerning custodianship. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:50, 6 October 2024 (UTC) == Vandalism == Please block [[Special:Contributions/76.121.217.240]]. --[[User:Tres Libras|Tres Libras]] ([[User talk:Tres Libras|discuss]] • [[Special:Contributions/Tres Libras|contribs]]) 03:09, 24 November 2024 (UTC) :Not a local sysop here but {{done}}. --[[User:SHB2000|SHB2000]] ([[User talk:SHB2000|discuss]] • [[Special:Contributions/SHB2000|contribs]]) 03:18, 24 November 2024 (UTC) :: {{comment}} Page deleted. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:21, 24 November 2024 (UTC) :::Hi it's me again. Please block [[Special:Contributions/216.186.51.108]] &mdash; @[[User:MathXplore|MathXplore]] @[[User:SHB2000|SHB2000]]. [[User:Tres Libras|Tres Libras]] ([[User talk:Tres Libras|discuss]] • [[Special:Contributions/Tres Libras|contribs]]) 19:54, 2 December 2024 (UTC) ::::{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 19:58, 2 December 2024 (UTC) :::::Hello. Again and again. Please block [[Special:Contributions/152.22.75.23]]. [[User:Tres Libras|Tres Libras]] ([[User talk:Tres Libras|discuss]] • [[Special:Contributions/Tres Libras|contribs]]) 19:48, 3 December 2024 (UTC) ::::::Already done by Aramil. I think Wikiversity should take action against proxies. I suspect that these IPs are proxies and belong to the same person. [[User:Tres Libras|Tres Libras]] ([[User talk:Tres Libras|discuss]] • [[Special:Contributions/Tres Libras|contribs]]) 19:52, 3 December 2024 (UTC) :::::::Possibly {{ping|Mu301}} has some experience dealing with proxies? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:58, 4 December 2024 (UTC) c708cwaqfrv8omfozi2z5taixz9qsiu 2690286 2690285 2024-12-04T16:59:13Z Atcovi 276019 /* Vandal: User:Ceoalphonso */ Reply 2690286 wikitext text/x-wiki {{Shortcut|WV:NOTICE|WV:AN}} {{/Header}} == [[Special:Contribs/84.126.165.13]] == Vandalism. – [[User:Tryvix1509|Tryvix1509]] ([[User talk:Tryvix1509|discuss]] • [[Special:Contributions/Tryvix1509|contribs]]) 14:42, 11 January 2023 (UTC) :CC {{ping|Antandrus}} since you were deleted a vandalism page. – [[User:Tryvix1509|Tryvix1509]] ([[User talk:Tryvix1509|discuss]] • [[Special:Contributions/Tryvix1509|contribs]]) 15:23, 11 January 2023 (UTC) :I'm not able to block, only protect and delete - I can protect it from re-creation though. [[User:Antandrus|Antandrus]] ([[User talk:Antandrus|discuss]] • [[Special:Contributions/Antandrus|contribs]]) 15:25, 11 January 2023 (UTC) ::Blocked and hidden. Thank you, &mdash;Hasley&nbsp;[[user talk:Hasley|<span style="color: #0645AD; vertical-align: super; font-size: smaller;">talk</span>]] 16:10, 11 January 2023 (UTC) :::See also [[Special:Contributions/Goggdoggdetroyer]]. This is a long-term abuser who will edit from several IPs. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:54, 12 January 2023 (UTC) ::::[[Special:Contributions/201.146.32.131]] —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:55, 12 January 2023 (UTC) :::::Please make sure to revdel all of these as well. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:56, 12 January 2023 (UTC) ::::::[[Special:Contributions/189.129.121.186]] —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:17, 12 January 2023 (UTC) :::::::[[Special:Contributions/189.223.219.134]] —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:20, 12 January 2023 (UTC) ::::::::{{done}}. &mdash;Hasley&nbsp;[[user talk:Hasley|<span style="color: #0645AD; vertical-align: super; font-size: smaller;">talk</span>]] 15:16, 12 January 2023 (UTC) == [[Special:Contribs/41.214.18.231]] == Long-term abuse. [[User:Tryvix1509|Tryvix1509]] ([[User talk:Tryvix1509|discuss]] • [[Special:Contributions/Tryvix1509|contribs]]) 08:05, 24 January 2023 (UTC) :{{done}} by GV. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:33, 24 January 2023 (UTC) == [[Special:Contribs/ApprenticeFan]] == Long-term abuse. – [[Special:Contributions/2405:4802:1AB:E140:91F9:4320:8977:E239|2405:4802:1AB:E140:91F9:4320:8977:E239]] ([[User talk:2405:4802:1AB:E140:91F9:4320:8977:E239|discuss]]) 14:59, 28 January 2023 (UTC) :Globally locked. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 15:17, 28 January 2023 (UTC) == [[Special:Contribs/41.202.78.57]] == Long-term abuse: Globally+WMF banned: [[:w:en:WP:LTA/GRP]]. – [[Special:Contributions/2405:4802:36C:3B0:1C90:9B74:603F:E6F|2405:4802:36C:3B0:1C90:9B74:603F:E6F]] ([[User talk:2405:4802:36C:3B0:1C90:9B74:603F:E6F|discuss]]) 13:02, 30 January 2023 (UTC) :I will take no action on this because I am confused: One IP user is requesting that action be taken against another IP user. Who is [[User:2405:4802:36C:3B0:1C90:9B74:603F:E6F]]?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:43, 31 January 2023 (UTC) == [[Special:Contributions/2A02:1210:5493:6F00:8D60:CF27:AC3E:C328]] == Block and revdel, please. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:29, 13 February 2023 (UTC) :{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:59, 13 February 2023 (UTC) == [[Special:Contributions/KEMONO PANTSU X]] == Vandalism, please block. [[User:Blua lago|Blua lago]] ([[User talk:Blua lago|discuss]] • [[Special:Contributions/Blua lago|contribs]]) 08:29, 29 March 2023 (UTC) : {{done}} The user has been locked. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:15, 29 March 2023 (UTC) ==[[Special:Contributions/2600:1003:B03E:C660:0:19:5833:C401]]== Revdel, please. --[[User:Jan.Kamenicek|Jan Kameníček]] ([[User talk:Jan.Kamenicek|discuss]] • [[Special:Contributions/Jan.Kamenicek|contribs]]) 19:12, 23 April 2023 (UTC) :@[[User:Jan.Kamenicek|Jan.Kamenicek]]: {{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 16:13, 24 April 2023 (UTC) == GRP socks == * {{user|170.83.119.121}} * {{user|181.115.60.123}} Long-term abuse: [[w:WP:LTA/GRP]]. [[User:Leonidlednev|Leonidlednev]] ([[User talk:Leonidlednev|discuss]] • [[Special:Contributions/Leonidlednev|contribs]]) 02:13, 4 May 2023 (UTC) :{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 03:26, 4 May 2023 (UTC) ::Revdel request: [[Special:Contributions/154.246.203.175]] —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:44, 7 May 2023 (UTC) == [[special:contribs/189.223.144.31|189.223.144.31]] == * {{user|189.223.144.31}} * Report reason: Open proxy (used by [[w:WP:LTA/GRP|GRP]]), left an abusive edit summary on my talk page. [[User:Leonidlednev|Leonidlednev]] ([[User talk:Leonidlednev|discuss]] • [[Special:Contributions/Leonidlednev|contribs]]) 01:50, 14 May 2023 (UTC) :{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 02:09, 14 May 2023 (UTC) == [[special:contribs/2601:206:8580:35D0:714E:77DC:EAE3:DFE3|2601:206:8580:35D0:714E:77DC:EAE3:DFE3]] == * {{user|2601:206:8580:35D0:714E:77DC:EAE3:DFE3}} * Report reason: Edit warring on [[Types of Dinosaur]] (see [[special:diff/2521104|1]], [[special:diff/2521102|2]], [[special:diff/2521100|3]], [[special:diff/2521098|4]]) [[User:Leonidlednev|Leonidlednev]] ([[User talk:Leonidlednev|discuss]] • [[Special:Contributions/Leonidlednev|contribs]]) 23:10, 16 May 2023 (UTC) :Now blocked, thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:56, 17 May 2023 (UTC) == [[special:contribs/I Join john|I Join john]] == * {{user|I Join john}} * Report reason: spam [[User:Leonidlednev|Leonidlednev]] ([[User talk:Leonidlednev|discuss]] • [[Special:Contributions/Leonidlednev|contribs]]) 02:58, 10 June 2023 (UTC) :Late notice, but for recording purposes: blocked by Dave Braunschweig on June 17. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:52, 3 October 2023 (UTC) == New special page to fight spam == {{int:please-translate}} <div lang="en" dir="ltr" class="mw-content-ltr"> Hello, We are replacing most of the functionalities of [[MediaWiki:Spam-blacklist]] with a new special page called [[Special:BlockedExternalDomains]]. In this special page, admins can simply add a domain and notes on the block (usually reasoning and/or link to a discussion) and the added domain would automatically be blocked to be linked in Wikis anymore (including its subdomains). Content of this list is stored in [[MediaWiki:BlockedExternalDomains.json]]. You can see [[:w:fa:Special:BlockedExternalDomains]] as an example. Check [[phab:T337431|the phabricator ticket]] for more information. This would make fighting spam easier and safer without needing to know regex or accidentally breaking wikis while also addressing the need to have some notes next to each domain on why it’s blocked. It would also make the list of blocked domains searchable and would make editing Wikis in general faster by optimizing matching links added against the blocked list in every edit (see [[phab:T337431#8936498]] for some measurements). If you want to migrate your entries in [[MediaWiki:Spam-blacklist]], there is a python script in [[phab:P49299]] that would produce contents of [[MediaWiki:Spam-blacklist]] and [[MediaWiki:BlockedExternalDomains.json]] for you automatically migrating off simple regex cases. Note that this new feature doesn’t support regex (for complex cases) nor URL paths matching. Also it doesn’t support bypass by spam whitelist. For those, please either keep using [[MediaWiki:Spam-blacklist]] or switch to an abuse filter if possible. And adding a link to the list might take up to five minutes to be fully in effect (due to server-side caching, this is already the case with the old system) and admins and bots automatically bypass the blocked list. Let me know if you have any questions or encounter any issues. Happy editing. [[User:Ladsgroup|Amir]] ([[User talk:Ladsgroup|talk]]) 09:41, 19 June 2023 (UTC) </div> <!-- Message sent by User:Ladsgroup@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Ladsgroup/target_ANs&oldid=25167735 --> ==Vandal: [[User:Ceoalphonso]]== They were spamming on Wikipedia, later claimed that their account was hacked. They were blocked for having a compromised account. They have started editing Wikiversity. High risk of future vandalism. They should be blocked for having a compromised account, in spite of the fact that this is dubious, because of AGF. [[User:Janhrach|Janhrach]] ([[User talk:Janhrach|discuss]] • [[Special:Contributions/Janhrach|contribs]]) 07:45, 30 September 2023 (UTC) :This user hasn't edited since February. Unless their editing picks back up and becomes problematic, there's no action necessary at this time. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:47, 30 September 2023 (UTC) ::{{ping|Atcovi}} They have not been active here since February, but they have made several edits across other wikis in the meantime. Their edits include highly visible vandalism, such as spamming in TemplateData. I would highly recommend a block. [[User:Janhrach|Janhrach]] ([[User talk:Janhrach|discuss]] • [[Special:Contributions/Janhrach|contribs]]) 14:52, 30 September 2023 (UTC) :::Since I still personally disagree, I'll leave this another custodian to take whatever action they deem reasonable. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:06, 30 September 2023 (UTC) ::::{{not done}} user has not edited since February 2023, so for record-keeping purposes, I'm marking this request as stale. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 4 December 2024 (UTC) == [[Special:Contributions/2600:1003:B05E:6EC6:0:33:95BF:E701]] == * {{user|2600:1003:B05E:6EC6:0:33:95BF:E701}} Cross-wiki vandalism. --[[User:SHB2000|SHB2000]] ([[User talk:SHB2000|discuss]] • [[Special:Contributions/SHB2000|contribs]]) 01:21, 2 June 2024 (UTC) :{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:22, 2 June 2024 (UTC) ::Thank you, Koavf! --[[User:SHB2000|SHB2000]] ([[User talk:SHB2000|discuss]] • [[Special:Contributions/SHB2000|contribs]]) 01:26, 2 June 2024 (UTC) == Maintenance vs administration == Bumping a [[User_talk:Tule-hog#Wikiversity:Administration|redirect for discussion]] concerning custodianship. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:50, 6 October 2024 (UTC) == Vandalism == Please block [[Special:Contributions/76.121.217.240]]. --[[User:Tres Libras|Tres Libras]] ([[User talk:Tres Libras|discuss]] • [[Special:Contributions/Tres Libras|contribs]]) 03:09, 24 November 2024 (UTC) :Not a local sysop here but {{done}}. --[[User:SHB2000|SHB2000]] ([[User talk:SHB2000|discuss]] • [[Special:Contributions/SHB2000|contribs]]) 03:18, 24 November 2024 (UTC) :: {{comment}} Page deleted. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:21, 24 November 2024 (UTC) :::Hi it's me again. Please block [[Special:Contributions/216.186.51.108]] &mdash; @[[User:MathXplore|MathXplore]] @[[User:SHB2000|SHB2000]]. [[User:Tres Libras|Tres Libras]] ([[User talk:Tres Libras|discuss]] • [[Special:Contributions/Tres Libras|contribs]]) 19:54, 2 December 2024 (UTC) ::::{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 19:58, 2 December 2024 (UTC) :::::Hello. Again and again. Please block [[Special:Contributions/152.22.75.23]]. [[User:Tres Libras|Tres Libras]] ([[User talk:Tres Libras|discuss]] • [[Special:Contributions/Tres Libras|contribs]]) 19:48, 3 December 2024 (UTC) ::::::Already done by Aramil. I think Wikiversity should take action against proxies. I suspect that these IPs are proxies and belong to the same person. [[User:Tres Libras|Tres Libras]] ([[User talk:Tres Libras|discuss]] • [[Special:Contributions/Tres Libras|contribs]]) 19:52, 3 December 2024 (UTC) :::::::Possibly {{ping|Mu301}} has some experience dealing with proxies? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:58, 4 December 2024 (UTC) hy9im643ha1z2tf0wpw4e044xzyl0jc 10 3915 2690332 2666466 2024-12-05T03:21:10Z Atcovi 276019 fixing up red link 2690332 wikitext text/x-wiki {{Userbox | id = {{{id|[[File:{{#if: {{{image|}}} | {{{image|}}} | {{#if:{{{noannounce|}}}|Wikiversity logo.svg|Wikiversity Administrator.svg}} }}|{{{logo-size|43x43px}}}]]}}} | info-s = {{{info-s|{{{info-size|8}}}}}} | info = This user {{#if:{{{noannounce|}}}|knows [[Wikiversity:User access levels|user roles]] ≠ user ''value''.|is an '''[[Wikiversity:Custodians|custodian]]'''{{#if:{{{othertext|}}}|&nbsp;{{{othertext}}}|}} on the {{{1|[[Wikiversity:Introduction|English Wikiversity]]}}}. {{#if:{{{1|}}}|<span style="font-size:0.9em;" class="plainlinks">([{{fullurl:{{{lang_code|en}}}:Special:UserRights|user={{urlencode:{{{username|{{BASEPAGENAME}}}}}}}&uselang=en}} <span style="color:#5871C6;">verify{{#if:{{{lang_code|}}}|&nbsp;{{{lang_code}}}|}}</span>])</span>|<span style="font-size:0.9em;" class="plainlinks">{{toolbar|separator=dot|[[Special:UserRights/{{{username|{{BASEPAGENAME}}}}}|<span style="color:#5871C6;">verify</span>]]|{{#if:{{{RFA|{{{RfA|{{{rfa|}}}}}}}}}|[[{{{RFA|{{{RfA|{{{rfa}}}}}}}}}|RfA]]}}}}</span>}} }}}}<includeonly>{{#if:{{{1|}}}||{{category handler|user=[[Category:Wikiversity custodians]]|talk=[[Category:Wikiversity custodians]]|subpage=no|nocat={{{nocat}}}}}}}<!-- -->{{#if:{{yesno|{{{recall|}}}}}| [[Category:Wikiversity custodians open to recall]]}}</includeonly><noinclude> {{documentation}} </noinclude> qyyb5zuzadgtqfwik5poqfvjwum6psr 0 63210 2690269 2610925 2024-12-04T14:48:37Z 2003:E8:1731:F5B4:89BC:8779:A5ED:DCAA /* Advanced configuration */browser.tabs.insertAfterCurrent - This is essential. I always use it on desktop. I am surprised no setting exists for this on the graphical preferences menu. It makes browsing so much more comfortable. 2690269 wikitext text/x-wiki ==Setup== * [http://support.mozilla.com/en-US/kb/How+to+make+Firefox+the+default+browser How to make Firefox the default browser] ==Shortcuts== * CTRL+W closes a tab * CTRL+T opens a new tab * CTRL+SHIFT+T reopens the last closed tab * CTRL+↹Tabulator; CTRL+Shift+↹Tabulator; CTRL+PgUp/PgDn: switch through tabs. * CTRL+Shift+PgUp/PgDn: move opened tab left or right respectively. * F6 or Ctrl+L : Gets you right up into the Address/URL bar. * F5 : Reload the page. * CTRL+F5: Clear cache for the current page and reload the page. * / or Ctrl+F: Search/find text on page * Ctrl+K : Takes you to the Firefox search box. * Ctrl+U : View the page’s source code. * F11 : View the page in full-screen mode. * Ctrl+W : Closes the active tab. * Ctrl+<kbd>=</kbd> : Resets font size. * Ctrl+<kbd>+</kbd> : Increases font size. * Ctrl+<kbd>-</kbd> : Decreases font size. * Ctrl+S : Save page * Ctrl+Shift+I, F12: Open web development tools. == Downloading == There are four main ways to download an internet resource: * <kbd>Ctrl+S</kbd> – Download currently viewed page. In the file saving dialogue, "Web Page, complete" takes the loaded page from memory and stores resources such as multimedia and style sheets and JavaScripts as individual files inside a <code>_files</code> suffix directory. "Web Page, HTML only" saves the verbatim page source code downloaded from the server. * "Save Link as" from right-click context menu (saves verbatim page source code) * Dragging the URL or tab into the Download arrow in the tool bar. * Pasting an URL from the clipboard into the opened download history (in the "Library" window or <code>about:downloads</code>). Additionally, the page source code can be saved by copying it from the web development tools and pasted into a text editor. == Profiles == User profiles allow separating configuration, extensions, bookmarks, and history within Firefox. By default, there is one profile. The profile manager allows creating additional profiles. It can be opened by launching the Firefox process through a command line with the <code>-profilemanager</code> parameter, without spaces. Profile folders are typically located under <code>%AppData%\Mozilla\Firefox\</code> on Microsoft Windows,<code>~/Library/Application Support/Firefox/</code> on MacOS, and <code>~/.mozilla/firefox/</code> on Linux and BSD. The profile manager allows manually specifying custom profile directories.<ref>[https://support.mozilla.org/en-US/kb/profile-manager-create-remove-switch-firefox-profiles?redirectslug=profile-manager-create-and-remove-firefox-profiles&redirectlocale=en-US ''Profile Manager - Create, remove or switch Firefox profiles'' – Mozilla Support (knowledge base)]</ref> Relevant files in the profile folder are <code>places.sqlite</code> for browsing history and download history and bookmarks, <code>cookies.sqlite</code>, <code>prefs.js</code> for user preferences, and <code>containers.json</code> for configuration of multi-account containers. Sessions are stored in <code>sessionstore.jsonlz4</code> when closing the browser (before Firefox 56 of 2017, it was stored as plain JSON into <code>sessionstore.json</code>), and backed up to <code>sessionstore-backups/recovery.jsonlz4</code> and <code>sessionstore-backups/recovery.baklz4</code> while browsing for recovery in case of an unexpected process termination. The previous session is backed up to <code>sessionstore-backups/previous.jsonlz4</code>. <ref>[https://support.mozilla.org/en-US/kb/profiles-where-firefox-stores-user-data Profiles - Where Firefox stores your bookmarks, passwords and other user data – Mozilla Support]</ref> == Advanced configuration == On the desktop edition (and earlier mobile versions), the <code>about:config</code> page allows fine-tuning the browser. The search bar at the top facilitates finding properties. Notable properties are: * <code>browser.urlbar.trimURLs</code> – may hide the protocol from the URL bar. Deactivate to always show the full URL including the protocol. Activated by default. * <code>browser.tabs.insertAfterCurrent</code> – open new tabs next to the currently active tab instead of at the end of the tab list. * <code>javascript.enabled</code> – deactivating usually increases speed and decreases CPU usage of [[:w:Progressive enhancement|progressively enhancing]] sites, but features implemented using JavaScript such as [[:mw:Extension:WikiEditor|the toolbar of the wikitext form]] and JavaScript-based sites such as Twitter web app will not work. Activated by default. * <code>browser.backspace_action</code> – this parameter adjusts the function of the backspace key if no text input field is active. <code>0</code>: navigate to last page, <code>1</code>: scroll up, if ''Shift'' key is held then scroll down. <code>2</code>: deactivate. * <code>dom.event.contextmenu.enabled</code> – allows JavaScript to interfere with right-click context menu. Deactivating might interfere with some sites' functionality. Activated by default. * <code>dom.event.clipboardevents.enabled</code> – allows JavaScript to evaluate on-page text selection. Deactivating might interfere with some sites' functionality. Activated by default. * <code>browser.download.autohideButton</code> – hide download button if no downloads in current session. Activated by default. * <code>browser.download.alwaysOpenPanel</code> – Open list of recent downloads after a download finished. * <code>browser.download.lastDir</code>, <code>browser.open.lastDir</code> – last directory to which a file was downloaded or from which a file was opened through the file picker dialogue. * <code>browser.startup.homepage</code> – page displayed when starting a new session (if not restoring the previous session), and when pressing the button with the house icon. * <code>dom.ipc.processCount.web</code> – number of processes across which web content is distributed. Higher count increases performance but consumes more memory, thus recommended for computers with much RAM. Four by default. * <code>general.autoScroll</code> – enable scrolling using middle mouse button press. * <code>devtools.chrome.enabled</code> – enable write access to browser console, accessed using Ctrl+Shift+J. * <code>browser.cache.disk.capacity</code>, <code>browser.cache.offline.capacity</code> disk cache capacity, mainly used for static resources to save bandwidth. Difference between values unclear yet. – Values in Kilobytes. * <code>places.history.expiration.transient_current_max_pages</code>: number of entries retained in the browsing history database (<code>places.sqlite</code> file). According to a moderator of Mozilla's support forum, the value is read-only by default but can be adjusted through the <code>places.history.expiration.max_pages</code> property, which has to be added manually.<ref>[https://support.mozilla.org/en-US/questions/1275209#answer-1274292 Answer to ''FF 71. Cannot change value in setting places.history.expiration.transient_current_max_pages'' | Firefox Support Forum | Mozilla Support (December 17, 2019)]</ref> * <code>accessibility.blockautorefresh</code> – blocks "meta refreshes" and "meta redirects", meaning refreshes and redirects using the <code>meta http-equiv</code> tag, as well as redirects instructed by the <code>refresh</code> [[:w:HTTP header|HTTP header]]. However, it does not block JavaScript-based redirects and refreshes, i. e. <code>document.location.href</code>.<ref>[https://support.mozilla.org/en-US/questions/1327430 ''Enable/disable automatic redirect. | Firefox Support Forum | Mozilla Support'' (2/26/21)]</ref> * <code>security.csp.enable</code> – Claimed by Mozilla to increase security against "cross-site scripting" (XSS) attacks.<ref>[https://developer.mozilla.org/en-US/docs/Web/HTTP/CSP Content Security Policy (CSP) - HTTP] from Mozilla Developers Network</ref> User reports suggest it interferes with the function of [[:w:JavaScript bookmarklet|bookmarklet]]s.<ref>[https://superuser.com/questions/586063/how-to-disable-csp-in-firefox-for-just-bookmarklets ''How to disable CSP in Firefox for just bookmarklets?'' – Super User]</ref> * <code>permissions.default.image</code> – 1: Load images as usual (default); 2: Do not load images (usually for testing purposes or to save bandwidth); 3: Only load images from the same domain.<ref>https://support.mozilla.org/en-US/questions/981640</ref> * <code>media.videocontrols.picture-in-picture.video-toggle.enabled</code> (or similar): show the picture-in-picture button in video players. * <code>security.dialog_enable_delay</code> – delay in milliseconds until the confirmation button in some dialogue boxes like downloading an executable file or installing an extension gets unlocked. Until then, it is [[:w:Grayed_out|"grayed out"]]. Default value: 1000 (1 second).<ref>[http://kb.mozillazine.org/Disable_extension_install_delay_-_Firefox ''Disable extension install delay - Firefox - MozillaZine Knowledge Base'']</ref> * <code>browser.cache.disk.enable</code> – enable saving sites' resources locally to speed up loading. * <code>browser.cache.disk.capacity</code> – disk cache size in kilobytes. * <code>browser.urlbar.update2.engineAliasRefresh</code> – allow adding custom search engines.<ref>[https://superuser.com/questions/7327/how-to-add-a-custom-search-engine-to-firefox/1756774#1756774 How to add a custom search engine to Firefox? - Super User]</ref> == System pages == * <code>about:memory</code> – internal process viewer. * <code>about:processes</code> – newer internal process manager. Introduced with Firefox 67 in early 2019.<ref>https://www.ghacks.net/2019/03/01/firefox-67-automatically-unload-unused-tabs-to-improve-memory/</ref> * <code>about:performance</code> – similar to <code>about:processes</code>, but with more beginner-friendly worded user interface, such as "Energy impact" instead of "CPU". Introduced in late 2018.<ref>[https://www.ghacks.net/2018/10/11/this-is-firefoxs-upcoming-aboutperformance-page-huge-improvements/ This is Firefox's upcoming about:performance page (huge improvements) – Martin Brinkmann, 2018-10-11]</ref> ==See also== * [[w:Iceweasel|Iceweasel]] browser * [[devmo:|Firefox Development Wiki]] * [[MozillaZineKB:|Mozilla Knowledge-Base Wiki]] * [[links (browser)]], [[lynx (browser)]], [[Elinks (browser)]] command line browsers * [[CURL (software)|cURL]] ==External links== * [http://www.qedoc.org/en/index.php?title=Mozilla_Firefox Mozilla Firefox] - Interactive learning quiz at Qedoc.org [[Category:Firefox]] [[Category:Web browsers]] mhgsinmkpqx0zzu461gm05jj3gg2q44 3 106891 2690307 2687824 2024-12-04T20:40:47Z RockTransport 2992610 /* Page called Food Tests */ new section 2690307 wikitext text/x-wiki [[User:Atcovi/Archive 1|/Archive 1 (September 25, 2013 - November 15, 2013)]] • [[User talk:Atcovi/Archive 2|/Archive 2 (November 15, 2013 - November 27, 2013)]] • [[User talk:Atcovi/Archive 3|/Archive 3 (December 3, 2013 - December 25, 2013)]] • [[User talk:Atcovi/Archive 4|/Archive 4 (December 24, 2013 - January 1, 2014)]] • [[User talk:Atcovi/Archive 5|/Archive 5 (January 2, 2014 - January 20, 2014)]] • [[User talk:Atcovi/Archive 6|/Archive 6 (March 24, 2014 - April 14, 2014)]] • [[User talk:Atcovi/Archive 7|/Archive 7 (April 19, 2014 - September 8, 2014)]] • [[User talk:Atcovi/Archive 8|/Archive 8 (September 12, 2014 - November 3, 2014)]] • [[User talk:Atcovi/Archive 9|/Archive 9 (November 6, 2014 - January 26, 2015)]] • [[User talk:Atcovi/Archive 10|/Archive 10 (January 28, 2015 - March 11, 2015)]] • [[User talk:Atcovi/Archive 11|/Archive 11 (March 22, 2015 - June 25, 2016)]] • [[User talk:Atcovi/Archive 12 (June 26, 2016 - January 8, 2018)|/Archive 12 (June 26, 2016 - January 8, 2018)]] • [[User talk:Atcovi/Archive 13 (January 9, 2018 - April 14, 2023)|/Archive 13 (January 9, 2018 - April 14, 2023)]] :''Before 2013: [https://en.wikiversity.org/w/index.php?title=User_talk:Atcovi&diff=750617&oldid=740650 see this]'' {{tmbox |small = |image = [[Image:Busy desk.svg|{{#ifeq:|yes|40px|75x50px}}]] |text = This user is busy in [http://en.wikipedia.org/wiki/Real_life Real Life] {{#if:|until&nbsp;{{{end}}}&nbsp;}}{{#if:|due to&nbsp;{{{reason}}}&nbsp;}}and may not respond swiftly to queries.{{#if:|<P>{{{msg}}} }} | style = {{#if:|width: {{{width}}}px;}} {{#ifeq:{{{shadow}}}|yes|{{box-shadow|0px|2px|4px|rgba(0,0,0,0.2)}}|}} }} == Hydrangeas and Me (Short Story in Progress) == '''CHAPTER 1''': It was a rainy Monday morning. The overcast clouds brought about a wave of light gray over the world, allowing the rain its refuge from the harsh sun. I woke up earlier than usual, due to the incessant knocking of light rain on my window. I rose from my bed, spread open the curtains, and raised the blinds. I looked outside, admittedly a bit upset about the current state of weather, and stepped down to change into my school uniform. The walk to school would prove a bit more difficult today because of the light spray, so I prepared my best umbrella and opened the door, ready to face the world. Each of my steps pittered and pattered. Water began to soak through my sole to my foot, giving me quite an icky feeling. I walked, feeling as if sponges had been rubber banded to my feet. I saw other students making the commute as well. Some of them rose their feet in comedic ways to avoid all the water, to no avail. Others ignored the puddles and continued forward. ''Rather courageous…'' I thought to myself, smiling lightly. Little moments like these can bring happiness, even in this less-than-desired weather. Eventually, me and the other hoards of students successfully made their way to the front gate. Some teachers decided to move under the canopy of the school’s main entrance, instead of their typical formation at the school’s gate. As I entered the canopy, I closed my umbrella and shook off the rain on the tiles in front of the glass door. Some other students mimicked my behavior and I laughed, albeit embarrassingly. I walked into the school, put my umbrella on the stand and marched toward my shoe locker before changing into the inside schools that were required of us. ''Today… I hope…'' Unlike the other mundane days, today was a lot more exciting. She would be there, in the garden of hydrangeas. After the change, I noted the time. 7:30. I would have a good amount of time to see her. I rushed with a silly look on my face to the library. Toward the back, there were several large windows which looked out upon the gardening club’s area of operation. I made sure to pick a table that was close to the windows. I turned my head outside, and like clockwork, she stood there tending to the flowers. Today, because of the rain, she wore a little umbrella cap supplied to the club to fend off the rain. She stood and cut stems with tiny gardening scissors. Her look was focused, yet soft. She didn’t seem like she wanted to injure the flowers, as if they meant everything to her. Each movement and cut, every one seemed to take special care toward the hydrangeas. I sat holding my head up with one hand watching her. Today’s view was particularly beautiful. Maybe the rain isn’t so bad for today. The girl in the garden walked toward the shed to grab a few more tools. There were about 20 more minutes until the first bell would ring. Despite that, she continued to work. Her soft look and gaze had enraptured me, it reminded me of more innocent times. It reassured me, and it calmed me. Everytime I did this, I would feel so tired and relaxed. I wish I could watch for hours. Today, I want to change things. I don’t want to watch from afar anymore. I wish to know more about this girl who tends to the garden. What kind of flowers does she like? How long has she been tending to this garden? Many, many questions trying to flow out to reach their answered counterparts. 15 minutes left, if I want to do this I better act fast. Tens of days spent watching, relaxed, for it suddenly to turn to stress and anxiety. Would she accept me? Is this her own personal time I'm trying to infringe upon? Each doubt weighed my steps like ankle weights, but despite it all, I continued forward. I opened the glass door and went outside, with no umbrella. The rain was rather light, so it didn’t pose much of a threat to my clothing. Upon opening the door, the girl in the garden noticed my entrance. Suddenly the image became reality, like entering a painting. The smell of dirt and flowers, mixed with rain. Her appearance is much more real. “H-hello…” I stammer, honestly not knowing what else to say. “Hi! What brings you to the garden, were you interested in joining the club?” She directs that typical soft smile in my direction. It’s very hard to not turn away from such a bright light. The rain complemented her well, like a warm bowl of porridge. What should I say? I hadn’t intended on joining this club, but what do I say aside from that? However, I don’t want to lie… “I wanted… to talk to you. To ''see'' you.” Ehhh… that’s not right. The girl’s expression seems to lighten, giving me a good chance to make eye contact. She seems a bit confused. “Do we know each other? I don’t remember ever seeing you in class… I’m sorry…” “The hydrangeas, I mean. I want to talk to you about the hydrangeas.” Her quizzical expression turns even brighter than the one she showed me first. “Oh! What would you like to know?” she’s smiling brightly, and tilts forward a bit waiting for my response. A conversation…! We’re having a conversation! Ahhh… I’m so excited but I have to keep my cool. “You cut the flowers with those scissors, why? Does that not hurt them?” “Ah! It’s a process called pruning… “ She snips her scissors up toward me. She turns around and beckons me to one of the many bushes. I follow suit. “These… are buds. It’s where the flower grows from. We prune them for many reasons, but most of the time it’s because the flower or stem is diseased.” I’m smiling so much right now, I can’t contain myself. So much stimulation just from a little conversation. I laugh a bit, out loud. “Hmmm…?” She looks up at me, hearing my light laughter. I quickly blush, becoming a bit embarrassed. “S-sometimes… It’s the little things that bring me happiness…” My signature phrase comes out, resulting in a large smile formulating on my face. The girl looks at me, and reflects my smile. “Do you like hydrangeas, too?” “I-I think I do?” The girl erupts into laughter, and I can’t help but join in. “You think? Well, maybe you should think a little more… about the kinds of flowers you like. And maybe the kinds of things you say, as well?” She lightly teases me, before standing up. “Do you have any other questions about these flowers?” I think. An important question… yet familiar. “Do ''you'' like hydrangeas?” I reflect her question back on her. Her expression grows warm, as if I can see the colors of her mood changing before me. “I do… They bring me a lot of my own happiness, too.” I think back to what I said earlier and smile again. “In the rain… they are so beautiful too! It makes me think of…” She pauses. “Tears of joy, or maybe the feeling of contentment. They are so… simple. Y’know it’s not just people that can talk… These flowers can too. And when you listen close enough, you find out that they have a lot to say.” “I’m not sure I can understand…” “That’s okay, we all have to start somewhere.” She smiles mischievously, before walking toward the shed. “We have about 5 minutes to class, you know…? What’s your name?” I panic over the fact that we have so little time, partially because my class is on the opposite side of where we are now. “I-I’m Akio…” “Well, Akio, thanks for talking to me… I’ll see you around!” “Same to you!” I walk back to the glass door before looking toward her again. She’s working in the shed, putting her tools away. I open the door inside. For some reason, leaving that rainy hydrangea world hurt me when I walked inside. As if the dryness was worse than rain… —---------------------------------------------------------------------------------------------------------------------------- '''Date: 2018/07/21''' '''Today, when I was tending to the hydrangeas, a boy approached me and asked me about them. I was nervous at first, because I didn’t know what he wanted.''' '''He wanted to ask about the flowers I was tending to, and why I was pruning them. We talked a little bit after this. He seems interesting… yet suspicious! I will keep an eye out…''' '''Again, me and Mom fought. It hurts a lot when we fight, I say things that I don’t mean to say. She wants me to do so many things, but I’m just one person. I think she’s just living through me… I love her, but I want to be my own person. Making my own choices and decisions.''' '''School was bland, except for the gardening club (as usual)! We have a field trip to a sunflower field coming up. It’s always so beautiful to see those fields in the summer… I hope the skies are blue that day. Maybe me and the club members can take photographs.''' '''Ahh… I CAN’T WAIT!''' '''Sincerely (again),''' '''Hana''' That night when I got home, I plopped back on my bed. Hours of lecturing and note-taking always takes quite a toll on me. Well, mostly everyone else too. If you think any of that is fun, I must commend you. My bookbag slid onto the floor as I unbuttoned my uniform. I felt a bit too lazy to do anything else other than rest. I noticed a pencil was sticking from my shirt pocket, if left unnoticed it might’ve poked me. I removed it, and threw it across the room. Just cause? Downstairs, I could hear the door open and close. “Akio, we’re home!” My mother called. “Welcome back.” I didn’t really attempt to yell this so I’m not too certain as to if the both of them actually heard that. “Man…” I thought back to earlier this morning, when me and… And…? “Ah!” I exclaimed aloud, recognizing my mistake. I forgot to ask her name. I wonder? “Akane? Hmm, maybe too harsh…” A soft complexion, warm smile… a love of flowers? “Aki… Akemi…” Maybe I shouldn’t focus on A’s so much. Yet that letter seems like it would make some sense. I kept deliberating as to what her name could potentially be before my thinking was interrupted. I heard a knock at the door. “Akio?” “Yeah, Mom?” “I didn’t hear you welcome us, so I thought maybe you were out? You surprised me…” “Sorry. ‘Welcome Home!’” “Don’t tease! How was school?” “Better than usual… Aside from the rain maybe.” “Oho? Anything new happen?” I thought back to how I expressed meeting the girl in the garden as ‘entering a painting.’ “‘Things may look up in your favor’” I said, like reading a fortune cookie. “Very funny. Well, I’m glad…” My mom flashes me a warm smile, before leaving the room and closing the door. “Don’t forget, this weekend is the festival in town. Your father and I will be going, you can come along too if you’d like?” She speaks through the door. “Hmm… Maybe. Let me think about it.” “Okay.” I hear her footsteps as she makes her way down the stairs. I look up at the ceiling and use my body weight to launch myself from the comfort of my bed. I trudge over to my bookbag and collect a few slips of homework and textbooks. “Time for the long haul…” A few hours pass, as I complete sheet after sheet. I’m careful to review what we learned from our textbooks. My hand grows tired of writing and my eyes get weary. I look up at the clock. 10:32 PM. Yeesh, it’s rather late now. Best to get to sleep. I close my blinds, and shut the curtains before making my final approach to the blanket kingdom that is my bed. It’s warm and fuzzy, and makes me smile with contentment. I hope that this feeling lasts forever. Next week… Can’t come any sooner… Maybe… Sleep takes me before I can finish my thought. '''CHAPTER 2''': “...” My classroom door is shut, it seems the teacher has yet to arrive. I really have to stop getting here so early. The hallway seems rather barren this morning. Yet again, the rain continues and won’t let up. Even if it’s monsoon season, I want at least one sunny day, y’know? I sit down next to the door and put my bookbag next to me, I feel rather sleepy. The fatigue from last night’s battle still sits on my shoulders. “Hahhh…” “Tired today, Akio?” A voice stirs me, yet my eyes remain closed. “Mmm…” Wait. Who’s voice is that? I open my eyes and turn to see who spoke. “Ah!” I yelp, accidentally. I quickly sit up, and try to straighten myself out. “H-hello! Yes, last night, so much homework… Not much sleep either.” It was the girl from the garden, and I was very startled. I couldn’t handle the cognitive dissonance of seeing her in the hallway. It was a first, and the fact that she was talking to me exacerbated this feeling a hundredfold. “I see. Well, I’m going to my classroom. See ya..!” The girl makes her way up the hallway, marching with great purpose. Mgghhh. A brief exchange, yet… “W-wait…!” She turns around. She seems a little startled by the amount of force I put into my voice. The hallway was rather empty, save for us two, after all. “I never… got your name. Might I-I ask what it is?” “Oh! You’re right! It’s Hana… Can’t believe I never told you. Apologies.” Hana. Hana. A lot of A’s, huh? Eventually, the teacher and a few students march up the opposite end of the hallway to our classroom. “Early today, Akio?” The teacher says, as he unlocks the door with her key. “Something like that…” I responded. I’m admittedly a little embarrassed by what just happened in the hallway, even though it went rather well. An after-effect of love, I suppose. The door finally opens and the smell of the classroom fills my nose. It’s rather comforting, but also fills me with boredom. The brief period of movement I get from door to chair is simply too little for my antsy legs. Maybe if they made every desk a treadmill it would alleviate the problem. I chuckle to myself at the thought of some of my classmates trying to keep up. I walk and sit down at my desk, awaiting today’s lesson. A friend of mine, Jiro, turns to me. He had found his desk a bit before me. “Oi, Akiooo! So early, today? What gives?” “I’m always early.” “Pshhh! Sure… Tell that to yourself last week, Mr. 5-minutes-after-the-bell” “I’ll have you know that is not my last name.” Me and Jiro have been friends since the start of the school year, we haven’t really known each other long, but due to our desks being in close proximity and our great chemistry, we’ve been able to get really close. Jiro is rather eccentric and energetic. It’s a great match for my rather calm personality. We’d make a perfect straight man/funny man duo, if we were to pursue comedy. “Yo, Akio, the festival is coming to town. And… We’ve never even hung out!? Let’s change that!” Jiro flashes a corporate smile, as if he was pitching his company’s product to a stern investor. “Hmm… My parents asked me about the same thing…” “It’s me or it’s them, Akio. No in betweens! I thought we had something…” Jiro pretends to tearfully sob, putting me in a rather awkward situation. Other students settling in begin to look at us. “Ok, ok… Maybe I’ll go with you, I had a choice to begin with. I’ll have an answer for you by tomorrow.” “Perfect! It’s a date, then!” Jiro attempts to make his most masculine face, but I can’t help but start laughing. “Good joke?” Jiro says, smiling lightly. “More like a good face, that got me.” “Nihihi!” Jiro’s face reminds me of a cat when he does this. After this exchange, the teacher begins to give some announcements before class. Some of them pertaining to festival safety, seeing as a great majority of us would be in attendance. Others included information about clubs, the coming exams, and career documents. All of us were in 12th grade, so after this, it was time to move on to the big world and get to work. Honestly, anytime I would hear anything about career forms or schooling after high school would always scare me. I was worried about putting myself out there in the world, afraid of what might come from that. Knowing that everyone else would have to face the same thing, it did make me feel a bit better. That light knot in my stomach still wouldn’t go away, however. The class passed by, uneventfully. The teacher talked in great detail about some wars, called on a few clueless students, and sighed a great many times. The typical class experience. Jiro got the golden ticket today, he was sleeping when he was called on. I’ve never seen a stick of chalk fly so fast and hit so hard. I could’ve sworn that he would be sent flying, but he started holding his head in pain. I laughed, but quickly fixed my expression when he looked at me with “the stare of a thousand deaths.” As we all began to walk out of the classroom after the bell, the hallways were flooding with oceans of students. Luckily, I had a boat and paddle to bear the waves. Or, maybe just experience? '''*''' The rest of the school day passed by, the classes got faster with each one that came and went. I put my shoes on, and walked out from the entrance. It was cloudy today, but there was no rain. Maybe it’s really starting to grow on me. The rain usually maddens and annoys me, but this year I want to see it more than ever. Is it because of her? What even are these feelings of mine? Is it appropriate to attribute them to a real person? I would stare and stare at her in that garden, as if admiring a beautiful landscape portrait in a museum. When we talked those two times, I was so nervous I could barely keep it together. I don’t view myself as someone who’s lacking in confidence, or has low self-esteem, but her presence is almost choking. And it’s my own fault. The more I pedestalize her, the worse those interactions will get. I’ve already confirmed it myself, in that garden. Hana is as real as those flowers. As real as the rain. Flesh, blood, bone. Breathing, beating, human. She gets sick, uses the bathroom, and makes mistakes. “Maybe I’ve been thinking about this wrong…” I say aloud, albeit accidentally. I think it’s okay to be curious about who she is. I have to leave this mindset behind if I want to keep talking to her. I don’t really have much of a reason though, except for that I just want to. Life is funny, in that way. Sometimes we just want to do something, and we don’t really have to explain why. Rather comforting. On a whim, I decided to visit the garden today before I made the trek home. Walking to the side gate this time, I opened it and once again found myself at the whim of these flowers. I was half expecting people to be here, seeing as it is after-school, but there was no one to be found. It was rather ethereal. The scene was similar to yesterday’s, but it possessed much more weight. Perhaps it was because I was alone. I could be off my guard, in this beautiful place. I walked by the hydrangea bushes, and touched their leaves. They can talk, huh? I move my hand gently to the flower and feel the petals. They are silky and soft, and a bit warm. Not too far off from human skin. The flower was a bright, light purple. It swayed with each touch, despite my light movements. As a joke, I put my ear up to the flower, as if to listen to what it was trying to tell me. A gust of wind blows through, and the flower hits my head a few times. Maybe it doesn’t like me too much? Upon realizing this, I burst into laughter. “Silly little flowers, maybe we’re equals after all?” Before my exit, I wave and bow courteously for the flowers and open the garden gate. Much to my surprise, a few gardening club members stood on the other side of the opening. This moment was admittedly, rather embarrassing. For one, I really hope they didn’t hear me talking to those flowers. For two, am I allowed back here? I didn’t ever really think about that. I gaze at each of their startled faces, and yet, Hana’s isn’t there. I am a bit disappointed, I let out a little sigh. “W-what brings you to the garden?” One of the members perks up, a small boy who appears to be a bit nervous about the encounter. The other two girls look toward him and then back at me, expectant of an answer. “I hope I wasn’t breaking any rules by being in there, I was just ta- …loo-king at the flowers? Is that okay?” I didn’t mean to sound so apologetic, but it came out rather emotional anyway. “No, that won’t be an issue. As long as all flowers are accounted for.” One of the two girls speaks, and gives a bit of a stern answer. She still seems suspicious about my presence. “Mei… No need to be so harsh, he doesn’t seem so bad after all.” The last girl speaks softly to the more brusque one, and gives a warm smile. I don’t know how to explain it, but I’m thanking her to high heaven right now. Who knows what would be in store for me had she not been here. My ears probably would’ve fallen off. “S-sorry again, I wasn't trying to cause any trouble.” I make my way past the exit, and attempt to pass by the three of them. I’m successful, but I stop and turn around hesitantly. “I love the garden, please don’t stop talking with those flowers.” I realize that’s not exactly what I wanted to say, and in my intense embarrassment I turn around and continue home. From behind me, I could hear the group laughing a bit. I could’ve swore I heard her name too… —---------------------------------------------------------------------------------------------------------------------------- '''Date: 2018/07/29''' '''A slower day than usual. Today was the last day of classes, meaning that we don’t come back until September. The festival will be starting tomorrow, as well. The week after, I FINALLY get to go on the sunflower trip. I am unbelievably excited for it.''' '''I don’t have much else to say. Thanks, I guess?''' '''Sincerely (again),''' '''Hana''' Today, we had the festival. I had inevitably let my parents know that instead of going with them, I would be going with Jiro. They didn’t seem disturbed by that fact, which was nice. Maybe even a little happy that they got to have some alone time. I guess this would be a win-win situation? I had called Jiro a little while ago to confirm where we would meet before making the commute. It would be about a ten minute walk. Thank the heavens the sun is setting, or I don’t think I would’ve made it to the 5 minute mark. I was thinking of coming in my kimono, but it was much too hot to wear something like that. Decidedly, I just went with my plain clothes. I waited a few more hours for 5:50 to come around before leaving my house. The festival started at 6:30, but we wanted to look around a bit before it began. I made the trek outside of my front door and walked to the side of the road. Despite the sun going down, it was still unbearably hot. I raised my hand to cover my eyes while walking in the sun’s gaze. A few neighbors appeared to be preparing themselves for the festival as well. We lived in a rather small town, so most of the people around here prepared themselves for things like these. [[User:IvoctA|IvoctA]] ([[User talk:IvoctA|discuss]] • [[Special:Contributions/IvoctA|contribs]]) 19:39, 16 September 2023 (UTC) == Hello! Wikijournal chat? == Happy new year :) Long time. I just saw you've been working with WikiJournal; how are you finding this format and process? What are the bottlenecks at the moment, what else might be possible? <span style="padding:0 2px 0 2px;background-color:white;color:#bbb;">&ndash;[[User:Sj|SJ]][[User Talk:Sj|<font style="color:#f90;">+</font>]]</span> 14:35, 25 January 2024 (UTC) :Hey Sj, good morning. :My submission for WJ has been quite slow due to IRL commitments, but I've had good experiences so far. One of the WJ reviewers was nice enough to give me and my buddy solid advice on how to improve our journal for submission, so I was pleased with that. The discussion at [[Talk:WikiJournal_User_Group#Current_status_of_WikiJournals]] is concerning, though, and I hope WJ will continue to be in operation. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:21, 25 January 2024 (UTC) == ''The Signpost'': 31 January 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/News and notes|Wikipedian Osama Khalid celebrated his 30th birthday in jail]] * Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Opinion|Until it happens to you]] * Disinformation report: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Disinformation report|How paid editors squeeze you dry]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/In the media|Katherine Maher new NPR CEO, go check Wikipedia, race in the race]] * In focus: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/In focus|The long road of a featured article candidate, part 2]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Recent research|Croatian takeover was enabled by "lack of bureaucratic openness and rules constraining [admins]"]] * Comix: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Comix|We've all got to start somewhere]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Traffic report|DJ, gonna burn this goddamn house right down]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 15:17, 31 January 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:Bri@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=26086360 --> == Invitation to discuss page deletion policy == A discussion that might interest you has been started at [[Wikiversity:Requests_for_Deletion#Wikiversity:Deletion_Convention_2024]]. -- [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:49, 15 February 2024 (UTC) :{{ping|Guy vandegrift}} Hello Guy vandegrift, I appreciate you thinking about me and reaching out to me for my thoughts. I'm afraid I will have to echo Dave's response and abstain from formulating any suggestions, since I do not have the needed time to review the discussion. Best of luck. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:35, 17 February 2024 (UTC) == Movie reviews == '''[[Paris, Texas]]''' is under a prod and when I found '''[[Book Reviews]]''' I thought of copying and creating a page called [[Movie reviews]]. I obviously don't need your permission (cc-by!), but was wondering what you thought of that idea. Any chance of attracting new movie reviews? ... Another idea would be to create a subspace under [[Essay]], with a subpage that links to [[Book reviews]]. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:54, 1 March 2024 (UTC)--Afterthought: See '''[[Essay/Collection]]'''-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:09, 1 March 2024 (UTC) :Movie reviews would be interesting. It can be educational, but also promotional. Personally, I don't have interest in producing movie reviews - but setting it up for future users in the hopes it is developed in accordance with our guidelines does not sound like a bad idea. I think putting [[Book Reviews]] under an "Essay" subspace would be redundant (and provide exceptionally long page titles), and that may be the same for Movie reviews. I would lean towards making it a stand alone project and, if issues arise, we can address them when we cross that bridge. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:15, 2 March 2024 (UTC) ::I am astonished by the success of [[Book Reviews]], but noticed one odd features: Certain titles are not shown on the front page. One example is [[Book_Reviews/A_Hero_of_Our_Time|A Hero for Our Time]]. I traced the problem to the dynamicpagelist and its count variable. I think I can fix that feature. Do you want me to?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 11:23, 2 March 2024 (UTC) :::Of course, go for it! Thank you in advance. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:30, 2 March 2024 (UTC) == ''The Signpost'': 4 September 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/News and notes|WikiCup enters final round, MCDC wraps up activities, 17-year-old hoax article unmasked]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/In the media|AI is not playing games anymore. Is Wikipedia ready?]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Recent research|Simulated Wikipedia seen as less credible than ChatGPT and Alexa in experiment]] * News from the WMF: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/News from the WMF|Meet the 12 candidates running in the WMF Board of Trustees election]] * Wikimania: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Wikimania|A month after Wikimania 2024]] * Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Serendipity|What it's like to be Wikimedian of the Year]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Traffic report|After the gold rush]] * Humour: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Humour|Local man halfway through rude reply no longer able to recall why he hates other editor]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 13:32, 4 September 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27280376 --> == ''The Signpost'': 26 September 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/In the media|Indian courts order Wikipedia to take down name of crime victim, and give up names of editors]] * Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Serendipity|A Wikipedian at the 2024 Paralympics]] * Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Opinion|asilvering's RfA debriefing]] * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/News and notes|Are you ready for admin elections?]] * Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Gallery|Are Ludd''ai''tes defending the English Wikipedia?]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Recent research|Article-writing AI is less "prone to reasoning errors (or hallucinations)" than human Wikipedia editors]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Traffic report|Jump in the line, rock your body in time]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 20:16, 26 September 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27463206 --> == ''The Signpost'': 19 October 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/News and notes|One election's end, another election's beginning]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Recent research|"As many as 5%" of new English Wikipedia articles "contain significant AI-generated content", says paper]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/In the media|Off to the races! Wikipedia wins!]] * Contest: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Contest|A WikiCup for the underdeveloped world]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Traffic report|A scream breaks the still of the night]] * Book review: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Book review|''The Editors'']] * Humour: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Humour|The Newspaper Editors]] * Crossword: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Crossword|Spilled Coffee Mug]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 11:18, 19 October 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27606308 --> == ''The Signpost'': 6 November 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * From the editors: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/From the editors|Editing Wikipedia should not be a crime]] * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/News and notes|Wikimedia Foundation shares ANI lawsuit updates; first admin elections appoint eleven sysops; first admin recalls opened; temporary accounts coming soon?]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/In the media|An old scrimmage, politics and purported libel]] * Special report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Special report|Wikipedia editors face litigation, censorship]] * Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Gallery|Why you should take more photos and upload them]] * In focus: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/In focus|Questions and answers about the court case]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Traffic report|Twisted tricks or tempting treats?]] * Technology report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Technology report|Wikimedia tech, the Asian News International case, and the ultra-rare BLACKLOCK]] * Humour: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Humour|Man quietly slinks away from talk page argument after realizing his argument dumb, wrong]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 08:09, 6 November 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27684085 --> == ''The Signpost'': 18 November 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/News and notes|Open letter to WMF about court case breaks one thousand signatures, big arb case declined, U4C begins accepting cases]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/In the media|Summons issued for Wikipedia editors by Indian court, "Gaza genocide" RfC close in news, old admin Gwern now big AI guy, and a "spectrum of reluctance" over Australian place names]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/Recent research|SPINACH: AI help for asking Wikidata "challenging real-world questions"]] * News from the WMF: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/News from the WMF|Wikimedia Foundation and Wikimedia Endowment audit reports: FY 2023–2024]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/Traffic report|Well, let us share with you our knowledge, about the electoral college]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 23:46, 18 November 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27733567 --> == Page called Food Tests == Hello there, I have contributed to the Food Tests page and I would like to inform you that we are currently expanding the page, and we are constantly adding new features to it. Please do not nominate the page for deletion any more throughout its improvement. This was just a kind notice. [[Food Tests]] Kind regards, Rock [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 20:40, 4 December 2024 (UTC) cut5d8ype5f2oujttwg5nro0clciofj 2690314 2690307 2024-12-04T23:06:54Z Atcovi 276019 /* Page called Food Tests */ Reply 2690314 wikitext text/x-wiki [[User:Atcovi/Archive 1|/Archive 1 (September 25, 2013 - November 15, 2013)]] • [[User talk:Atcovi/Archive 2|/Archive 2 (November 15, 2013 - November 27, 2013)]] • [[User talk:Atcovi/Archive 3|/Archive 3 (December 3, 2013 - December 25, 2013)]] • [[User talk:Atcovi/Archive 4|/Archive 4 (December 24, 2013 - January 1, 2014)]] • [[User talk:Atcovi/Archive 5|/Archive 5 (January 2, 2014 - January 20, 2014)]] • [[User talk:Atcovi/Archive 6|/Archive 6 (March 24, 2014 - April 14, 2014)]] • [[User talk:Atcovi/Archive 7|/Archive 7 (April 19, 2014 - September 8, 2014)]] • [[User talk:Atcovi/Archive 8|/Archive 8 (September 12, 2014 - November 3, 2014)]] • [[User talk:Atcovi/Archive 9|/Archive 9 (November 6, 2014 - January 26, 2015)]] • [[User talk:Atcovi/Archive 10|/Archive 10 (January 28, 2015 - March 11, 2015)]] • [[User talk:Atcovi/Archive 11|/Archive 11 (March 22, 2015 - June 25, 2016)]] • [[User talk:Atcovi/Archive 12 (June 26, 2016 - January 8, 2018)|/Archive 12 (June 26, 2016 - January 8, 2018)]] • [[User talk:Atcovi/Archive 13 (January 9, 2018 - April 14, 2023)|/Archive 13 (January 9, 2018 - April 14, 2023)]] :''Before 2013: [https://en.wikiversity.org/w/index.php?title=User_talk:Atcovi&diff=750617&oldid=740650 see this]'' {{tmbox |small = |image = [[Image:Busy desk.svg|{{#ifeq:|yes|40px|75x50px}}]] |text = This user is busy in [http://en.wikipedia.org/wiki/Real_life Real Life] {{#if:|until&nbsp;{{{end}}}&nbsp;}}{{#if:|due to&nbsp;{{{reason}}}&nbsp;}}and may not respond swiftly to queries.{{#if:|<P>{{{msg}}} }} | style = {{#if:|width: {{{width}}}px;}} {{#ifeq:{{{shadow}}}|yes|{{box-shadow|0px|2px|4px|rgba(0,0,0,0.2)}}|}} }} == Hydrangeas and Me (Short Story in Progress) == '''CHAPTER 1''': It was a rainy Monday morning. The overcast clouds brought about a wave of light gray over the world, allowing the rain its refuge from the harsh sun. I woke up earlier than usual, due to the incessant knocking of light rain on my window. I rose from my bed, spread open the curtains, and raised the blinds. I looked outside, admittedly a bit upset about the current state of weather, and stepped down to change into my school uniform. The walk to school would prove a bit more difficult today because of the light spray, so I prepared my best umbrella and opened the door, ready to face the world. Each of my steps pittered and pattered. Water began to soak through my sole to my foot, giving me quite an icky feeling. I walked, feeling as if sponges had been rubber banded to my feet. I saw other students making the commute as well. Some of them rose their feet in comedic ways to avoid all the water, to no avail. Others ignored the puddles and continued forward. ''Rather courageous…'' I thought to myself, smiling lightly. Little moments like these can bring happiness, even in this less-than-desired weather. Eventually, me and the other hoards of students successfully made their way to the front gate. Some teachers decided to move under the canopy of the school’s main entrance, instead of their typical formation at the school’s gate. As I entered the canopy, I closed my umbrella and shook off the rain on the tiles in front of the glass door. Some other students mimicked my behavior and I laughed, albeit embarrassingly. I walked into the school, put my umbrella on the stand and marched toward my shoe locker before changing into the inside schools that were required of us. ''Today… I hope…'' Unlike the other mundane days, today was a lot more exciting. She would be there, in the garden of hydrangeas. After the change, I noted the time. 7:30. I would have a good amount of time to see her. I rushed with a silly look on my face to the library. Toward the back, there were several large windows which looked out upon the gardening club’s area of operation. I made sure to pick a table that was close to the windows. I turned my head outside, and like clockwork, she stood there tending to the flowers. Today, because of the rain, she wore a little umbrella cap supplied to the club to fend off the rain. She stood and cut stems with tiny gardening scissors. Her look was focused, yet soft. She didn’t seem like she wanted to injure the flowers, as if they meant everything to her. Each movement and cut, every one seemed to take special care toward the hydrangeas. I sat holding my head up with one hand watching her. Today’s view was particularly beautiful. Maybe the rain isn’t so bad for today. The girl in the garden walked toward the shed to grab a few more tools. There were about 20 more minutes until the first bell would ring. Despite that, she continued to work. Her soft look and gaze had enraptured me, it reminded me of more innocent times. It reassured me, and it calmed me. Everytime I did this, I would feel so tired and relaxed. I wish I could watch for hours. Today, I want to change things. I don’t want to watch from afar anymore. I wish to know more about this girl who tends to the garden. What kind of flowers does she like? How long has she been tending to this garden? Many, many questions trying to flow out to reach their answered counterparts. 15 minutes left, if I want to do this I better act fast. Tens of days spent watching, relaxed, for it suddenly to turn to stress and anxiety. Would she accept me? Is this her own personal time I'm trying to infringe upon? Each doubt weighed my steps like ankle weights, but despite it all, I continued forward. I opened the glass door and went outside, with no umbrella. The rain was rather light, so it didn’t pose much of a threat to my clothing. Upon opening the door, the girl in the garden noticed my entrance. Suddenly the image became reality, like entering a painting. The smell of dirt and flowers, mixed with rain. Her appearance is much more real. “H-hello…” I stammer, honestly not knowing what else to say. “Hi! What brings you to the garden, were you interested in joining the club?” She directs that typical soft smile in my direction. It’s very hard to not turn away from such a bright light. The rain complemented her well, like a warm bowl of porridge. What should I say? I hadn’t intended on joining this club, but what do I say aside from that? However, I don’t want to lie… “I wanted… to talk to you. To ''see'' you.” Ehhh… that’s not right. The girl’s expression seems to lighten, giving me a good chance to make eye contact. She seems a bit confused. “Do we know each other? I don’t remember ever seeing you in class… I’m sorry…” “The hydrangeas, I mean. I want to talk to you about the hydrangeas.” Her quizzical expression turns even brighter than the one she showed me first. “Oh! What would you like to know?” she’s smiling brightly, and tilts forward a bit waiting for my response. A conversation…! We’re having a conversation! Ahhh… I’m so excited but I have to keep my cool. “You cut the flowers with those scissors, why? Does that not hurt them?” “Ah! It’s a process called pruning… “ She snips her scissors up toward me. She turns around and beckons me to one of the many bushes. I follow suit. “These… are buds. It’s where the flower grows from. We prune them for many reasons, but most of the time it’s because the flower or stem is diseased.” I’m smiling so much right now, I can’t contain myself. So much stimulation just from a little conversation. I laugh a bit, out loud. “Hmmm…?” She looks up at me, hearing my light laughter. I quickly blush, becoming a bit embarrassed. “S-sometimes… It’s the little things that bring me happiness…” My signature phrase comes out, resulting in a large smile formulating on my face. The girl looks at me, and reflects my smile. “Do you like hydrangeas, too?” “I-I think I do?” The girl erupts into laughter, and I can’t help but join in. “You think? Well, maybe you should think a little more… about the kinds of flowers you like. And maybe the kinds of things you say, as well?” She lightly teases me, before standing up. “Do you have any other questions about these flowers?” I think. An important question… yet familiar. “Do ''you'' like hydrangeas?” I reflect her question back on her. Her expression grows warm, as if I can see the colors of her mood changing before me. “I do… They bring me a lot of my own happiness, too.” I think back to what I said earlier and smile again. “In the rain… they are so beautiful too! It makes me think of…” She pauses. “Tears of joy, or maybe the feeling of contentment. They are so… simple. Y’know it’s not just people that can talk… These flowers can too. And when you listen close enough, you find out that they have a lot to say.” “I’m not sure I can understand…” “That’s okay, we all have to start somewhere.” She smiles mischievously, before walking toward the shed. “We have about 5 minutes to class, you know…? What’s your name?” I panic over the fact that we have so little time, partially because my class is on the opposite side of where we are now. “I-I’m Akio…” “Well, Akio, thanks for talking to me… I’ll see you around!” “Same to you!” I walk back to the glass door before looking toward her again. She’s working in the shed, putting her tools away. I open the door inside. For some reason, leaving that rainy hydrangea world hurt me when I walked inside. As if the dryness was worse than rain… —---------------------------------------------------------------------------------------------------------------------------- '''Date: 2018/07/21''' '''Today, when I was tending to the hydrangeas, a boy approached me and asked me about them. I was nervous at first, because I didn’t know what he wanted.''' '''He wanted to ask about the flowers I was tending to, and why I was pruning them. We talked a little bit after this. He seems interesting… yet suspicious! I will keep an eye out…''' '''Again, me and Mom fought. It hurts a lot when we fight, I say things that I don’t mean to say. She wants me to do so many things, but I’m just one person. I think she’s just living through me… I love her, but I want to be my own person. Making my own choices and decisions.''' '''School was bland, except for the gardening club (as usual)! We have a field trip to a sunflower field coming up. It’s always so beautiful to see those fields in the summer… I hope the skies are blue that day. Maybe me and the club members can take photographs.''' '''Ahh… I CAN’T WAIT!''' '''Sincerely (again),''' '''Hana''' That night when I got home, I plopped back on my bed. Hours of lecturing and note-taking always takes quite a toll on me. Well, mostly everyone else too. If you think any of that is fun, I must commend you. My bookbag slid onto the floor as I unbuttoned my uniform. I felt a bit too lazy to do anything else other than rest. I noticed a pencil was sticking from my shirt pocket, if left unnoticed it might’ve poked me. I removed it, and threw it across the room. Just cause? Downstairs, I could hear the door open and close. “Akio, we’re home!” My mother called. “Welcome back.” I didn’t really attempt to yell this so I’m not too certain as to if the both of them actually heard that. “Man…” I thought back to earlier this morning, when me and… And…? “Ah!” I exclaimed aloud, recognizing my mistake. I forgot to ask her name. I wonder? “Akane? Hmm, maybe too harsh…” A soft complexion, warm smile… a love of flowers? “Aki… Akemi…” Maybe I shouldn’t focus on A’s so much. Yet that letter seems like it would make some sense. I kept deliberating as to what her name could potentially be before my thinking was interrupted. I heard a knock at the door. “Akio?” “Yeah, Mom?” “I didn’t hear you welcome us, so I thought maybe you were out? You surprised me…” “Sorry. ‘Welcome Home!’” “Don’t tease! How was school?” “Better than usual… Aside from the rain maybe.” “Oho? Anything new happen?” I thought back to how I expressed meeting the girl in the garden as ‘entering a painting.’ “‘Things may look up in your favor’” I said, like reading a fortune cookie. “Very funny. Well, I’m glad…” My mom flashes me a warm smile, before leaving the room and closing the door. “Don’t forget, this weekend is the festival in town. Your father and I will be going, you can come along too if you’d like?” She speaks through the door. “Hmm… Maybe. Let me think about it.” “Okay.” I hear her footsteps as she makes her way down the stairs. I look up at the ceiling and use my body weight to launch myself from the comfort of my bed. I trudge over to my bookbag and collect a few slips of homework and textbooks. “Time for the long haul…” A few hours pass, as I complete sheet after sheet. I’m careful to review what we learned from our textbooks. My hand grows tired of writing and my eyes get weary. I look up at the clock. 10:32 PM. Yeesh, it’s rather late now. Best to get to sleep. I close my blinds, and shut the curtains before making my final approach to the blanket kingdom that is my bed. It’s warm and fuzzy, and makes me smile with contentment. I hope that this feeling lasts forever. Next week… Can’t come any sooner… Maybe… Sleep takes me before I can finish my thought. '''CHAPTER 2''': “...” My classroom door is shut, it seems the teacher has yet to arrive. I really have to stop getting here so early. The hallway seems rather barren this morning. Yet again, the rain continues and won’t let up. Even if it’s monsoon season, I want at least one sunny day, y’know? I sit down next to the door and put my bookbag next to me, I feel rather sleepy. The fatigue from last night’s battle still sits on my shoulders. “Hahhh…” “Tired today, Akio?” A voice stirs me, yet my eyes remain closed. “Mmm…” Wait. Who’s voice is that? I open my eyes and turn to see who spoke. “Ah!” I yelp, accidentally. I quickly sit up, and try to straighten myself out. “H-hello! Yes, last night, so much homework… Not much sleep either.” It was the girl from the garden, and I was very startled. I couldn’t handle the cognitive dissonance of seeing her in the hallway. It was a first, and the fact that she was talking to me exacerbated this feeling a hundredfold. “I see. Well, I’m going to my classroom. See ya..!” The girl makes her way up the hallway, marching with great purpose. Mgghhh. A brief exchange, yet… “W-wait…!” She turns around. She seems a little startled by the amount of force I put into my voice. The hallway was rather empty, save for us two, after all. “I never… got your name. Might I-I ask what it is?” “Oh! You’re right! It’s Hana… Can’t believe I never told you. Apologies.” Hana. Hana. A lot of A’s, huh? Eventually, the teacher and a few students march up the opposite end of the hallway to our classroom. “Early today, Akio?” The teacher says, as he unlocks the door with her key. “Something like that…” I responded. I’m admittedly a little embarrassed by what just happened in the hallway, even though it went rather well. An after-effect of love, I suppose. The door finally opens and the smell of the classroom fills my nose. It’s rather comforting, but also fills me with boredom. The brief period of movement I get from door to chair is simply too little for my antsy legs. Maybe if they made every desk a treadmill it would alleviate the problem. I chuckle to myself at the thought of some of my classmates trying to keep up. I walk and sit down at my desk, awaiting today’s lesson. A friend of mine, Jiro, turns to me. He had found his desk a bit before me. “Oi, Akiooo! So early, today? What gives?” “I’m always early.” “Pshhh! Sure… Tell that to yourself last week, Mr. 5-minutes-after-the-bell” “I’ll have you know that is not my last name.” Me and Jiro have been friends since the start of the school year, we haven’t really known each other long, but due to our desks being in close proximity and our great chemistry, we’ve been able to get really close. Jiro is rather eccentric and energetic. It’s a great match for my rather calm personality. We’d make a perfect straight man/funny man duo, if we were to pursue comedy. “Yo, Akio, the festival is coming to town. And… We’ve never even hung out!? Let’s change that!” Jiro flashes a corporate smile, as if he was pitching his company’s product to a stern investor. “Hmm… My parents asked me about the same thing…” “It’s me or it’s them, Akio. No in betweens! I thought we had something…” Jiro pretends to tearfully sob, putting me in a rather awkward situation. Other students settling in begin to look at us. “Ok, ok… Maybe I’ll go with you, I had a choice to begin with. I’ll have an answer for you by tomorrow.” “Perfect! It’s a date, then!” Jiro attempts to make his most masculine face, but I can’t help but start laughing. “Good joke?” Jiro says, smiling lightly. “More like a good face, that got me.” “Nihihi!” Jiro’s face reminds me of a cat when he does this. After this exchange, the teacher begins to give some announcements before class. Some of them pertaining to festival safety, seeing as a great majority of us would be in attendance. Others included information about clubs, the coming exams, and career documents. All of us were in 12th grade, so after this, it was time to move on to the big world and get to work. Honestly, anytime I would hear anything about career forms or schooling after high school would always scare me. I was worried about putting myself out there in the world, afraid of what might come from that. Knowing that everyone else would have to face the same thing, it did make me feel a bit better. That light knot in my stomach still wouldn’t go away, however. The class passed by, uneventfully. The teacher talked in great detail about some wars, called on a few clueless students, and sighed a great many times. The typical class experience. Jiro got the golden ticket today, he was sleeping when he was called on. I’ve never seen a stick of chalk fly so fast and hit so hard. I could’ve sworn that he would be sent flying, but he started holding his head in pain. I laughed, but quickly fixed my expression when he looked at me with “the stare of a thousand deaths.” As we all began to walk out of the classroom after the bell, the hallways were flooding with oceans of students. Luckily, I had a boat and paddle to bear the waves. Or, maybe just experience? '''*''' The rest of the school day passed by, the classes got faster with each one that came and went. I put my shoes on, and walked out from the entrance. It was cloudy today, but there was no rain. Maybe it’s really starting to grow on me. The rain usually maddens and annoys me, but this year I want to see it more than ever. Is it because of her? What even are these feelings of mine? Is it appropriate to attribute them to a real person? I would stare and stare at her in that garden, as if admiring a beautiful landscape portrait in a museum. When we talked those two times, I was so nervous I could barely keep it together. I don’t view myself as someone who’s lacking in confidence, or has low self-esteem, but her presence is almost choking. And it’s my own fault. The more I pedestalize her, the worse those interactions will get. I’ve already confirmed it myself, in that garden. Hana is as real as those flowers. As real as the rain. Flesh, blood, bone. Breathing, beating, human. She gets sick, uses the bathroom, and makes mistakes. “Maybe I’ve been thinking about this wrong…” I say aloud, albeit accidentally. I think it’s okay to be curious about who she is. I have to leave this mindset behind if I want to keep talking to her. I don’t really have much of a reason though, except for that I just want to. Life is funny, in that way. Sometimes we just want to do something, and we don’t really have to explain why. Rather comforting. On a whim, I decided to visit the garden today before I made the trek home. Walking to the side gate this time, I opened it and once again found myself at the whim of these flowers. I was half expecting people to be here, seeing as it is after-school, but there was no one to be found. It was rather ethereal. The scene was similar to yesterday’s, but it possessed much more weight. Perhaps it was because I was alone. I could be off my guard, in this beautiful place. I walked by the hydrangea bushes, and touched their leaves. They can talk, huh? I move my hand gently to the flower and feel the petals. They are silky and soft, and a bit warm. Not too far off from human skin. The flower was a bright, light purple. It swayed with each touch, despite my light movements. As a joke, I put my ear up to the flower, as if to listen to what it was trying to tell me. A gust of wind blows through, and the flower hits my head a few times. Maybe it doesn’t like me too much? Upon realizing this, I burst into laughter. “Silly little flowers, maybe we’re equals after all?” Before my exit, I wave and bow courteously for the flowers and open the garden gate. Much to my surprise, a few gardening club members stood on the other side of the opening. This moment was admittedly, rather embarrassing. For one, I really hope they didn’t hear me talking to those flowers. For two, am I allowed back here? I didn’t ever really think about that. I gaze at each of their startled faces, and yet, Hana’s isn’t there. I am a bit disappointed, I let out a little sigh. “W-what brings you to the garden?” One of the members perks up, a small boy who appears to be a bit nervous about the encounter. The other two girls look toward him and then back at me, expectant of an answer. “I hope I wasn’t breaking any rules by being in there, I was just ta- …loo-king at the flowers? Is that okay?” I didn’t mean to sound so apologetic, but it came out rather emotional anyway. “No, that won’t be an issue. As long as all flowers are accounted for.” One of the two girls speaks, and gives a bit of a stern answer. She still seems suspicious about my presence. “Mei… No need to be so harsh, he doesn’t seem so bad after all.” The last girl speaks softly to the more brusque one, and gives a warm smile. I don’t know how to explain it, but I’m thanking her to high heaven right now. Who knows what would be in store for me had she not been here. My ears probably would’ve fallen off. “S-sorry again, I wasn't trying to cause any trouble.” I make my way past the exit, and attempt to pass by the three of them. I’m successful, but I stop and turn around hesitantly. “I love the garden, please don’t stop talking with those flowers.” I realize that’s not exactly what I wanted to say, and in my intense embarrassment I turn around and continue home. From behind me, I could hear the group laughing a bit. I could’ve swore I heard her name too… —---------------------------------------------------------------------------------------------------------------------------- '''Date: 2018/07/29''' '''A slower day than usual. Today was the last day of classes, meaning that we don’t come back until September. The festival will be starting tomorrow, as well. The week after, I FINALLY get to go on the sunflower trip. I am unbelievably excited for it.''' '''I don’t have much else to say. Thanks, I guess?''' '''Sincerely (again),''' '''Hana''' Today, we had the festival. I had inevitably let my parents know that instead of going with them, I would be going with Jiro. They didn’t seem disturbed by that fact, which was nice. Maybe even a little happy that they got to have some alone time. I guess this would be a win-win situation? I had called Jiro a little while ago to confirm where we would meet before making the commute. It would be about a ten minute walk. Thank the heavens the sun is setting, or I don’t think I would’ve made it to the 5 minute mark. I was thinking of coming in my kimono, but it was much too hot to wear something like that. Decidedly, I just went with my plain clothes. I waited a few more hours for 5:50 to come around before leaving my house. The festival started at 6:30, but we wanted to look around a bit before it began. I made the trek outside of my front door and walked to the side of the road. Despite the sun going down, it was still unbearably hot. I raised my hand to cover my eyes while walking in the sun’s gaze. A few neighbors appeared to be preparing themselves for the festival as well. We lived in a rather small town, so most of the people around here prepared themselves for things like these. [[User:IvoctA|IvoctA]] ([[User talk:IvoctA|discuss]] • [[Special:Contributions/IvoctA|contribs]]) 19:39, 16 September 2023 (UTC) == Hello! Wikijournal chat? == Happy new year :) Long time. I just saw you've been working with WikiJournal; how are you finding this format and process? What are the bottlenecks at the moment, what else might be possible? <span style="padding:0 2px 0 2px;background-color:white;color:#bbb;">&ndash;[[User:Sj|SJ]][[User Talk:Sj|<font style="color:#f90;">+</font>]]</span> 14:35, 25 January 2024 (UTC) :Hey Sj, good morning. :My submission for WJ has been quite slow due to IRL commitments, but I've had good experiences so far. One of the WJ reviewers was nice enough to give me and my buddy solid advice on how to improve our journal for submission, so I was pleased with that. The discussion at [[Talk:WikiJournal_User_Group#Current_status_of_WikiJournals]] is concerning, though, and I hope WJ will continue to be in operation. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:21, 25 January 2024 (UTC) == ''The Signpost'': 31 January 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/News and notes|Wikipedian Osama Khalid celebrated his 30th birthday in jail]] * Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Opinion|Until it happens to you]] * Disinformation report: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Disinformation report|How paid editors squeeze you dry]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/In the media|Katherine Maher new NPR CEO, go check Wikipedia, race in the race]] * In focus: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/In focus|The long road of a featured article candidate, part 2]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Recent research|Croatian takeover was enabled by "lack of bureaucratic openness and rules constraining [admins]"]] * Comix: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Comix|We've all got to start somewhere]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Traffic report|DJ, gonna burn this goddamn house right down]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 15:17, 31 January 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:Bri@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=26086360 --> == Invitation to discuss page deletion policy == A discussion that might interest you has been started at [[Wikiversity:Requests_for_Deletion#Wikiversity:Deletion_Convention_2024]]. -- [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:49, 15 February 2024 (UTC) :{{ping|Guy vandegrift}} Hello Guy vandegrift, I appreciate you thinking about me and reaching out to me for my thoughts. I'm afraid I will have to echo Dave's response and abstain from formulating any suggestions, since I do not have the needed time to review the discussion. Best of luck. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:35, 17 February 2024 (UTC) == Movie reviews == '''[[Paris, Texas]]''' is under a prod and when I found '''[[Book Reviews]]''' I thought of copying and creating a page called [[Movie reviews]]. I obviously don't need your permission (cc-by!), but was wondering what you thought of that idea. Any chance of attracting new movie reviews? ... Another idea would be to create a subspace under [[Essay]], with a subpage that links to [[Book reviews]]. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:54, 1 March 2024 (UTC)--Afterthought: See '''[[Essay/Collection]]'''-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:09, 1 March 2024 (UTC) :Movie reviews would be interesting. It can be educational, but also promotional. Personally, I don't have interest in producing movie reviews - but setting it up for future users in the hopes it is developed in accordance with our guidelines does not sound like a bad idea. I think putting [[Book Reviews]] under an "Essay" subspace would be redundant (and provide exceptionally long page titles), and that may be the same for Movie reviews. I would lean towards making it a stand alone project and, if issues arise, we can address them when we cross that bridge. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:15, 2 March 2024 (UTC) ::I am astonished by the success of [[Book Reviews]], but noticed one odd features: Certain titles are not shown on the front page. One example is [[Book_Reviews/A_Hero_of_Our_Time|A Hero for Our Time]]. I traced the problem to the dynamicpagelist and its count variable. I think I can fix that feature. Do you want me to?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 11:23, 2 March 2024 (UTC) :::Of course, go for it! Thank you in advance. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:30, 2 March 2024 (UTC) == ''The Signpost'': 4 September 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/News and notes|WikiCup enters final round, MCDC wraps up activities, 17-year-old hoax article unmasked]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/In the media|AI is not playing games anymore. Is Wikipedia ready?]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Recent research|Simulated Wikipedia seen as less credible than ChatGPT and Alexa in experiment]] * News from the WMF: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/News from the WMF|Meet the 12 candidates running in the WMF Board of Trustees election]] * Wikimania: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Wikimania|A month after Wikimania 2024]] * Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Serendipity|What it's like to be Wikimedian of the Year]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Traffic report|After the gold rush]] * Humour: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Humour|Local man halfway through rude reply no longer able to recall why he hates other editor]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 13:32, 4 September 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27280376 --> == ''The Signpost'': 26 September 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/In the media|Indian courts order Wikipedia to take down name of crime victim, and give up names of editors]] * Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Serendipity|A Wikipedian at the 2024 Paralympics]] * Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Opinion|asilvering's RfA debriefing]] * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/News and notes|Are you ready for admin elections?]] * Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Gallery|Are Ludd''ai''tes defending the English Wikipedia?]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Recent research|Article-writing AI is less "prone to reasoning errors (or hallucinations)" than human Wikipedia editors]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Traffic report|Jump in the line, rock your body in time]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 20:16, 26 September 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27463206 --> == ''The Signpost'': 19 October 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/News and notes|One election's end, another election's beginning]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Recent research|"As many as 5%" of new English Wikipedia articles "contain significant AI-generated content", says paper]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/In the media|Off to the races! Wikipedia wins!]] * Contest: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Contest|A WikiCup for the underdeveloped world]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Traffic report|A scream breaks the still of the night]] * Book review: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Book review|''The Editors'']] * Humour: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Humour|The Newspaper Editors]] * Crossword: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Crossword|Spilled Coffee Mug]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 11:18, 19 October 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27606308 --> == ''The Signpost'': 6 November 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * From the editors: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/From the editors|Editing Wikipedia should not be a crime]] * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/News and notes|Wikimedia Foundation shares ANI lawsuit updates; first admin elections appoint eleven sysops; first admin recalls opened; temporary accounts coming soon?]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/In the media|An old scrimmage, politics and purported libel]] * Special report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Special report|Wikipedia editors face litigation, censorship]] * Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Gallery|Why you should take more photos and upload them]] * In focus: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/In focus|Questions and answers about the court case]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Traffic report|Twisted tricks or tempting treats?]] * Technology report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Technology report|Wikimedia tech, the Asian News International case, and the ultra-rare BLACKLOCK]] * Humour: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Humour|Man quietly slinks away from talk page argument after realizing his argument dumb, wrong]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 08:09, 6 November 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27684085 --> == ''The Signpost'': 18 November 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/News and notes|Open letter to WMF about court case breaks one thousand signatures, big arb case declined, U4C begins accepting cases]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/In the media|Summons issued for Wikipedia editors by Indian court, "Gaza genocide" RfC close in news, old admin Gwern now big AI guy, and a "spectrum of reluctance" over Australian place names]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/Recent research|SPINACH: AI help for asking Wikidata "challenging real-world questions"]] * News from the WMF: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/News from the WMF|Wikimedia Foundation and Wikimedia Endowment audit reports: FY 2023–2024]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/Traffic report|Well, let us share with you our knowledge, about the electoral college]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 23:46, 18 November 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27733567 --> == Page called Food Tests == Hello there, I have contributed to the Food Tests page and I would like to inform you that we are currently expanding the page, and we are constantly adding new features to it. Please do not nominate the page for deletion any more throughout its improvement. This was just a kind notice. [[Food Tests]] Kind regards, Rock [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 20:40, 4 December 2024 (UTC) :{{ping|RockTransport}} Hey Rock, no need to worry. You have around 3 months or so to improve the content of the page, and I'm sure with your intentions stated here you'll be able to provide content to the page that would suffice [[Wikiversity:What is Wikiversity?|our objectives]]. Thanks and [[Template:Welcome|welcome to Wikiversity]] (make sure to check this page out for some useful tips as a beginner here)! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 23:06, 4 December 2024 (UTC) o4zgn6se13rvn4cez1xjwmf5t53d39y 2690316 2690314 2024-12-04T23:07:29Z Atcovi 276019 /* Page called Food Tests */ rewording 2690316 wikitext text/x-wiki [[User:Atcovi/Archive 1|/Archive 1 (September 25, 2013 - November 15, 2013)]] • [[User talk:Atcovi/Archive 2|/Archive 2 (November 15, 2013 - November 27, 2013)]] • [[User talk:Atcovi/Archive 3|/Archive 3 (December 3, 2013 - December 25, 2013)]] • [[User talk:Atcovi/Archive 4|/Archive 4 (December 24, 2013 - January 1, 2014)]] • [[User talk:Atcovi/Archive 5|/Archive 5 (January 2, 2014 - January 20, 2014)]] • [[User talk:Atcovi/Archive 6|/Archive 6 (March 24, 2014 - April 14, 2014)]] • [[User talk:Atcovi/Archive 7|/Archive 7 (April 19, 2014 - September 8, 2014)]] • [[User talk:Atcovi/Archive 8|/Archive 8 (September 12, 2014 - November 3, 2014)]] • [[User talk:Atcovi/Archive 9|/Archive 9 (November 6, 2014 - January 26, 2015)]] • [[User talk:Atcovi/Archive 10|/Archive 10 (January 28, 2015 - March 11, 2015)]] • [[User talk:Atcovi/Archive 11|/Archive 11 (March 22, 2015 - June 25, 2016)]] • [[User talk:Atcovi/Archive 12 (June 26, 2016 - January 8, 2018)|/Archive 12 (June 26, 2016 - January 8, 2018)]] • [[User talk:Atcovi/Archive 13 (January 9, 2018 - April 14, 2023)|/Archive 13 (January 9, 2018 - April 14, 2023)]] :''Before 2013: [https://en.wikiversity.org/w/index.php?title=User_talk:Atcovi&diff=750617&oldid=740650 see this]'' {{tmbox |small = |image = [[Image:Busy desk.svg|{{#ifeq:|yes|40px|75x50px}}]] |text = This user is busy in [http://en.wikipedia.org/wiki/Real_life Real Life] {{#if:|until&nbsp;{{{end}}}&nbsp;}}{{#if:|due to&nbsp;{{{reason}}}&nbsp;}}and may not respond swiftly to queries.{{#if:|<P>{{{msg}}} }} | style = {{#if:|width: {{{width}}}px;}} {{#ifeq:{{{shadow}}}|yes|{{box-shadow|0px|2px|4px|rgba(0,0,0,0.2)}}|}} }} == Hydrangeas and Me (Short Story in Progress) == '''CHAPTER 1''': It was a rainy Monday morning. The overcast clouds brought about a wave of light gray over the world, allowing the rain its refuge from the harsh sun. I woke up earlier than usual, due to the incessant knocking of light rain on my window. I rose from my bed, spread open the curtains, and raised the blinds. I looked outside, admittedly a bit upset about the current state of weather, and stepped down to change into my school uniform. The walk to school would prove a bit more difficult today because of the light spray, so I prepared my best umbrella and opened the door, ready to face the world. Each of my steps pittered and pattered. Water began to soak through my sole to my foot, giving me quite an icky feeling. I walked, feeling as if sponges had been rubber banded to my feet. I saw other students making the commute as well. Some of them rose their feet in comedic ways to avoid all the water, to no avail. Others ignored the puddles and continued forward. ''Rather courageous…'' I thought to myself, smiling lightly. Little moments like these can bring happiness, even in this less-than-desired weather. Eventually, me and the other hoards of students successfully made their way to the front gate. Some teachers decided to move under the canopy of the school’s main entrance, instead of their typical formation at the school’s gate. As I entered the canopy, I closed my umbrella and shook off the rain on the tiles in front of the glass door. Some other students mimicked my behavior and I laughed, albeit embarrassingly. I walked into the school, put my umbrella on the stand and marched toward my shoe locker before changing into the inside schools that were required of us. ''Today… I hope…'' Unlike the other mundane days, today was a lot more exciting. She would be there, in the garden of hydrangeas. After the change, I noted the time. 7:30. I would have a good amount of time to see her. I rushed with a silly look on my face to the library. Toward the back, there were several large windows which looked out upon the gardening club’s area of operation. I made sure to pick a table that was close to the windows. I turned my head outside, and like clockwork, she stood there tending to the flowers. Today, because of the rain, she wore a little umbrella cap supplied to the club to fend off the rain. She stood and cut stems with tiny gardening scissors. Her look was focused, yet soft. She didn’t seem like she wanted to injure the flowers, as if they meant everything to her. Each movement and cut, every one seemed to take special care toward the hydrangeas. I sat holding my head up with one hand watching her. Today’s view was particularly beautiful. Maybe the rain isn’t so bad for today. The girl in the garden walked toward the shed to grab a few more tools. There were about 20 more minutes until the first bell would ring. Despite that, she continued to work. Her soft look and gaze had enraptured me, it reminded me of more innocent times. It reassured me, and it calmed me. Everytime I did this, I would feel so tired and relaxed. I wish I could watch for hours. Today, I want to change things. I don’t want to watch from afar anymore. I wish to know more about this girl who tends to the garden. What kind of flowers does she like? How long has she been tending to this garden? Many, many questions trying to flow out to reach their answered counterparts. 15 minutes left, if I want to do this I better act fast. Tens of days spent watching, relaxed, for it suddenly to turn to stress and anxiety. Would she accept me? Is this her own personal time I'm trying to infringe upon? Each doubt weighed my steps like ankle weights, but despite it all, I continued forward. I opened the glass door and went outside, with no umbrella. The rain was rather light, so it didn’t pose much of a threat to my clothing. Upon opening the door, the girl in the garden noticed my entrance. Suddenly the image became reality, like entering a painting. The smell of dirt and flowers, mixed with rain. Her appearance is much more real. “H-hello…” I stammer, honestly not knowing what else to say. “Hi! What brings you to the garden, were you interested in joining the club?” She directs that typical soft smile in my direction. It’s very hard to not turn away from such a bright light. The rain complemented her well, like a warm bowl of porridge. What should I say? I hadn’t intended on joining this club, but what do I say aside from that? However, I don’t want to lie… “I wanted… to talk to you. To ''see'' you.” Ehhh… that’s not right. The girl’s expression seems to lighten, giving me a good chance to make eye contact. She seems a bit confused. “Do we know each other? I don’t remember ever seeing you in class… I’m sorry…” “The hydrangeas, I mean. I want to talk to you about the hydrangeas.” Her quizzical expression turns even brighter than the one she showed me first. “Oh! What would you like to know?” she’s smiling brightly, and tilts forward a bit waiting for my response. A conversation…! We’re having a conversation! Ahhh… I’m so excited but I have to keep my cool. “You cut the flowers with those scissors, why? Does that not hurt them?” “Ah! It’s a process called pruning… “ She snips her scissors up toward me. She turns around and beckons me to one of the many bushes. I follow suit. “These… are buds. It’s where the flower grows from. We prune them for many reasons, but most of the time it’s because the flower or stem is diseased.” I’m smiling so much right now, I can’t contain myself. So much stimulation just from a little conversation. I laugh a bit, out loud. “Hmmm…?” She looks up at me, hearing my light laughter. I quickly blush, becoming a bit embarrassed. “S-sometimes… It’s the little things that bring me happiness…” My signature phrase comes out, resulting in a large smile formulating on my face. The girl looks at me, and reflects my smile. “Do you like hydrangeas, too?” “I-I think I do?” The girl erupts into laughter, and I can’t help but join in. “You think? Well, maybe you should think a little more… about the kinds of flowers you like. And maybe the kinds of things you say, as well?” She lightly teases me, before standing up. “Do you have any other questions about these flowers?” I think. An important question… yet familiar. “Do ''you'' like hydrangeas?” I reflect her question back on her. Her expression grows warm, as if I can see the colors of her mood changing before me. “I do… They bring me a lot of my own happiness, too.” I think back to what I said earlier and smile again. “In the rain… they are so beautiful too! It makes me think of…” She pauses. “Tears of joy, or maybe the feeling of contentment. They are so… simple. Y’know it’s not just people that can talk… These flowers can too. And when you listen close enough, you find out that they have a lot to say.” “I’m not sure I can understand…” “That’s okay, we all have to start somewhere.” She smiles mischievously, before walking toward the shed. “We have about 5 minutes to class, you know…? What’s your name?” I panic over the fact that we have so little time, partially because my class is on the opposite side of where we are now. “I-I’m Akio…” “Well, Akio, thanks for talking to me… I’ll see you around!” “Same to you!” I walk back to the glass door before looking toward her again. She’s working in the shed, putting her tools away. I open the door inside. For some reason, leaving that rainy hydrangea world hurt me when I walked inside. As if the dryness was worse than rain… —---------------------------------------------------------------------------------------------------------------------------- '''Date: 2018/07/21''' '''Today, when I was tending to the hydrangeas, a boy approached me and asked me about them. I was nervous at first, because I didn’t know what he wanted.''' '''He wanted to ask about the flowers I was tending to, and why I was pruning them. We talked a little bit after this. He seems interesting… yet suspicious! I will keep an eye out…''' '''Again, me and Mom fought. It hurts a lot when we fight, I say things that I don’t mean to say. She wants me to do so many things, but I’m just one person. I think she’s just living through me… I love her, but I want to be my own person. Making my own choices and decisions.''' '''School was bland, except for the gardening club (as usual)! We have a field trip to a sunflower field coming up. It’s always so beautiful to see those fields in the summer… I hope the skies are blue that day. Maybe me and the club members can take photographs.''' '''Ahh… I CAN’T WAIT!''' '''Sincerely (again),''' '''Hana''' That night when I got home, I plopped back on my bed. Hours of lecturing and note-taking always takes quite a toll on me. Well, mostly everyone else too. If you think any of that is fun, I must commend you. My bookbag slid onto the floor as I unbuttoned my uniform. I felt a bit too lazy to do anything else other than rest. I noticed a pencil was sticking from my shirt pocket, if left unnoticed it might’ve poked me. I removed it, and threw it across the room. Just cause? Downstairs, I could hear the door open and close. “Akio, we’re home!” My mother called. “Welcome back.” I didn’t really attempt to yell this so I’m not too certain as to if the both of them actually heard that. “Man…” I thought back to earlier this morning, when me and… And…? “Ah!” I exclaimed aloud, recognizing my mistake. I forgot to ask her name. I wonder? “Akane? Hmm, maybe too harsh…” A soft complexion, warm smile… a love of flowers? “Aki… Akemi…” Maybe I shouldn’t focus on A’s so much. Yet that letter seems like it would make some sense. I kept deliberating as to what her name could potentially be before my thinking was interrupted. I heard a knock at the door. “Akio?” “Yeah, Mom?” “I didn’t hear you welcome us, so I thought maybe you were out? You surprised me…” “Sorry. ‘Welcome Home!’” “Don’t tease! How was school?” “Better than usual… Aside from the rain maybe.” “Oho? Anything new happen?” I thought back to how I expressed meeting the girl in the garden as ‘entering a painting.’ “‘Things may look up in your favor’” I said, like reading a fortune cookie. “Very funny. Well, I’m glad…” My mom flashes me a warm smile, before leaving the room and closing the door. “Don’t forget, this weekend is the festival in town. Your father and I will be going, you can come along too if you’d like?” She speaks through the door. “Hmm… Maybe. Let me think about it.” “Okay.” I hear her footsteps as she makes her way down the stairs. I look up at the ceiling and use my body weight to launch myself from the comfort of my bed. I trudge over to my bookbag and collect a few slips of homework and textbooks. “Time for the long haul…” A few hours pass, as I complete sheet after sheet. I’m careful to review what we learned from our textbooks. My hand grows tired of writing and my eyes get weary. I look up at the clock. 10:32 PM. Yeesh, it’s rather late now. Best to get to sleep. I close my blinds, and shut the curtains before making my final approach to the blanket kingdom that is my bed. It’s warm and fuzzy, and makes me smile with contentment. I hope that this feeling lasts forever. Next week… Can’t come any sooner… Maybe… Sleep takes me before I can finish my thought. '''CHAPTER 2''': “...” My classroom door is shut, it seems the teacher has yet to arrive. I really have to stop getting here so early. The hallway seems rather barren this morning. Yet again, the rain continues and won’t let up. Even if it’s monsoon season, I want at least one sunny day, y’know? I sit down next to the door and put my bookbag next to me, I feel rather sleepy. The fatigue from last night’s battle still sits on my shoulders. “Hahhh…” “Tired today, Akio?” A voice stirs me, yet my eyes remain closed. “Mmm…” Wait. Who’s voice is that? I open my eyes and turn to see who spoke. “Ah!” I yelp, accidentally. I quickly sit up, and try to straighten myself out. “H-hello! Yes, last night, so much homework… Not much sleep either.” It was the girl from the garden, and I was very startled. I couldn’t handle the cognitive dissonance of seeing her in the hallway. It was a first, and the fact that she was talking to me exacerbated this feeling a hundredfold. “I see. Well, I’m going to my classroom. See ya..!” The girl makes her way up the hallway, marching with great purpose. Mgghhh. A brief exchange, yet… “W-wait…!” She turns around. She seems a little startled by the amount of force I put into my voice. The hallway was rather empty, save for us two, after all. “I never… got your name. Might I-I ask what it is?” “Oh! You’re right! It’s Hana… Can’t believe I never told you. Apologies.” Hana. Hana. A lot of A’s, huh? Eventually, the teacher and a few students march up the opposite end of the hallway to our classroom. “Early today, Akio?” The teacher says, as he unlocks the door with her key. “Something like that…” I responded. I’m admittedly a little embarrassed by what just happened in the hallway, even though it went rather well. An after-effect of love, I suppose. The door finally opens and the smell of the classroom fills my nose. It’s rather comforting, but also fills me with boredom. The brief period of movement I get from door to chair is simply too little for my antsy legs. Maybe if they made every desk a treadmill it would alleviate the problem. I chuckle to myself at the thought of some of my classmates trying to keep up. I walk and sit down at my desk, awaiting today’s lesson. A friend of mine, Jiro, turns to me. He had found his desk a bit before me. “Oi, Akiooo! So early, today? What gives?” “I’m always early.” “Pshhh! Sure… Tell that to yourself last week, Mr. 5-minutes-after-the-bell” “I’ll have you know that is not my last name.” Me and Jiro have been friends since the start of the school year, we haven’t really known each other long, but due to our desks being in close proximity and our great chemistry, we’ve been able to get really close. Jiro is rather eccentric and energetic. It’s a great match for my rather calm personality. We’d make a perfect straight man/funny man duo, if we were to pursue comedy. “Yo, Akio, the festival is coming to town. And… We’ve never even hung out!? Let’s change that!” Jiro flashes a corporate smile, as if he was pitching his company’s product to a stern investor. “Hmm… My parents asked me about the same thing…” “It’s me or it’s them, Akio. No in betweens! I thought we had something…” Jiro pretends to tearfully sob, putting me in a rather awkward situation. Other students settling in begin to look at us. “Ok, ok… Maybe I’ll go with you, I had a choice to begin with. I’ll have an answer for you by tomorrow.” “Perfect! It’s a date, then!” Jiro attempts to make his most masculine face, but I can’t help but start laughing. “Good joke?” Jiro says, smiling lightly. “More like a good face, that got me.” “Nihihi!” Jiro’s face reminds me of a cat when he does this. After this exchange, the teacher begins to give some announcements before class. Some of them pertaining to festival safety, seeing as a great majority of us would be in attendance. Others included information about clubs, the coming exams, and career documents. All of us were in 12th grade, so after this, it was time to move on to the big world and get to work. Honestly, anytime I would hear anything about career forms or schooling after high school would always scare me. I was worried about putting myself out there in the world, afraid of what might come from that. Knowing that everyone else would have to face the same thing, it did make me feel a bit better. That light knot in my stomach still wouldn’t go away, however. The class passed by, uneventfully. The teacher talked in great detail about some wars, called on a few clueless students, and sighed a great many times. The typical class experience. Jiro got the golden ticket today, he was sleeping when he was called on. I’ve never seen a stick of chalk fly so fast and hit so hard. I could’ve sworn that he would be sent flying, but he started holding his head in pain. I laughed, but quickly fixed my expression when he looked at me with “the stare of a thousand deaths.” As we all began to walk out of the classroom after the bell, the hallways were flooding with oceans of students. Luckily, I had a boat and paddle to bear the waves. Or, maybe just experience? '''*''' The rest of the school day passed by, the classes got faster with each one that came and went. I put my shoes on, and walked out from the entrance. It was cloudy today, but there was no rain. Maybe it’s really starting to grow on me. The rain usually maddens and annoys me, but this year I want to see it more than ever. Is it because of her? What even are these feelings of mine? Is it appropriate to attribute them to a real person? I would stare and stare at her in that garden, as if admiring a beautiful landscape portrait in a museum. When we talked those two times, I was so nervous I could barely keep it together. I don’t view myself as someone who’s lacking in confidence, or has low self-esteem, but her presence is almost choking. And it’s my own fault. The more I pedestalize her, the worse those interactions will get. I’ve already confirmed it myself, in that garden. Hana is as real as those flowers. As real as the rain. Flesh, blood, bone. Breathing, beating, human. She gets sick, uses the bathroom, and makes mistakes. “Maybe I’ve been thinking about this wrong…” I say aloud, albeit accidentally. I think it’s okay to be curious about who she is. I have to leave this mindset behind if I want to keep talking to her. I don’t really have much of a reason though, except for that I just want to. Life is funny, in that way. Sometimes we just want to do something, and we don’t really have to explain why. Rather comforting. On a whim, I decided to visit the garden today before I made the trek home. Walking to the side gate this time, I opened it and once again found myself at the whim of these flowers. I was half expecting people to be here, seeing as it is after-school, but there was no one to be found. It was rather ethereal. The scene was similar to yesterday’s, but it possessed much more weight. Perhaps it was because I was alone. I could be off my guard, in this beautiful place. I walked by the hydrangea bushes, and touched their leaves. They can talk, huh? I move my hand gently to the flower and feel the petals. They are silky and soft, and a bit warm. Not too far off from human skin. The flower was a bright, light purple. It swayed with each touch, despite my light movements. As a joke, I put my ear up to the flower, as if to listen to what it was trying to tell me. A gust of wind blows through, and the flower hits my head a few times. Maybe it doesn’t like me too much? Upon realizing this, I burst into laughter. “Silly little flowers, maybe we’re equals after all?” Before my exit, I wave and bow courteously for the flowers and open the garden gate. Much to my surprise, a few gardening club members stood on the other side of the opening. This moment was admittedly, rather embarrassing. For one, I really hope they didn’t hear me talking to those flowers. For two, am I allowed back here? I didn’t ever really think about that. I gaze at each of their startled faces, and yet, Hana’s isn’t there. I am a bit disappointed, I let out a little sigh. “W-what brings you to the garden?” One of the members perks up, a small boy who appears to be a bit nervous about the encounter. The other two girls look toward him and then back at me, expectant of an answer. “I hope I wasn’t breaking any rules by being in there, I was just ta- …loo-king at the flowers? Is that okay?” I didn’t mean to sound so apologetic, but it came out rather emotional anyway. “No, that won’t be an issue. As long as all flowers are accounted for.” One of the two girls speaks, and gives a bit of a stern answer. She still seems suspicious about my presence. “Mei… No need to be so harsh, he doesn’t seem so bad after all.” The last girl speaks softly to the more brusque one, and gives a warm smile. I don’t know how to explain it, but I’m thanking her to high heaven right now. Who knows what would be in store for me had she not been here. My ears probably would’ve fallen off. “S-sorry again, I wasn't trying to cause any trouble.” I make my way past the exit, and attempt to pass by the three of them. I’m successful, but I stop and turn around hesitantly. “I love the garden, please don’t stop talking with those flowers.” I realize that’s not exactly what I wanted to say, and in my intense embarrassment I turn around and continue home. From behind me, I could hear the group laughing a bit. I could’ve swore I heard her name too… —---------------------------------------------------------------------------------------------------------------------------- '''Date: 2018/07/29''' '''A slower day than usual. Today was the last day of classes, meaning that we don’t come back until September. The festival will be starting tomorrow, as well. The week after, I FINALLY get to go on the sunflower trip. I am unbelievably excited for it.''' '''I don’t have much else to say. Thanks, I guess?''' '''Sincerely (again),''' '''Hana''' Today, we had the festival. I had inevitably let my parents know that instead of going with them, I would be going with Jiro. They didn’t seem disturbed by that fact, which was nice. Maybe even a little happy that they got to have some alone time. I guess this would be a win-win situation? I had called Jiro a little while ago to confirm where we would meet before making the commute. It would be about a ten minute walk. Thank the heavens the sun is setting, or I don’t think I would’ve made it to the 5 minute mark. I was thinking of coming in my kimono, but it was much too hot to wear something like that. Decidedly, I just went with my plain clothes. I waited a few more hours for 5:50 to come around before leaving my house. The festival started at 6:30, but we wanted to look around a bit before it began. I made the trek outside of my front door and walked to the side of the road. Despite the sun going down, it was still unbearably hot. I raised my hand to cover my eyes while walking in the sun’s gaze. A few neighbors appeared to be preparing themselves for the festival as well. We lived in a rather small town, so most of the people around here prepared themselves for things like these. [[User:IvoctA|IvoctA]] ([[User talk:IvoctA|discuss]] • [[Special:Contributions/IvoctA|contribs]]) 19:39, 16 September 2023 (UTC) == Hello! Wikijournal chat? == Happy new year :) Long time. I just saw you've been working with WikiJournal; how are you finding this format and process? What are the bottlenecks at the moment, what else might be possible? <span style="padding:0 2px 0 2px;background-color:white;color:#bbb;">&ndash;[[User:Sj|SJ]][[User Talk:Sj|<font style="color:#f90;">+</font>]]</span> 14:35, 25 January 2024 (UTC) :Hey Sj, good morning. :My submission for WJ has been quite slow due to IRL commitments, but I've had good experiences so far. One of the WJ reviewers was nice enough to give me and my buddy solid advice on how to improve our journal for submission, so I was pleased with that. The discussion at [[Talk:WikiJournal_User_Group#Current_status_of_WikiJournals]] is concerning, though, and I hope WJ will continue to be in operation. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:21, 25 January 2024 (UTC) == ''The Signpost'': 31 January 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/News and notes|Wikipedian Osama Khalid celebrated his 30th birthday in jail]] * Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Opinion|Until it happens to you]] * Disinformation report: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Disinformation report|How paid editors squeeze you dry]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/In the media|Katherine Maher new NPR CEO, go check Wikipedia, race in the race]] * In focus: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/In focus|The long road of a featured article candidate, part 2]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Recent research|Croatian takeover was enabled by "lack of bureaucratic openness and rules constraining [admins]"]] * Comix: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Comix|We've all got to start somewhere]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-01-31/Traffic report|DJ, gonna burn this goddamn house right down]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 15:17, 31 January 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:Bri@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=26086360 --> == Invitation to discuss page deletion policy == A discussion that might interest you has been started at [[Wikiversity:Requests_for_Deletion#Wikiversity:Deletion_Convention_2024]]. -- [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:49, 15 February 2024 (UTC) :{{ping|Guy vandegrift}} Hello Guy vandegrift, I appreciate you thinking about me and reaching out to me for my thoughts. I'm afraid I will have to echo Dave's response and abstain from formulating any suggestions, since I do not have the needed time to review the discussion. Best of luck. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:35, 17 February 2024 (UTC) == Movie reviews == '''[[Paris, Texas]]''' is under a prod and when I found '''[[Book Reviews]]''' I thought of copying and creating a page called [[Movie reviews]]. I obviously don't need your permission (cc-by!), but was wondering what you thought of that idea. Any chance of attracting new movie reviews? ... Another idea would be to create a subspace under [[Essay]], with a subpage that links to [[Book reviews]]. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 09:54, 1 March 2024 (UTC)--Afterthought: See '''[[Essay/Collection]]'''-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:09, 1 March 2024 (UTC) :Movie reviews would be interesting. It can be educational, but also promotional. Personally, I don't have interest in producing movie reviews - but setting it up for future users in the hopes it is developed in accordance with our guidelines does not sound like a bad idea. I think putting [[Book Reviews]] under an "Essay" subspace would be redundant (and provide exceptionally long page titles), and that may be the same for Movie reviews. I would lean towards making it a stand alone project and, if issues arise, we can address them when we cross that bridge. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:15, 2 March 2024 (UTC) ::I am astonished by the success of [[Book Reviews]], but noticed one odd features: Certain titles are not shown on the front page. One example is [[Book_Reviews/A_Hero_of_Our_Time|A Hero for Our Time]]. I traced the problem to the dynamicpagelist and its count variable. I think I can fix that feature. Do you want me to?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 11:23, 2 March 2024 (UTC) :::Of course, go for it! Thank you in advance. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:30, 2 March 2024 (UTC) == ''The Signpost'': 4 September 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/News and notes|WikiCup enters final round, MCDC wraps up activities, 17-year-old hoax article unmasked]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/In the media|AI is not playing games anymore. Is Wikipedia ready?]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Recent research|Simulated Wikipedia seen as less credible than ChatGPT and Alexa in experiment]] * News from the WMF: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/News from the WMF|Meet the 12 candidates running in the WMF Board of Trustees election]] * Wikimania: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Wikimania|A month after Wikimania 2024]] * Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Serendipity|What it's like to be Wikimedian of the Year]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Traffic report|After the gold rush]] * Humour: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-04/Humour|Local man halfway through rude reply no longer able to recall why he hates other editor]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 13:32, 4 September 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27280376 --> == ''The Signpost'': 26 September 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/In the media|Indian courts order Wikipedia to take down name of crime victim, and give up names of editors]] * Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Serendipity|A Wikipedian at the 2024 Paralympics]] * Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Opinion|asilvering's RfA debriefing]] * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/News and notes|Are you ready for admin elections?]] * Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Gallery|Are Ludd''ai''tes defending the English Wikipedia?]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Recent research|Article-writing AI is less "prone to reasoning errors (or hallucinations)" than human Wikipedia editors]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-09-26/Traffic report|Jump in the line, rock your body in time]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 20:16, 26 September 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27463206 --> == ''The Signpost'': 19 October 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/News and notes|One election's end, another election's beginning]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Recent research|"As many as 5%" of new English Wikipedia articles "contain significant AI-generated content", says paper]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/In the media|Off to the races! Wikipedia wins!]] * Contest: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Contest|A WikiCup for the underdeveloped world]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Traffic report|A scream breaks the still of the night]] * Book review: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Book review|''The Editors'']] * Humour: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Humour|The Newspaper Editors]] * Crossword: [[w:en:Wikipedia:Wikipedia Signpost/2024-10-19/Crossword|Spilled Coffee Mug]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 11:18, 19 October 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27606308 --> == ''The Signpost'': 6 November 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * From the editors: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/From the editors|Editing Wikipedia should not be a crime]] * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/News and notes|Wikimedia Foundation shares ANI lawsuit updates; first admin elections appoint eleven sysops; first admin recalls opened; temporary accounts coming soon?]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/In the media|An old scrimmage, politics and purported libel]] * Special report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Special report|Wikipedia editors face litigation, censorship]] * Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Gallery|Why you should take more photos and upload them]] * In focus: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/In focus|Questions and answers about the court case]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Traffic report|Twisted tricks or tempting treats?]] * Technology report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Technology report|Wikimedia tech, the Asian News International case, and the ultra-rare BLACKLOCK]] * Humour: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-06/Humour|Man quietly slinks away from talk page argument after realizing his argument dumb, wrong]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 08:09, 6 November 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27684085 --> == ''The Signpost'': 18 November 2024 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/News and notes|Open letter to WMF about court case breaks one thousand signatures, big arb case declined, U4C begins accepting cases]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/In the media|Summons issued for Wikipedia editors by Indian court, "Gaza genocide" RfC close in news, old admin Gwern now big AI guy, and a "spectrum of reluctance" over Australian place names]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/Recent research|SPINACH: AI help for asking Wikidata "challenging real-world questions"]] * News from the WMF: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/News from the WMF|Wikimedia Foundation and Wikimedia Endowment audit reports: FY 2023–2024]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2024-11-18/Traffic report|Well, let us share with you our knowledge, about the electoral college]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 23:46, 18 November 2024 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:JPxG@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=27733567 --> == Page called Food Tests == Hello there, I have contributed to the Food Tests page and I would like to inform you that we are currently expanding the page, and we are constantly adding new features to it. Please do not nominate the page for deletion any more throughout its improvement. This was just a kind notice. [[Food Tests]] Kind regards, Rock [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 20:40, 4 December 2024 (UTC) :{{ping|RockTransport}} Hey Rock, no need to worry. You have around 3 months or so to improve the content of the page, and I'm sure with your intentions stated here you'll be able to provide content to the page that would suffice [[Wikiversity:What is Wikiversity?|our objectives]] by the 'deadline'. Thank you for your contributions and [[Template:Welcome|welcome to Wikiversity]] (make sure to check this page out for some useful tips for beginners)! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 23:06, 4 December 2024 (UTC) fn1b6x4w13cy4e5gwoz6yqul2m7pyyt 2 136588 2690281 2529686 2024-12-04T16:29:42Z CommonsDelinker 9184 Removing [[:c:File:20040514_large_hail_5.25".jpg|20040514_large_hail_5.25".jpg]], it has been deleted from Commons by [[:c:User:Krd|Krd]] because: per [[:c:Commons:Deletion requests/File:20040514 large hail 5.25".jpg|]]. 2690281 wikitext text/x-wiki [[Image:Willamette Meteorite AMNH.jpg|thumb|right|250px|The Williamette Meteorite is on display at the American Museum of Natural History in New York City. Credit: [[w:User:Dante Alighieri|Dante Alighieri]].]] A '''meteorite''' is a natural object that survives impact with the Earth's surface. Common objects falling from the sky are rain drops, snow flakes, and hail. Less often meteors are stones and chunks of metal. A '''megacryometeor''' is a very large chunk of ice ... sometimes called huge hailstones, but do not need to form in thunderstorms. Rocks found on the Moon have also been determined to be meteoritic; i.e., not of lunar origin, have survived impact with the lunar surface, probably came from "outer space". This suggests a more general definition of '''meteorite''' is needed. {{clear}} ==Astronomy== {{main|Radiation astronomy/Astronomy}} [[Image:Widmanstatten patterns 2.jpg|thumb|left|250px|This image is a cross-section of the Laguna Manantiales meteorite showing Widmanstätten patterns. Credit: [[commons:User:Aramgutang|Aram Dulyan]].]] A '''meteorite''' is a natural object originating in outer space that survives impact with the Earth's surface. Most meteorites derive from small astronomical objects called meteoroids. When a meteoroid enters the atmosphere, frictional, pressure, and chemical interactions with the atmospheric gasses cause the body to heat up and emit light, thus forming a fireball, also known as a meteor or '''shooting/falling star'''. Meteorites have been found on the Moon<ref name=McSween>{{cite journal |last=McSween Jr. |first=Harry Y. |year=1976 |title=A new type of chondritic meteorite found in lunar soil |journal=Earth and Planetary Science Letters |volume=31 |issue=2 |pages=193–9 |doi=10.1016/0012-821X(76)90211-9 |bibcode=1976E&PSL..31..193M }}</ref><ref name=Rubin>{{ cite journal |last=Rubin |first=Alan E. |year=1997 |title=The Hadley Rille enstatite chondrite and its agglutinate-like rim: Impact melting during accretion to the Moon |journal=Meteoritics & Planetary Science |volume=32 |issue=1 |pages=135–41 |bibcode=1997M&PS...32..135R |doi=10.1111/j.1945-5100.1997.tb01248.x }}</ref> and Mars.<ref name=Rover>{{ cite book | title=Opportunity Rover Finds an Iron Meteorite on Mars | publisher=JPL | date=January 19, 2005 | url=http://marsrovers.jpl.nasa.gov/newsroom/pressreleases/20050119a.html | accessdate=2006-12-12 }}</ref> '''Widmanstätten patterns''', also called '''Thomson structures''', are unique figures of long nickel-iron crystals, found in the octahedrite iron meteorites and some pallasites. They consist of a fine interleaving of kamacite and taenite bands or ribbons called ''lamellæ''. Commonly, in gaps between the lamellæ, a fine-grained mixture of kamacite and taenite called plessite can be found. {{clear}} ==Radiation== {{main|Radiation}} [[Image:Mackinac Island.jpg|thumb|right|250px|This is a natural color image of the weathered iron meteorite "Mackinac Island". Credit: NASA.]] [[Image:PIA07269-Mars Rover Opportunity-Iron Meteorite.jpg|thumb|left|250px|NASA's Mars Exploration Rover Opportunity has found this iron meteorite on Mars. This is the first meteorite of any type ever identified on another planet. Credit: NASA/JPL/Cornell.]] Martian meteors are thought to be from Mars because they have elemental and isotopic compositions that are similar to rocks and atmosphere gases analyzed by spacecraft on Mars.<ref name=Treiman>{{ cite journal |author=A.H. Treiman ''et al.'' |title=The SNC meteorites are from Mars |journal=Planetary and Space Science |volume=48 |issue=12–14 |month=October |year=2000 |pages=1213–30 |bibcode=2000P&SS...48.1213T |doi=10.1016/S0032-0633(00)00105-7 }}</ref> The image at right is of the Mackinac Island meteorite, discovered on Mars by the NASA {{w|Opportunity rover}} on October 13, 2009. At top left is "the first meteorite of any type ever identified on another planet. The pitted, basketball-size object is mostly made of iron and nickel. Readings from spectrometers on the rover determined that composition. Opportunity used its panoramic camera to take the images used in this approximately true-color composite on the rover's 339th martian day, or sol (Jan. 6, 2005). This composite combines images taken through the panoramic camera's 600-nanometer (red), 530-nanometer (green), and 480-nanometer (blue) filters."<ref name=NASA05>{{ cite book |author=NASA |title=PIA07269: Iron Meteorite on Mars |publisher=NASA |location=Pasadena, California USA |date=January 19, 2005 |url=http://photojournal.jpl.nasa.gov/catalog/PIA07269 |accessdate=2013-02-16 }}</ref> Comparison of the two meteorites shown here suggests that the left one is a much more recent fall. Meteorites have been determined to occur on Earth, the Moon, and Mars. Radiation in a general sense is an entity, source, or object moving fast relative to local entities, sources, or objects that are or appear relatively motionless such as the ground, an atmosphere (although the gases within may move fast), or a nearby mountain. The larger the entity, source, or object that is radiating, the larger is the impact, or impact crater, and perhaps the surviving pieces, which can still be called a meteorite. Objects larger than a molecule, that have been radiated, can be termed meteors. A meteor, or meteoroid, could range continuously to larger particle sizes to a maximum on the order of a galaxy cluster. {{clear}} ==Planetary sciences== {{main|Planets/Sciences|Planetary sciences}} [[Image:Willamette-meteorite-2.jpg|thumb|right|250px|The American Museum of Natural History ensures access to the Willamette Meteorite. Credit: Ellen V. Futter.{{tlx|fairuse}}]] [[Image:Ahnighito AMNH, 34 tons meteorite.jpg|thumb|left|250px|Ahnighito fragment of the Cape York meteorite weighs 34 tons and is in the AMNH. Credit: [http://www.flickr.com/people/34811070@N05 Mike Cassano].{{tlx|free media}}]] "Willamette Meteorite (Tomanowos iron meteorite) [is] another massive iron meteorite that slammed into the earth and left no crater, yet is the largest meteorite in the USA!"<ref name=Willamette>{{ cite book |author=GeUlogy |title=Willamette iron meteorite (Tomanowos) |publisher=Geulogy.com |location= |date=12 November 2012 |url=http://geulogy.com/willamette-iron-meteorite-tomanowos/ |accessdate=2015-01-11 }}</ref> "Cape York iron meteorites are separate lumps of iron but have been grouped together as fragments of the same iron meteorite, as they are found around the same location. The iron pieces known as the Women and the Dog were found about 25 meters from each other on the mainland and Ahnighito was found on an island. They were found above the ground and with no visible crater around them, even for the largest one called Ahnighito."<ref name=York>{{ cite book |author=Geulogy |title=Cape York iron meteorite |publisher=Geulogy.com |location= |date=29 November 2012 |url=http://geulogy.com/cape-york-iron-meteorite/ |accessdate=2015-01-11 }}</ref> Cape York is in Savissivik, Northwwest Greenland. Ahnighito (the Tent) weighs 31 metric tons; the Woman, weighs 3 metric tons; the Dog, weighs 400 kilograms, Savik I 3.4 tons, Savik II 7.8 kg, and Agpalilik about 15 tons.<ref name=Buchwald/> "The meteorite lay on an ice-free slope 500 m from the shore and was partly covered with gneiss boulders. There was no crater and no crushing of rocks discovered."<ref name=Buchwald>{{ cite journal |author=Vagn Buchwald |title=Discovery of Cape York (Agpalilik) Iron Meteorite, Northwest Greenland |publisher=USRA |location=Moscow, USSR |month=October |year=1963 |journal=The Meteoritical Bulletin |volume=10 |issue=28 |url=http://www.lpi.usra.edu/meteor/docs/mb28.pdf |accessdate=2015-01-11 }}</ref> {{clear}} ==Minerals== {{main|Astrominerals}} [[Image:Panguite-allende-meteorite-1.jpg|thumb|right|250px|In this image the mineral panguite occurs with the scandium-rich silicate davisite embedded in a piece of the Allende meteorite. Credit: Caltech/Chi Ma.]] [[Mineral astronomy]] is the use of various astronomical techniques to locate and identify minerals and mineral deposits, especially on astronomical rocky objects. At right is a thin-section image of a slice through the Allende meteorite. The Allende meteorite "lit up Mexico's skies in 1969 [and] scattered thousands of meteorite bits across the northern Mexico state of Chihuahua. ... Panguite [a titanium dioxide mineral] is believed to be among the oldest minerals in the solar system, which is [estimated to be] about 4.5 billion years old. Panguite belongs to a class of refractory minerals that could have formed only under the extreme temperatures and conditions present in the infant solar system."<ref name=Bryner>{{ cite book |author=Jeanna Bryner |title=1969 Fireball Meteorite Reveals New Ancient Mineral |publisher=LiveScience |location= |date=June 26, 2012 |url=http://www.livescience.com/21197-allende-meteorite-panguite-mineral.html |accessdate=2013-11-01 }}</ref> {{clear}} ==Theoretical meteorites== '''Def.''' "a meteor that reaches the surface of the Earth without being completely vaporized" is called a '''meteorite'''.<ref name=Gove>{{ cite book |author= |title=Webster's Seventh New Collegiate Dictionary |publisher=G. & C. Merriam Company |location=Springfield, Massachusetts |date=1963 |editor=Philip B. Gove |pages=1221 |url=https://archive.org/details/webstersseventhn00unse |bibcode= |doi= |pmid= |isbn= |accessdate=2011-08-26 }}</ref> '''Def.''' a "fast-moving streak of light in the night sky caused by the entry of extraterrestrial matter into the earth's atmosphere"<ref name=MeteorWikt>{{ cite book |author=[[wikt:User:Xed~enwiktionary|Xed~enwiktionary]] |title=meteor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=3 September 2004 |url=https://en.wiktionary.org/wiki/meteor |accessdate=26 June 2019 }}</ref> is called a '''meteor'''. Here's a theoretical definition: '''Def.''' "any natural object radiating through a portion or all of the Earth's or another natural, astronomical object's atmosphere"<ref name=Marshallsumter>{{ cite book |author=[[User:Marshallsumter|Marshallsumter]] |title=meteor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California USA |date=September 24, 2011 |url=http://en.wikiversity.org/wiki/Radiation/Meteors |accessdate=2018-01-24 }}</ref> is called a '''meteor'''. '''Def.''' "a relatively small (sand- to boulder-sized) fragment of debris in a solar system"<ref name=MeteoroidWikt>{{ cite book |author=[[wikt:User:SnoopY|SnoopY]] |title=meteoroid |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 December 2005 |url=https://en.wiktionary.org/wiki/meteoroid |accessdate=2016-02-06 }}</ref> is called a '''meteoroid'''. '''Def.''' a "metallic or stony object or body that [is the remains of a meteor]<ref name=MeteoriteWikt1>{{ cite book |author=[[wikt:User:SemperBlotto|SemperBlotto]] |title=meteorite |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=28 December 2007 |url=http://en.wiktionary.org/wiki/meteorite |accessdate=2015-03-28 }}</ref>oid]<ref name=MeteoriteWikt2>{{ cite book |author=[[wikt:User:186.74.9.130|186.74.9.130]] |title=meteorite |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=28 May 2019 |url=http://en.wiktionary.org/wiki/meteorite |accessdate=2015-03-28 }}</ref> [or] has fallen to the surface of the Earth from outer space"<ref name=MeteoriteWikt>{{ cite book |author=[[wikt:User:SnoopY|SnoopY]] |title=meteorite |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 January 2006 |url=http://en.wiktionary.org/wiki/meteorite |accessdate=2015-03-28 }}</ref> is called a '''meteorite'''. These are usage notes: #Such an object may be as small as an electron or much larger. # Astronomical objects that are atoms, nuclei, or subatomic particles are part of [[Radiation/Cosmic rays|cosmic-ray astronomy]]. # Astronomical objects larger than atoms, nuclei, or subatomic particles that are fast-moving relative to perceived, almost motionless objects, radiating through another natural object's atmosphere or gaseous environment are also here referred to as meteors. # These can be a high-velocity star moving through the [[interstellar medium]] or a larger object moving through an [[intergalactic medium]]. # At the extreme a meteor can be a galaxy cluster moving relative to apparently stationary clusters in its neighborhood of the universe. '''Def.''' for a meteor that strikes another entity, source, or object, which appears relatively motionless, and leaves behind a rock, that rock is called a '''meteorite'''. ==Meteoroids== {{main|Radiation/Meteors}} A meteoroid is a suggested term for a sand- to boulder-sized particle of debris in the [[Solar System]]. The visible path of a meteoroid that enters the Earth's atmosphere (or another body's) atmosphere is called a '''''meteor''''', or colloquially a '''''shooting star''''' or '''''falling star'''''. If a meteoroid reaches the ground and survives impact, then it is called a meteorite. "As of 2011 the International Astronomical Union officially defines a meteoroid as a solid object moving in interplanetary space, of a size considerably smaller than an asteroid and considerably larger than an atom".<ref name=Millman>{{ cite journal |author=Peter M. Millman |year=1961 |title=A report on meteor terminology |journal=JRASC |volume=55 |pages=265–267 |bibcode=1961JRASC..55..265M }}</ref><ref name=Glossary>{{ cite book |url=http://www.imo.net/glossary |title=Glossary International Meteor Organization |publisher=Imo.net |date=2008-11-18 |accessdate=2011-09-16}}</ref> Beech and Steel, writing in ''Quarterly Journal of the Royal Astronomical Society'', proposed a new definition where a meteoroid is between 100 µm and 10 m across.<ref name=Beach>{{ cite journal |author=Martin Beech, Duncan Steel |year=1995 |month=September |title=On the Definition of the Term Meteoroid |journal=Quarterly Journal of the Royal Astronomical Society |volume=36 |issue=3 |pages=281–284 |bibcode=1995QJRAS..36..281B }})</ref> Following the discovery and naming of asteroids below 10 m in size (e.g., 2008 TC3), Rubin and Grossman refined the Beech and Steel definition of meteoroid to objects between 10&nbsp;µm and 1 m in diameter.<ref name=Rubin2010>{{Cite journal |author=Rubin, A.E. |coauthors=Grossman, J.N. |year=2010 |month=January |title=Meteorite and meteoroid: New comprehensive definitions |journal=Meteoritics & Planetary Science |volume=45 |issue=1 |pages=114–122 |bibcode=2010M&PS...45..114R |doi = 10.1111/j.1945-5100.2009.01009.x }})</ref> The near-Earth object (NEO) definition includes larger objects, up to 50 m in diameter, in this category. Very small meteoroids are known as '''micrometeoroids''' (see also interplanetary dust). The composition of meteoroids can be determined as they pass through Earth's atmosphere from their trajectories and the light spectra of the resulting meteor. Their effects on radio signals also give information, especially useful for daytime meteors which are otherwise very difficult to observe. The light spectra, combined with trajectory and light curve measurements, have yielded various compositions and densities, ranging from fragile snowball-like objects with density about a quarter that of ice,<ref name=Povenmire>Povenmire, H. [http://www.lpi.usra.edu/meetings/lpsc2000/pdf/1183.pdf PHYSICAL DYNAMICS OF THE UPSILON PEGASID FIREBLL – EUROPEAN NETWORK 190882A]. Florida Institute of Technology</ref> to nickel-iron rich dense rocks. "The silicate spheres are the most dominant group."<ref name=Blanchard>{{ cite journal |author=M.B. Blanchard |author2=D.E. Brownlee |author3=T.E. Bunch |author4=P.W. Hodge |author5=F.T. Kyte |title=Meteoroid ablation spheres from deep-sea sediments |journal=Earth and Planetary Science Letters |month=January |year=1980 |volume=46 |issue=2 |pages=178-90 |url=http://www.sciencedirect.com/science/article/pii/0012821X80900047 |arxiv= |bibcode= |doi=10.1016/0012-821X(80)90004-7 |pmid= |accessdate=2012-01-02 }}</ref> From these trajectory measurements, meteoroids have been found to have many different orbits, some clustering in streams (see meteor showers) often associated with a parent comet, others apparently sporadic. Debris from meteoroid streams may eventually be scattered into other orbits. ... Meteoroids travel around the Sun in a variety of orbits and at various velocities. The fastest ones move at about 26 miles per second (42 kilometers per second) through space in the vicinity of Earth's orbit. The Earth travels at about 18 miles per second (29 kilometers per second). Thus, when meteoroids meet the Earth's atmosphere head-on (which would only occur if the meteors were in a retrograde orbit), the combined speed may reach about 44 miles per second (71 kilometers per second). Meteoroids moving through the earth's orbital space average about 20&nbsp;km/s.<ref name=OrbitalDebris>{{ cite book |title=Report on Orbital Debris |url=http://hdl.handle.net/2060/19900003319 |publisher=NASA Technical Reports Server |accessdate=1 September 2012 }}</ref> {{clear}} ==Meteors== {{main|Radiation/Meteors}} [[Image:Leonid Meteor.jpg|thumb|right|250px|The photograph shows the meteor, afterglow, and wake as distinct components of a meteor during the peak of the 2009 Leonid Meteor Shower. Credit: [[commons:User:Navicore|Navicore]].]] [[Image:Meteor burst.jpg|thumb|right|250px|This picture is of the Alpha-Monocerotid meteor outburst in 1995. It is a timed exposure where the meteors have actually occurred several seconds to several minutes apart. Credit: NASA Ames Research Center/S. Molau and P. Jenniskens.]] A '''meteor''' is the visible path of a meteoroid that has entered the Earth's atmosphere. Meteors typically occur in the mesosphere, and most range in altitude from 75&nbsp;km to 100&nbsp;km.<ref name=Erickson>{{ cite book |url=http://www.haystack.mit.edu/~pje/meteors/ | title = Millstone Hill UHF Meteor Observations: Preliminary Results | author = Philip J. Erickson }}</ref> Millions of meteors occur in the Earth's atmosphere every day. Most meteoroids that cause meteors are about the size of a pebble. Although there are many definitions of a meteor ranging from any atmospheric phenomenon to a fast-moving streak of light in the night sky caused by the entry of extraterrestrial matter into the earth's atmosphere: A shooting star or falling star, for this resource, an alternative definition is proposed. '''Def.''' any natural object radiating through a portion or all of the [[Earth]]'s or another natural object's atmosphere is called a '''meteor'''. Such an object may be as small as an electron or much larger. "The distribution of photographic meteors in iron, stony, and porous meteors is given in this paper".<ref name=Ceplecha/> "[A]mong all the 217 meteors for which we know the beginning there are 70 iron meteors, i. e. about 32 p. c., and 147 stony meteors, i. e. 68 p. c."<ref name=Ceplecha/> The meteor streams: Perseids, Geminids, Taurids, Lyrids, κ Cygnids and Virginids, are quite stony.<ref name=Ceplecha/> "The dominant group in all cases are stony meteors."<ref name=Ceplecha>{{ cite journal |author=Zd. Ceplecha |title=On the composition of meteors |journal=Bulletin of the Astronomical Institutes of Czechoslovakia |month= |year=1958 |volume=9 |issue= |pages=154-9 |url= |bibcode=1958BAICz...9..154C |doi= |pmid= |accessdate=2011-08-10 }}</ref> Meteors become visible between about 75 to 120 kilometers (34 - 70 miles) above the Earth. They disintegrate at altitudes of 50 to 95 kilometers (31-51 miles). Meteors have roughly a fifty percent chance of a daylight (or&nbsp;near daylight) collision with the Earth. Most meteors are, however, observed at night, when darkness allows fainter objects to be recognized. Most meteors glow for about a second. A relatively small percentage of meteoroids hit the Earth's atmosphere and then pass out again: these are termed Earth-grazing fireballs (for example The Great Daylight 1972 Fireball). Meteors may occur in showers, which arise when the Earth passes through a trail of debris left by a comet, or as "random" or "sporadic" meteors, not associated with a specific single cause. A number of specific meteors have been observed, largely by members of the public and largely by accident, but with enough detail that orbits of the meteoroids producing the meteors have been calculated. All of the orbits passed through the asteroid belt.<ref name=Uregina>{{ cite book |url=http://uregina.ca/~astro/mb_5.html |title=Diagram 2: the orbit of the Peekskill meteorite along with the orbits derived for several other meteorite falls |publisher=Uregina.ca |date= |accessdate=2011-09-16 }}</ref> {{clear}} ==Fireballs== [[Image:Paola-Castillo.jpg|thumb|right|250px|This image taken October 17, 2012, is prior to the meteorite fall on the same day. Credit: Paola-Castillo; and Petrus M. Jenniskens, SETI Institute/NASA ARC.]] A '''fireball''' is a brighter-than-usual meteor. The International Astronomical Union defines a fireball as "a meteor brighter than any of the planets" (magnitude &minus;4 or greater).<ref name=MeteorObs>{{ cite book |url=http://www.meteorobs.org/maillist/msg13871.html |title=MeteorObs Explanations and Definitions (states IAU definition of a fireball) |publisher=Meteorobs.org |date=1999-07-09 |accessdate=2011-09-16 }}</ref> The International Meteor Organization (an amateur organization that studies meteors) has a more rigid definition. It defines a fireball as a meteor that would have a magnitude of &minus;3 or brighter if seen at zenith. This definition corrects for the greater distance between an observer and a meteor near the horizon. For example, a meteor of magnitude &minus;1 at 5 degrees above the horizon would be classified as a fireball because if the observer had been directly below the meteor it would have appeared as magnitude &minus;6.<ref name=IMO>{{ cite book |url=http://www.imo.net/fireball |title=International Meteor Organization - Fireball Observations |publisher=Imo.net |date=2004-10-12 |accessdate=2011-09-16 }}</ref> For 2011 there are 4589 fireballs records at the American Meteor Society.<ref name="Fireballs2011">{{ cite book |title=Fireball Report: 4589 records found between 2011-01-01 and 2011-12-31 |publisher=American Meteor Society |url=http://www.amsmeteors.org/fireball2/public.php?start_date=2011-01-01&end_date=2011-12-31&submit=Find+Reports |accessdate=2012-04-24 }}</ref> At right is cell phone camera image of the green fireball over San Mateo, California, that left meteorite fragments. "The asteroid entered at a speed of 14 km/s, typical but on the slow side of other meteorite falls for which orbits were determined. ... The orbit in space is also rather typical: perihelion distance close to Earth's orbit (q = 0.987 AU) and a low-inclination orbit (about 5 degrees). ... 2012, October 17 - At 7:44:29 pm PDT this evening, a bright fireball was seen in the San Francisco Bay Area."<ref name=Jenniskens/> ==Bolides== '''Def.''' a fireball reaching magnitude &minus;14 or brighter.<ref name=Belton>{{ cite book | author = MJS Belton | title = Mitigation of hazardous comets and asteroids | publisher = Cambridge University Press | date = 2004 | location = | pages = | url = http://books.google.com/?id=Dw0A7T0fy6AC | doi = | isbn = 0-521-82764-7 }}:156</ref> is called a '''bolide'''. '''Def.''' a fireball reaching an magnitude &minus;17 or brighter is called a '''superbolide'''. ==Meteor showers== {{main|Radiation astronomy/Showers}} [[Image:Novato-Webber-Jenniskens.jpg|thumb|right|250px|This image is a fragment of the October 17, 2012, fireball over San Mateo, California. Credit: Petrus M. Jenniskens, SETI Institute/NASA ARC.]] [[Image:Novato2a.jpg|thumb|right|250px|This is a second fragment from the fireball of October 17, 2012. Credit: Petrus M. Jenniskens, SETI Institute/NASA ARC.]] [[Image:Leonid meteor shower as seen from space (1997).jpg|thumb|right|250px|This photograph shows the Leonids as many begin contacting the Earth's atmosphere. Credit: NASA.]] The Perseid meteor shower, usually the richest meteor shower of the year, peaks in August. Over the course of an hour, a person watching a clear sky from a dark location might see as many as 50-100 meteors. Most meteors are actually pieces of rock that have broken off a comet and continue to orbit the Sun. The Earth travels through the comet debris in its orbit. As the small pieces enter the Earth's atmosphere, friction causes them to burn up. "The Orionid meteor shower [leftover bits of Halley's Comet] is scheduled to reach its maximum before sunrise on Sunday morning (Oct. 21 [2012]). This will be an excellent year to look for the Orionids, since the moon will set around 11 p.m. local time on Saturday night (Oct. 20) and will not be a hindrance at all ... The orbit of Halley's Comet closely approaches the Earth's orbit at two places. One point is in the early part of May producing a meteor display known as the Eta Aquarids. The other point comes in the middle to latter part of October, producing the Orionids."<ref name=Rao>{{ cite book |author=Joe Rao |title=Orionid Meteor Shower Spawned by Halley's Comet Peaks This Weekend |publisher=SPACE.com |location= |date=October 19, 2012 |url=http://news.yahoo.com/orionid-meteor-shower-spawned-halleys-comet-peaks-weekend-160214151.html |accessdate=2012-10-19 }}</ref> "At 66 kilometers (41 miles) per second, they appear as fast streaks, faster by a hair than their sisters, the Eta Aquarids of May. And like the Eta Aquarids, the brightest of family tend to leave long-lasting trains. Fireballs are possible three days after maximum."<ref name=Levy>{{ cite book |author=David Levy |author2=Stephen Edberg |title=Observe: Meteors |publisher=Astronomical League |location= |date= |editor= |pages= |url= |arxiv= |bibcode=1986obse.book.....L |doi= |pmid= |isbn= }}</ref> At right is a meteorite fragment from the October 17, 2012, green fireball over San Mateo, California, USA. "It is 63 grams, dense (feels heavy) and responds to a magnet (note: better to keep magnets away from meteorites to preserve the natural magnetic field)."<ref name=Jenniskens>{{ cite book |author=Petrus M. Jenniskens |title=2012, October 20 - FIRST METEORITE FOUND! |publisher=NASA Ames Research Center |location=San Francisco, California |date=October 20, 2012 |url=http://cams.seti.org/ |accessdate=2012-10-22 }}</ref> "The meteorite looks very unusual, because much of the fusion crust had come off. ... The meteorite appears to be a breccia, with light and dark parts."<ref name=Jenniskens/> The first meteorite from the San Mateo, California, fireball was looked at with a petrographic microscope and concluded it was not a meteorite. The crust appeared to be a product of weathering. The find of a second meteorite with the same crust confirms the first and second to be ordinary chondrites.<ref name=Jenniskens2>{{ cite book |author=Petrus M. Jenniskens |title=2012, October 24 - SECOND METEORITE CORROBORATES LISA'S FIND! |publisher=NASA Ames Research Center |location=San Francisco, California |date=October 24, 2012 |url=http://cams.seti.org/ |accessdate=2012-10-27 }}</ref> The photo on the right side shows the second meteorite cut in two.<ref name=Jenniskens2/> "The Leonid meteor shower peaked early Saturday (Nov. 17 [2012]), and some night sky watchers caught a great view. The Leonids are a yearly meteor display of shooting stars that appear to radiate out of the constellation Leo. They are created when Earth crosses the path of debris from the comet Tempel-Tuttle, which swings through the inner solar system every 33 years."<ref name=Moskowitz>{{ cite book |author=Clara Moskowitz |title=Amazing Leonid Meteor Shower Photos Captured By Stargazers |publisher=SPACE.com |location= |date=November 17, 2012 |url=http://news.yahoo.com/amazing-leonid-meteor-shower-photos-captured-stargazers-163450853.html |accessdate=2012-11-18 }}</ref> {{clear}} ==Cosmic rays== {{main|Radiation/Cosmic rays}} [[Image:Micrometeorite.jpg|thumb|right|250px|This is a micrometeorite collected from the antarctic snow. Credit: NASA.]] Micrometeorite is often abbreviated as MM. Most MMs are broadly chondritic in composition, meaning "that major elemental abundance ratios are within about 50% of those observed in carbonaceous chondrites."<ref name=Taylor/> Some MMs are chondrites, (basaltic) howardite, eucrite, and diogenite (HED) meteorites or Martian basalts, but not lunar samples.<ref name=Taylor/> "[T]he comparative mechanical weakness of carbonaceous precursor materials tends to encourage spherule formation."<ref name=Taylor/> From the number of different asteroidal precursors, the approximate fraction in MMs is 70 % carbonaceous.<ref name=Taylor/> "[T]he carbonaceous material [is] known from observation to dominate the terrestrial MM flux."<ref name=Taylor/> The "H, L, and E chondritic compositions" are "dominant among meteorites but rare among micrometeorites."<ref name=Taylor/> "Ureilites occur about half as often as eucrites (Krot et al. 2003), are relatively friable, have less a wide range of cosmic-ray exposure ages including two less than 1 Myr, and, like the dominant group of MM precursors, contain carbon."<ref name=Taylor>{{ cite journal |author=Susan Taylor |author2=Gregory F. Herzog |author3=Jeremy S. Delaney |title=Crumbs from the crust of Vesta: Achondritic cosmic spherules from the South Pole water well |journal=Meteoritics & Planetary Science |month= |year=2007 |volume=42 |issue=2 |pages=223-33 |url= |bibcode=2007M&PS...42..223T |doi=10.1111/j.1945-5100.2007.tb00229.x |pmid= |accessdate=2011-08-07 }}</ref> {{clear}} ==Liquid objects== {{main|Liquids/Liquid objects|Liquid objects}} '''Def.''' a flammable liquid ranging in color from clear to very dark brown and black, consisting mainly of hydrocarbons is called '''petroleum'''. ==Rocky objects== {{main|Rocks/Rocky objects}} [[Image:Olivina.png|thumb|right|250px|This is an image of an olivine rock. Credit: [[commons:User:Canica|Canica]].]] [[Image:Cristobalite-Fayalite-40048.jpg|thumb|right|250px|Cristobalite spheres appear within obsidian. Credit: Rob Lavinsky.]] [[Image:Mullite, Cordierite, Tridymite - Bohemia, Czech Republic.jpg|thumb|left|250px|Specimen consists of "porcelainite" - a semivitrified chert- or jasper-like rock composed of cordierite, mullite and tridymite, admixture of corundum, and subordinate K-feldspar. Credit: [http://www.mindat.org/user-13767.html#0 John Krygier].]] A '''rock''' is a naturally occurring solid aggregate of one or more minerals or mineraloids. '''Def.''' a solid, homogeneous, crystalline chemical element or compound that results from natural inorganic processes is called a '''mineral'''. Alpha-quartz (space group ''P''3₁21, no. 152, or ''P''3₂21, no. 154) under a high pressure of 2-3 gigapascals and a moderately high temperature of 700°C changes space group to monoclinic ''C''2/c, no. 15, and becomes the mineral coesite. It is found in extreme conditions such as the impact craters of meteorites. '''Def.''' a high-temperature (above 1470°C) polymorph of α-quartz with cubic, Fd{{overline|3}}m, space group no. 227, and a tetragonal form (P4₁2₁2, space group no. 92) is called '''cristobalite'''. '''Def.''' a polymorph of α-quartz formed by pressures > 100 kbar or 10 GPa and temperatures > 1200 °C is called '''stishovite'''. '''Def.''' a polymorph of α-quartz formed at an estimated minimum pressure of 35 GPa up to pressures above 40 GPa with a orthorhombic space group ''P''mmm no. 47 is called '''seifertite'''. '''Def.''' a polymorph of α-quartz formed at temperatures from 22-460°C with at least seven space groups for its forms with tabular crystals is called '''tridymite'''. '''Def.''' a substance that resembles a mineral but does not exhibit crystallinity is called a '''mineraloid'''. '''Def.''' a small, round, dark glassy object, composed of silicates is called a '''tektite'''. '''Def.''' any natural material with a distinctive composition of minerals is called a '''rock'''. Shocked quartz is associated with two high pressure polymorphs of silicon dioxide: coesite and stishovite. These polymorphs have a crystal structure different from standard quartz. Again, this structure can only be formed by intense pressure, but moderate temperatures. High temperatures would anneal the quartz back to its standard form. Stishovite may be formed by an instantaneous over pressure such as by an impact or nuclear explosion type event. So far several of the polymorphs of α-quartz formed at high temperature and pressure occur with rock types away from meteorite impact craters. '''Def.''' full of, or abounding in, rocks; consisting of rocks is called '''rocky'''. A division of astronomical objects between rocky objects and gaseous objects (including gas giants and stars) may be natural and informative. This division allows moons like Io to be viewed as rocky objects like [[Earth]] as part of rocky object science rather than as a natural satellite around a gaseous object like [[Jupiter]], or a plasma object like a [[coronal cloud]]. Astronomical objects that radiate, reflect, or fluoresce may range in size from individual atoms or subatomic particles to rocky objects. A rocky object may be radiation, radiated, or irradiated. {{clear}} ==Bombs== [[Image:Crmo volcanic bomb 20070516123632.jpg|thumb|left|250px|This image is of a bomb in the Craters of the Moon National Monument and Preserve, Idaho, USA. Credit: National Park Service.]] [[Image:Vulcanic bomb 008.jpg|thumb|right|250px|This is a volcanic bomb found in a shield volcano near Kladno. The rock hammer is a size gauge. Credit: [[commons:User:Chmee2|Chmee2]].]] [[Image:Puu Oo - boulder Royal Gardens 1983.jpg|thumb|right|250px|This is an accretionary lava ball. Credit: J. D. Griggs, USGS HVO.]] [[Image:Lava-bomb-01.jpg|thumb|right|200px|This is a fusiform lava bomb from Capelinhos Vulcano, Faial Island, Azores. Credit: M. Hollunder ([[commons:User:Apollo 8|Apollo 8]]).]] [[Image:VolcanicBombMojaveDesert.JPG|thumb|right|250px|This is a volcanic bomb found in the Mojave Desert National Preserve by Rob McConnell. Credit: [[commons:User:Wilson44691|Wilson44691]].]] [[Image:VulcaniaBombeVolcanique.JPG|thumb|right|250px|This is a volcanic bomb at Vulcania (Puy-de-Dôme). Credit: [[commons:User:Ji-Elle|Ji-Elle]].]] [[Image:Vulkanbombeneinschlag.png|thumb|right|250px|A volcanic bomb has deformed the rock strata. Credit: [[:de:User:Drucker03|Drucker03]].]] [[Image:Volcanic bomb from Hekla.jpg|thumb|right|250px|Volcanic bomb was found in the lava-fields of Mt. Hekla (Iceland). Credit: [[commons:User:Roger McLassus|Roger McLassus]].]] [[Image:Vulkanbombe strohn 20080722.jpg|thumb|right|250px|This is a picture of a lavabomb at Strohn Germany. Credit: [[commons:User:Jhintzbe|Jhintzbe]].]] [[Image:2011-07-09 gasometer 02.JPG|thumb|right|250px|A volcanic bomb from 350,000 BC. Credit: [[commons:User:Ziko van Dijk|Ziko van Dijk]].]] [[Image:Vulkaaniline pomm.jpg|thumb|right|250px|This volcanic bomb has a diameter of about 10 cm. Credit: [[:et:User:Siim|Siim Sepp]].]] '''Def.''' "distinctively shaped [natural] projectiles ... which acquired their shape essentially before landing"<ref name=Walker>{{ cite journal |author=G. P. L. Walker |title=The breaking of magma |journal=Geological Magazine |month=April |year=1969 |volume=106 |issue=02 |pages=166-73 |url=http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=4626560 |arxiv= |bibcode= |doi=10.1017/S0016756800051979 |pmid= |accessdate=2012-10-13 }}</ref> are called '''bombs'''. Volcanic bombs are thrown into the sky and travel some distance before returning to the ground. '''Def.''' a bomb "ejected from a volcanic vent"<ref name=Walker/> is called a '''volcanic bomb'''. "Confirmation of the source of excess argon comes from step-heating experiments on multiple anorthoclase aliquots separated from two phenocrysts and one glass aliquot prepared from the matrix of a volcanic bomb."<ref name=Esser>{{ cite journal |author=RP Esser |author2=WC McIntosh |author3=MT Heizler |author4=PR Kyle |title=Excess argon in melt inclusions in zero-age anorthoclase feldspar from Mt. Erebus, Antarctica, as revealed by the ⁴⁰Ar/³⁹Ar method |journal=Geochimica et Cosmochimica Acta |month=September |year=1997 |volume=61 |issue=18 |pages=3789-3801 |url=http://www.sciencedirect.com/science/article/pii/S0016703797002871 |arxiv= |bibcode= |doi=10.1016/S0016-7037(97)00287-1 |pmid= |accessdate=2012-10-13 }}</ref> {{lang|cs|Sopečná puma, též sopečná bomba, je těleso, které vzniká při explozi vulkánu. Jedná se o pyroklastický materiál, který je během exploze vymrštěn do okolí vulkánu}}. The second image at right is a volcanic bomb found in a shield volcano near Kladno. The rock hammer is a size gauge. Volcanic bombs can be thrown many kilometres from an erupting vent, and often acquire aerodynamic shapes during their flight. The third image at right is an "[a]ccretionary lava ball comes to rest on the grass after rolling off the top of an ‘a‘a flow in Royal Gardens subdivision. Accretionary lava balls form as viscous lava is molded around a core of already solidified lava."<ref name=Griggs>{{ cite book |author=J. D. Griggs |title=File:Puu Oo - boulder Royal Gardens 1983.jpg |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=April 27, 2012 |url=http://commons.wikimedia.org/wiki/File:Puu_Oo_-_boulder_Royal_Gardens_1983.jpg |accessdate=2012-10-13 }}</ref> Volcanic bombs cool into solid fragments before they reach the ground. Because volcanic bombs cool after they leave the volcano, they do not have grains making them extrusive igneous rocks. Volcanic bombs can be thrown many kilometres from an erupting vent, and often acquire aerodynamic shapes during their flight. Volcanic bombs can be extremely large; the 1935 eruption of Mount Asama in Japan expelled bombs measuring 5–6 m in diameter up to 600 m from the vent. Volcanic bombs are known to occasionally explode from internal gas pressure as they cool, but ... explosions are rare ... Bomb explosions are most often observed in 'bread-crust' type bombs. ''Ribbon or cylindrical bombs'' form from highly to moderately fluid magma, ejected as irregular strings and blobs. The strings break up into small segments which fall to the ground intact and look like ribbons. Hence, the name "ribbon bombs". These bombs are circular or flattened in cross section, are fluted along their length, and have tabular vesicles. ''Spherical bombs'' also form from high to moderately fluid magma. In the case of spherical bombs, surface tension plays a major role in pulling the ejecta into spheres. ''Spindle, fusiform, or almond/rotational bombs'' are formed by the same processes as spherical bombs, though the major difference being the partial nature of the spherical shape. Spinning during flight leaves these bombs looking elongated or almond shaped; the spinning theory behind these bombs' development has also given them the name 'fusiform bombs'. Spindle bombs are characterised by longitudinal fluting, one side slightly smoother and broader than the other. This smooth side represents the underside of the bomb as it fell through the air. ''Cow pie bombs'' are formed when highly fluid magma falls from moderate height; so the bombs do not solidify before impact (they are still liquid when they strike the ground). They consequently flatten or splash and form irregular roundish disks, which resemble cow-dung. ''Bread-crust bombs'' are formed if the outside of the lava bombs solidifies during their flights. They may develop cracked outer surfaces as the interiors continue to expand. ''Cored bombs'' are bombs that have rinds of lava enclosing a core of previously consolidated lava. The core consists of accessory fragments of an earlier eruption, accidental fragments of country rock or, in rare cases, bits of lava formed earlier during the same eruption. Natural bombs may produce impact craters and deform rock strata. "[I]ron oxide melts can exist in nature [as indicated] by describing a volcanic bomb composed of magnetite from El Laco."<ref name=Henriquez>{{ cite journal |author=Fernando Henríquez |author2=Jan Olov Nyström |title=Magnetite bombs at El Laco volcano, Chile |journal=GFF |month= |year=1998 |volume=120 |issue=3 |pages=269-71 |url=http://www.tandfonline.com/doi/abs/10.1080/11035899809453216 |arxiv= |bibcode= |doi=10.1080/11035899809453216 |pmid= |accessdate=2012-10-13 }}</ref> "[E]xistence of ballistic volcanic bombs composed of radiating porous aggregates of magnetite crystals in some of the orebodies, demonstrates that apatite iron ores can form directly from a melt."<ref name=Henriquez/> "The sulfur isotope ratio as well as the sulfur content were found to be uniform within a single unit of lava flow and a volcanic bomb."<ref name=Ueda>{{ cite journal |author=Akira Ueda, Hitoshi Sakai |title=Sulfur isotope study of Quaternary volcanic rocks from the Japanese Islands Arc |journal=Geochimica et Cosmochimica Acta |month=September |year=1984 |volume=48 |issue=9 |pages=1837-48 |url=http://www.sciencedirect.com/science/article/pii/0016703784900371 |arxiv= |bibcode= |doi=10.1016/0016-7037(84)90037-1 |pmid= |accessdate=2012-10-13 }}</ref> "Volcanic bombs with a distinctive shape are produced by post-impact mechanical rounding processes while traveling at high speed down the slopes of the scoria cone of the Pacaya Volcano in Guatemala. The name “cannonball bombs” is proposed for bombs formed by this mechanism."<ref name=Francis>{{ cite journal |author=P. W. Francis |title=Cannonball bombs, a new kind of volcanic bomb from the Pacaya volcano, Guatemala |journal=Geological Society of America Bulletin |month=August |year=1973 |volume=84 |issue=8 |pages=2791-4 |url=http://gsabulletin.gsapubs.org/content/84/8/2791.full.pdf |arxiv= |bibcode= |doi=10.1130/​0016-7606(1973)​84<2791:CBANKO>​2.0.CO;2 |pmid= |accessdate=2012-10-13 }}</ref> "There have been recorded in all periods of historic time ... showers of one kind or another of animals and plants or their products -- showers of hay, of grain, of manna, of blood, of fishes, of frogs, and even of rats. ... so many wonderful things occur in nature that negation of any observation is dangerous; it is better to preserve a judicial attitude and regard all (authentic) information that comes to hand as so much evidence, some of it supporting one side, some the other, of a given problem."<ref name=McAtre>{{ cite journal |author=Waldo L. McAtre |title=Showers of Organic Matter |journal=Monthly Weather Review |month=May |year=1917 |volume=45 |issue=5 |pages=217-24 |url=http://docs.lib.noaa.gov/rescue/mwr/045/mwr-045-05-0217.pdf |arxiv= |bibcode= |doi= |pmid= |accessdate=2013-02-18 }}</ref> "Mr. A. N. Caudell, of the United States Bureau of Entomology ... relates that at his former home in Oklahoma, on one occasion after a brief shower during an otherwise dry and hot period, numerous earthworms were found on the seat of an open buggy standing in the yard."<ref name=McAtre/> "[T]he statement by the famous French scientist, Francis Castlenau, ... he had seen fishes rain down in Singapore in such numbers that the natives went about picking them up by the basketful".<ref name=McAtre/> "By the tornado at Beauregard, Miss., April 22, 1883, the solid iron screw of a cotton press, weighing 675 pounds, was carried 900 feet."<ref name=McAtre/> "In the tornado at Mount Carmel, Ill., June 4, 1877, a piece of tin roof was carried 15 miles and a church spire 17 miles."<ref name=McAtre/> {{clear}} ==Iron meteorites== {{main|Radiation astronomy/Alloys#Iron_meteorites}} ==Petrologic types== The degree to which a meteorite has been affected by the secondary processes of thermal metamorphism and aqueous alteration on the parent asteroid is indicated by its petrologic type, which appears as a number following the group name (e.g., an LL5 chondrite belongs to the LL group and has a petrologic type of 5).<ref name="Van Schmus">{{cite journal |last1=Van Schmus |first1=W. R. |last2=Wood |first2=J. A. |year=1967 |title=A chemical-petrologic classification for the chondritic meteorites |journal=Geochimica et Cosmochimica Acta |volume=31 |pages=747–765 |doi=10.1016/S0016-7037(67)80030-9 |bibcode=1967GeCoA..31..747V |issue=5}}</ref> ==Type 1== Current usage of type 1 is simply to indicate meteorites that have experienced extensive aqueous alteration, to the point that most of their olivine and pyroxene have been altered to hydrous phases. This alteration took place at temperatures of 50 to 150 °C, so type 1 chondrites were warm, but not hot enough to experience thermal metamorphism. ==Type 2== Chondrites have experienced extensive aqueous alteration, but still contain recognizable chondrules as well as primary, unaltered olivine and/or pyroxene. The fine-grained matrix is generally fully hydrated and minerals inside chondrules may show variable degrees of hydration. This alteration probably occurred at temperatures below 20 °C, and again, these meteorites are not thermally metamorphosed. Almost all CM and CR chondrites are petrologic type 2; with the exception of some ungrouped carbonaceous chondrites, no other chondrites are type 2. ==Type 3== Low degrees of metamorphism, often referred to as unequilibrated chondrites because minerals such as olivine and pyroxene show a wide range of compositions, reflecting formation under a wide variety of conditions in the solar nebula. Chondrites that remain in nearly pristine condition, with all components (chondrules, matrix, etc.) having nearly the same composition and mineralogy as when they accreted to the parent asteroid, are designated type 3.0. As petrologic type increases from type 3.1 through 3.9, profound mineralogical changes occur, starting in the dusty matrix, and then increasingly affecting the coarser-grained components like chondrules. Type 3.9 chondrites still look superficially unchanged because chondrules retain their original appearances, but all of the minerals have been affected, mostly due to diffusion of elements between grains of different composition. ==Type 4== Chondrites have been increasingly altered by thermal metamorphism. These are equilibrated chondrites, in which the compositions of most minerals have become quite homogeneous due to high temperatures. By type 4, the matrix has thoroughly recrystallized and coarsened in grain size. ==Type 5== By type 5, chondrules begin to become indistinct and matrix cannot be discerned. ==Type 6== In type 6 chondrites, chondrules begin to integrate with what was once matrix, and small chondrules may no longer be recognizable. As metamorphism proceeds, many minerals coarsen and new, metamorphic minerals such as feldspar form. ==Type 7== Type 7 chondrites have experienced the highest temperatures possible, short of that required to produce melting. Should the onset of melting occur the meteorite would probably be classified as a primitive achondrite instead of a chondrite. ==Stony-iron meteorites== [[Image:Esquel.jpg|thumb|right|250px|A slice of the Esquel meteorite shows the mixture of meteoric iron and silicates that is typical of this division. Credit: [https://www.flickr.com/photos/16533652@N00 Doug Bowman].{{tlx|free media}}]] Stony-iron meteorites or siderolites are meteorites that consist of nearly equal parts of meteoric iron and silicates. This distinguishes them from the stony meteorites, that are mostly silicates, and the iron meteorites, that are mostly meteoric iron.<ref name="McSween1999">{{cite book|last=McSween|first=Harry Y.|title=Meteorites and their parent planets|date=1999|publisher=Cambridge Univ. Press|location=Cambridge|{{isbn|978-0521587518}}|edition=Sec.}}</ref> Stony-irons or siderolites are all differentiated, meaning that they show signs of alteration and are achondrites. The stony-irons are divided into mesosiderites and pallasites. Pallasites have a matrix of meteoric iron with embedded silicates (most of it olivine).<ref name=Buseck>{{cite journal |last=Buseck |first=P.R.|year=1977 |title=Pallasite meteorites: mineralogy, petrology, and geochemistry |journal=Geochimica et Cosmochimica Acta |volume=41 |pages=711–740|doi=10.1016/0016-7037(77)90044-8|bibcode = 1977GeCoA..41..711B|issue=6 }}</ref> Mesosiderites are breccias which show signs of metamorphism. The meteoric iron occurs in clasts instead of a matrix.<ref name=Heide>F. Heide, F. Wlotzka: Meteorites, Messengers from Space. Springer Verlag 1985.</ref><ref name=Turekian>Karl K. Turekian. ''Meteorites, comets, and planets'',[https://books.google.com/books?id=kYtksEUxw0oC&lpg=PA112&dq=mesosiderite&pg=PA112#v=onepage&q=mesosiderite&f=falsePage 112]</ref> The meteoric iron of stony-irons is similar to that of iron meteorites, consisting mostly of kamacite and taenite in different proportions. The silicates are dominated by olivine. Accessory minerals that also include non-silicates are: carlsbergite, chromite, cohenite, daubréelite, feldspar, graphite, ilmenite, merrillite, low-calcium pyroxene, schreibersite, tridymite and troilite. There are specific categories for mixed-composition meteorites, in which iron and 'stony' materials are combined. * II) '''Stony–iron meteorites''' ** Pallasites *** Main group pallasites *** Eagle station pallasite grouplet *** Pyroxene Pallasite grouplet ** Mesosiderite group {{clear}} ==Eagle Station group== [[Image:Eagle Station pallasite, Mineralogisches Museum, Bonn.jpg|thumb|right|250px|Eagle Station meteorite is the type specimen for the group. Credit: Elke Wetzig Elya.{{tlx|free media}}]] The Eagle Station group (abbreviated PES - Pallasite Eagle Station) is a set of pallasite meteorite specimen that don't fit into any of the other defined pallasite groups. In meteorite classification five meteorites have to be found, so they can be defined as their own group.<ref name=Weisberg2006>{{cite book|editor=D.S. Lauretta, H.Y. McSween, Jr.; foreword by Richard P. Binze|title=Meteorites and the early solar system II, In: ''Systematics and Evaluation of Meteorite Classification''|date=2006|publisher=University of Arizona Press|location=Tucson|{{isbn|978-0816525621}}|url=http://haroldconnolly.com/EES%20716%20Fall%2009%20Reading/Lecture%201/Background%20reading/Weisberg_etal_MESSII.pdf|author1=M. K. Weisberg|author2=T. J. McCoy|author3= A. N. Krot|accessdate=15 December 2012|pages=19–52 }}</ref> Currently only five Eagle Station type meteorites have been found, which is just enough for a separate group.<ref name="Meteoritical Bulletin Database">{{cite book|title=Meteoritical Bulletin Database|url=https://www.lpi.usra.edu/meteor/metbull.php?sea=Pallasite,%20PES&sfor=types&stype=exact|publisher=Meteoritical Society}}</ref> The Eagle Station group is named after the Eagle Station meteorite, the type specimen of the group. It is in turned named after Eagle Station, Carroll County Kentucky where it was found.<ref name="Met Eagle Station">{{cite book|title=Eagle Station|url=http://www.lpi.usra.edu/meteor/metbull.php?sea=pes&sfor=types&ants=&falls=&valids=&stype=contains&lrec=50&map=ge&browse=&country=All&srt=name&categ=All&mblist=All&rect=&phot=&snew=0&pnt=Normal%20table&code=7761|publisher=Meteoritical Society}}</ref> The Eagle Station group has a composition similar to Main group pallasites. Diagnostic differences are that the olivine is richer in iron and calcium. The group also has a distinct oxygen isotope signature.<ref name=Weisberg2006/> The meteoric iron is similar to the IIF iron meteorites. This might indicate that Eagle station group and IIF formed close to each other in the solar nebula.<ref name=Weisberg2006/> The trace elements in the phosphates of the Eagle Station group are distinct from other pallasites. Most pallasites are believed to be derived from the core-mantle boundary. Trace elements indicate that the Eagle Station group came from shallower depths of their parent body.<ref name=Davis1991>{{cite journal|last=Davis|first=Andrew M.|author2=Olsen, Edward J.|title=Phosphates in pallasite meteorites as probes of mantle processes in small planetary bodies |journal=Nature |date=17 October 1991|volume=353|issue=6345|pages=637–640|doi=10.1038/353637a0}}</ref> Only five specimen have been found so far:<ref name="Meteoritical Bulletin Database" /> * Cold Bay meteorite. * Eagle Station meteorite (type specimen). * Itzawisis meteorite. * Karavannoe meteorite. * Oued Bourdim 001. # Type: Stony-iron. # Class: Pallasite. # Composition: Meteoric iron, silicate minerals. {{clear}} ==Eagle Station meteorites== [[Image:Eagle Station pallasite, Mineralogisches Museum, Bonn.jpg|thumb|right|250px|Pallasite is from: Eagle Station, Kentucky/USA, Mineralogisches Museum Bonn. Credit: Elke Wetzig Elya.{{tlx|free media}}]] Eagle station pallasite grouplet (PES): 5 specimens known. They are related to IIF irons. {{clear}} ==Main group pallasites== ==Esquel meteorties== [[Image:Pallasite-Esquel-RoyalOntarioMuseum-Jan18-09.jpg|thumb|250px|right|An example of a Pallasite meteorite (from the Esquel fall) on display in the Vale Inco Limited Gallery of Minerals at the Royal Ontario Museum. Credit: [[c:user:Captmondo|Captmondo]].{{tlx|free media}}]] Esquel is regarded as one of the most beautiful meteorites ever found and is also one of the most desirable pallasites among meteorite collectors. It is a main group pallasite (MGP). {{clear}} ==Pallasite ungrouped== Pallasites ungrouped (P-ung): Specimens that don't fit into any groups or grouplets. ==Pyroxene Pallasite grouplet== Pyroxene Pallasite (PPX) counts only Vermillion and Yamato 8451. They take their name from the high orthopyroxene content (about 5%). Metal matrix shows a fine octahedrite Widmanstätten pattern. ==Vermillion meteorites== The Vermillion meteorite is a pallasite (stony-iron) meteorite and one of two members of the pyroxene pallasite grouplet.<ref name=Weisberg2006/> It was recognized as a meteorite and first described in 1995.<ref name=MetDBVermillion>{{cite book|title=Vermillion|url=http://www.lpi.usra.edu/meteor/metbull.php?sea=Pallasite&sfor=types&ants=&falls=&valids=&stype=contains&lrec=50&map=ge&browse=&country=All&srt=name&categ=All&mblist=All&rect=&phot=&snew=0&pnt=Normal%20table&code=24167|publisher=Meteoritical Society }}</ref> The silicates include olivine (93% of silicates), orthopyroxene (5%), chromite (1.5%) and merrillite (0.5%).<ref name=Boesenberg1995>{{cite journal|last=Boesenberg|first=J. S.|author2=M. Prinz |author3=M. K. Weisberg |author4=A. M. Davis |author5=R. N. Clyton |author6=T. K. Mayeda |title=Pyroxene Pallasites: A New Pallasite Grouplet|journal=Meteoritics|date=1995|volume=30|pages=488–489|url=http://adsabs.harvard.edu/full/1995Metic..30R.488B|bibcode=1995Metic..30R.488B|accessdate=29 December 2012}}</ref> Other accessory minerals include troilite, whitlockite,<ref name=Weisberg2006/> and cohenite.<ref name=Boesenberg2000>{{cite journal|last=Boesenberg|first=Joseph S.|author2=Davis, Andrew M. |author3=Prinz, Martin |author4=Weisberg, Michael K. |author5=Clayton, Robert N. |author6= Mayeda, Toshiko K.|title=The pyroxene pallasites, Vermillion and Yamato 8451: Not quite a couple|journal=Meteoritics & Planetary Science|date=1 July 2000|volume=35|issue=4|pages=757–769|doi=10.1111/j.1945-5100.2000.tb01460.x|bibcode=2000M&PS...35..757B }}</ref> The Vermillion meteorite is classified as a pyroxene pallasite because it contains pyroxene as an accessory mineral and shares a distinct oxygen isotope signature with Yamato 8451 meteorite.<ref name=Weisberg2006/> Some studies also object to this grouping, referring to the differences in siderophile trace elements and the occurrence of cohenite in the Vermillion meteorite.<ref name=Boesenberg2000/> # Type: Achondrite, pallasite. # Grouplet: Pyroxene pallasite grouplet. # Composition: Meteoric iron (~86%) silicates (~14%). # Country: United States. # Region: Kansas. # Coordinates: {{coord|39|44|11|N|96|21|41|W}}. # Observed fall: No. # Found date: 1991. # TKW: {{convert|34.36|kg}}. ==Achondrites== [[Image:NWA 6926 - 1.57g.jpg|thumb|right|250px|This image shows a unique and beautiful achondrite meteorite. Credit: [http://www.flickr.com/people/48082563@N08 Jon Taylor].{{tlx|free media}}]] [[Image:MillbillillieMeteorite.jpg|thumb|right|250px|This image shows a meteorite from the Millbillillie meteorite shower. Credit: H. Raab ([[c:User:Vesta|Vest]]).{{tlx|free media}}]] An '''achondrite''' is a stony meteorite that does not contain chondrules. It consists of material similar to terrestrial basalts or plutonic rocks and has been differentiated and reprocessed to a lesser or greater degree due to melting and recrystallization on or within meteorite parent bodies.<ref name="Sahijpal2007">{{ cite journal |author=Sahijpal, S. |coauthors=Soni, P.;Gagan, G. |title=Numerical simulations of the differentiation of accreting planetesimals with ²⁶Al and ⁶⁰Fe as the heat sources |journal=Meteoritics & Planetary Science |volume=42 |pages=1529–1548 |year=2007 |doi=10.1111/j.1945-5100.2007.tb00589.x |issue=9 |bibcode=2007M&PS...42.1529S}}</ref><ref name="Gupta2010">{{ cite journal |author=Gupta, G. |coauthors=Sahijpal, S. |title=Differentiation of Vesta and the parent bodies of other achondrites |journal=J. Geophys. Res. (Planets) |year=2010 |doi=10.1029/2009JE003525 |volume=115 |bibcode=2010JGRE..11508001G}}</ref> As a result, achondrites have distinct textures and mineralogies indicative of igneous processes.<ref name=Mason>{{ cite book |author=Brian Mason |title=Meteorites |publisher=John Wiley |location=New York |date=1962 |bibcode=1962Sci...138..887M }}</ref> Achondrites account for about 8% of meteorites overall, and the majority (about two thirds) of them are HED meteorites, originating from the crust of asteroid 4 Vesta. Other types include Martian, Lunar, and several types thought to originate from as-yet unidentified asteroids other than Vesta. These groups have been determined on the basis of e.g. the Fe/Mn chemical ratio and the {{chem|17|O}}/{{chem|18|O}} oxygen isotope ratios, thought to be characteristic "fingerprints" for each parent body.<ref name=Mittlefehldt>{{ cite journal |last1=Mittlefehldt |first1=David W. |last2=McCoy |first2=Timothy J. |last3=Goodrich |first3=Cyrena Anne |last4=Kracher |first4=Alfred |title=Non-chondritic Meteorites from Asteroidal Bodies |journal=Reviews in Mineralogy and Geochemistry |volume=36 |issue=1 |year=1998 |pages=4.1–4.195 |doi= |url=http://rimg.geoscienceworld.org/cgi/content/abstract/36/1/4.1 }}</ref> There is a "precise mixing required to create oxygen-18" in "silica grains".<ref name=Moskowitz2013>{{ cite book |author=Clara Moskowitz |title=Rare Meteorite Grains May be from Supernova That Sparked Solar System |publisher=Space.com |location= |date=April 24, 2013 |url=http://www.space.com/20797-meteorite-supernova-solar-system.html |accessdate=2013-04-25 }}</ref> "[S]ilica (SiO₂) grains in the primitive carbonaceous chondrites LaPaZ 031117 and Grove Mountains 021710 ... are characterized by moderate enrichments in ¹⁸O relative to solar, indicating that they originated in Type II supernova ejecta."<ref name=Haenecour>{{ cite journal |author=Pierre Haenecour |author2=Xuchao Zhao |author3=Christine Floss |author4=Yangting Lin |author5=Ernst Zinner |title=First Laboratory Observation of Silica Grains from Core Collapse Supernovae |journal=The Astrophysical Journal Letters |month=May 1, |year=2013 |volume=768 |issue=1 |pages=L17 |url=http://iopscience.iop.org/2041-8205/768/1/L17 |arxiv= |bibcode= |doi=10.1088/2041-8205/768/1/L17 |pmid= |accessdate=2013-04-25 }}</ref> The second image at right is a 175g individual of the Millbillillie meteorite shower, a eucrite achondrite that fell in Australia in 1960. This specimen is approx. 6 centimeters wide. Note the shiny black fusion crust with flow lines. The chip at lower right allows one to see the light-gray interior. The orange staining at top is a result of weathering, as these stones were not recovered until many years after they fell. Achondrites are classified into the following groups:<ref name=Norton>O. Richard Norton. The Cambridge encyclopedia of meteorites. UK, Cambridge University Press, 2002. {{ISBN|0-521-62143-7}}.</ref> * Primitive achondrites. * Asteroidal achondrites. * Lunar meteorites. * Martian meteorites. Primitive achondrites, also called PAC group, are so-called because their chemical composition is ''primitive'' in the sense that it is similar to the composition of chondrites, but their texture is igneous, indicative of melting processes. To this group belong:<ref name=Norton2002/> * Acapulcoites (after the meteorite Acapulco, Mexico) * Brachinites (after the meteorite Brachina) * Lodranites (after the meteorite Lodran) * Winonaites (after the meteorite Winona) * Ureilites (after the meteorite Novy Ureii, Russia) {{clear}} ==Asteroidal achondrites== {{main|Astrominerals}} [[Image:NWA 2999 meteorite, angrite.jpg|thumb|right|250px|A slice of "NWA 2999", an angrite, is similar to a terrestrial basalt. Credit: [https://www.flickr.com/people/48082563@N08 Jon Taylor].{{tlx|free media}}]] [[Image:Cumberland Falls meteorite.jpg|thumb|right|250px|Cumberland Falls meteorite is an aubrite. Credit: [https://flickr.com/people/8435962@N06 Claire H.].{{tlx|free media}}]] [[Image:Shallowater meteorite.jpg|thumb|right|250px|Shallowater meteorite is an aubrite. Credit: [https://flickr.com/people/8435962@N06 Claire H.].{{tlx|free media}}]] Asteroidal achondrites, also called evolved achondrites, are so-called because they have been ''differentiated'' on a parent body. This means that their mineralogical and chemical composition was changed by melting and crystallization processes. They are divided several groups:<ref name=Norton/> * HED meteorites (Vesta). They may have originated on the asteroid 4 Vesta, because their reflection spectra are very similar.<ref name=Drake>{{cite journal |first=M. J. |last=Drake |title=The eucrite/Vesta story |journal=Meteoritics and Planetary Science |volume=36 |issue=4 |date=2001 |pages=501–513 |doi=10.1111/j.1945-5100.2001.tb01892.x |bibcode=2001M&PS...36..501D }}</ref> They are named after the initial letters of the three subgroups: ** Howardites ** Eucrites ** Diogenites * Angrites: a rare group of achondrites consisting mostly of the mineral augite with some olivine, anorthite and troilite. The group is named for the Angra dos Reis meteorite. Angrites are basaltic rocks, often having porosity, with vesicle diameters of up to 2.5 centimetres (0.98 in). They are the oldest igneous rocks, with crystallization ages of around 4.55 billion years. By comparing the reflectance spectra of the angrites to that of several main belt asteroids, two potential parent bodies have been identified: 289 Nenetta and 3819 Robinson. Angrites could represent ejecta from Mercury, however later work has cast significant doubt upon these claims.<ref>{{Cite journal|last1=Irving|first1= A. J. |last2=Kuehner|first2= S. M. |last3=Rumble|first3= D. |last4=Bunch|first4= T. E. |last5=Wittke|first5= J. H. |title = Unique Angrite NWA 2999: The Case For Samples From Mercury| journal = American Geophysical Union, Fall Meeting 2005, abstract (2005)|date = December 2005|pages = P51A-0898|url = https://ui.adsabs.harvard.edu/abs/2005AGUFM.P51A0898I/abstract |bibcode=2005AGUFM.P51A0898I }}</ref> * Aubrites: a group of meteorites named for Aubres, a small achondrite meteorite that fell near Nyons, France, in 1836, primarily composed of the orthopyroxene enstatite, often called enstatite achondrites, with igneous origin separating them from primitive enstatite achondrites and means they originated in an asteroid. Aubrites are typically light-colored with a brownish fusion crust. Most aubrites are heavily brecciated; they are often said to look "lunar" in origin. Aubrites are primarily composed of large white crystals of the Fe-poor, Mg-rich orthopyroxene, or enstatite, with minor phases of olivine, nickel-iron metal, and troilite, indicating a magmatic formation under extremely reducing conditions. The severe brecciation of most aubrites attests to a violent history for their parent body. Since some aubrites contain chondritic xenoliths, it is likely that the aubrite parent body collided with an asteroid of “F-chondritic” composition. Comparisons of aubrite spectra to the spectra of asteroids have revealed striking similarities between the aubrite group and the E-type asteroids of the Nysa family. A small near-Earth object, 3103 Eger, is also often suggested as the parent body of the aubrites.<ref name=Gaffey1992>{{cite journal |last1=Gaffey, Michael J.; Reed, Kevin L.; Kelley, Michael S. |title=Relationship of E-type Apollo asteroid 3103 (1982 BB) to the enstatite achondrite meteorites and the Hungaria asteroids |journal=Icarus |date=November 1992 |volume=100 |issue=1 |pages=95–109 |doi=10.1016/0019-1035(92)90021-X |url=https://ui.adsabs.harvard.edu/abs/1992Icar..100...95G/abstract |accessdate=14 May 2021}}</ref> {{clear}} ==Martian meteorites== {{main|Areiominerals}} Martian meteorites<ref name=Treiman/> are divided into three main groups, with two exceptions (see last two entries): * Shergottites. * Nakhlites. * Chassignites. * OPX martian meteorites (Allan Hills 84001). * Regolith/Soil samples (Northwest Africa 7034). ==Enstatite chondrites== The E stands for Enstatite, H indicates a high metallic iron content of approximately 30%, and L low. The number refers to alteration. Enstatite chondrites (also known as E-type chondrites) are a rare form of meteorite thought to comprise only about 2% of the chondrites that fall to Earth.<ref name=Norton2008>Norton, O.R. and Chitwood, L.A. Field Guide to Meteors and Meteorites, Springer-Verlag, London 2008</ref> Only about 200 E-Type chondrites are currently known.<ref name=Norton2008/> The majority of enstatite chondrites have either been recovered in Antarctica or have been collected by the American National Weather Association, and they tend to be high in the mineral enstatite (MgSiO₃), from which they derive their name.<ref name=Norton2008/> E-type chondrites are among the most chemically reduced rocks known, with most of their iron taking the form of metal or sulfide rather than as an oxide, suggesting that they were formed in an area that lacked oxygen, probably within the orbit of Mercury.<ref name="new">{{cite book |title = Meteorlab|accessdate = 22 April 2009|author = New England Meteoritical Services|archive-date = 21 February 2009 |url = https://web.archive.org/web/20090221114126/http://meteorlab.com/METEORLAB2001dev/Open1.htm }}</ref> Enstatite chondrites contain sufficient hydrogen to have delivered to Earth at least three times the mass of water in its oceans.<ref>[https://science-sciencemag-org.insu.bib.cnrs.fr/content/369/6507/1110 Earth’s water may have been inherited from material similar to enstatite chondrite meteorites]</ref> Metallic Fe-Ni (iron-nickel) and Fe-bearing sulfide minerals contain nearly all of the iron in this type of meteorite. Enstatite chondrites contain a variety of unusual minerals that can only form in extremely reducing conditions, including oldhamite (CaS), niningerite (MgS), perryite (Fe-Ni silicide), and alkali sulfides (e.g., djerfisherite and caswellsilverite). All enstatite chondrites are dominantly composed of enstatite-rich chondrules plus abundant grains of metal and sulfide minerals. Dusty matrix material is uncommon and refractory inclusions are very rare. Chemically, enstatite chondrites are very low in refractory lithophile elements. Their oxygen isotopic compositions are intermediate between ordinary and carbonaceous chondrites, and are similar to rocks found on the Earth and Moon. Their lack of oxygen content may mean that they were originally formed near the center of the solar nebula that created the [[Solar System]], possibly within the orbit of Mercury. Most enstatite chondrites have experienced thermal metamorphism on the parent asteroids and are divided into two groups:<ref>{{Cite book|url=https://www.lpi.usra.edu/meteor/metbullclass.php?sea=EH|title=Meteoritical Bulletin: Recommended classifications|website=Lunar and Planetary Institute|accessdate=2020-04-24}}</ref><ref>{{Cite book|url=https://www.lpi.usra.edu/meteor/metbullclass.php?sea=EL5|title=Meteoritical Bulletin: Recommended classifications|website=Lunar and Planetary Institute|accessdate=2020-04-24}}</ref> * EH (high-iron) chondrites contain small chondrules (~{{convert|0.2|mm}}) and high ratios of siderophile elements to silicon. Somewhat more than 10% of the rock is composed of metal grains. A diagnostic feature of EH chondrites is that the Fe-Ni metal contains ~3 wt% elemental silicon. * EL (low-iron) chondrites contain larger chondrules (>{{convert|0.5|mm}}), and low ratios of siderophile elements to silicon. Fe-Ni metal contains ~1 wt% silicon. ==EH chondrites== ==EL chondrites== ==E3 Abundant chondrites== ==E4 Distinct chondrites== ==Abee meteorites== [[Image:Abee meteorite, partial slice 5.45g.jpg|thumb|right|250px|The only mass recovered from the Abee meteorite is a brecciated enstatite chondrite now on display at the American Museum of Natural History. Credit: [https://www.flickr.com/people/48082563@N08 Jon Taylor].{{tlx|free media}}]] The Abee meteorite is the only example in the world of an EH4 impact-melt breccia meteorite.<ref>[http://www.meteorlab.com/METEORLAB2001dev/abee.htm Abee Enstatite Chondrite]</ref> "Constraints are reported on the thermal history of the constituents of the Abee enstatite chondrite. From thermal experiments on laboratory-prepared alloys, and on actual samples of the meteorite, it is concluded that the metal phase of Abee cooled from above 700°C to room temperature in less than ten hours."<ref name=Herndon>{{ cite journal |author=J.M. Herndon and M.L. Rudee |title=Thermal history of the Abee enstatite chondrite |journal=Earth and Planetary Science Letters |month=September |year=1978 |volume=41 |issue=1 |pages=101-6 |url=http://www.sciencedirect.com/science/article/pii/0012821X78900468 |arxiv= |bibcode= |doi=10.1016/0012-821X(78)90046-8 |pmid= |accessdate=2015-10-15 }}</ref> "The early thermal history of planets [is evidenced] from meteorites."<ref name=Herndon/> The "early thermal history of chondritic asteroids [can be] derived by ²⁴⁴Pu fission track thermometry."<ref name=Herndon/> {{clear}} ==E5 Less distinct chondrites== ==Saint-Sauveur meteorites== [[Image:Météorite de Saint Sauveur MHNT2.jpg|thumb|right|250px|Saint-Sauveur meteorite, Chondrite EH5, 14Kg, fell in 1914, two views of same specimen. Credit: [[c:User:Archaeodontosaurus|Didier Descouens]].{{tlx|free media}}]] {{clear}} ==E6 Indistinct chondrites== ==E7 Melted chondrites== ==Ordinary chondrites== [[Image:NWA869Meteorite.jpg|thumb|right|250px|This is an image of a 700 g piece of the NWA 869 meteorite. Credit: H. Raab ([[c:User:Vesta|Vesta]]).{{tlx|free media}}]] [[Image:Bruderheim meteorite small.jpg|thumb|right|250px|This is an image of the Bruderheim meteorite. Credit: [[c:File:A_Meteorite_collection.jpg|A_Meteorite_collection.jpg]], derivative work by [[c:User:Basilicofresco|Basilicofresco]].{{tlx|free media}}]] Chondrites are stony meteorites that have not been modified due to melting or differentiation of the parent body. The ordinary chondrites (sometimes called the O chondrites) are a class of stony chondritic meteorites that are by far the most numerous group and comprise about 87% of all finds.<ref name=NaturalHistoryMuseum>{{cite book|url=https://www.nhm.ac.uk/our-science/data/metcat/search/metsPerGroup.dsml|title=The Catalogue of Meteorites|publisher=Natural History Museum|accessdate=28 May 2020}}</ref> Prominent among the components present in chondrites are the enigmatic chondrules, millimeter-sized objects, most of which are rich in the silicate minerals olivine and pyroxene. Chondrites also contain refractory inclusions (including Ca-Al Inclusions), particles rich in metallic Fe-Ni and sulfides, and isolated grains of silicate minerals. The remainder of chondrites consists of fine-grained (micrometer-sized or smaller) dust, which may either be present as the matrix of the rock or may form rims or mantles around individual chondrules and refractory inclusions. At right is a piece of the NWA 869 meteorite. "Chondrules and metal flakes can be seen on the cut and polished face of this specimen. NWA 869 is a ordinary chondrite (L4-6). ... The cut surface is about 65mm at it's widest point."<ref name=Raab>{{ cite book |author=H. Raab |title=NWA869Meteorite.jpg |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=September 9, 2011 |url=http://commons.wikimedia.org/wiki/File:NWA869Meteorite.jpg |accessdate=2012-10-19 }}</ref> The second image is of the Bruderheim (or Bruederheim) meteorite. "The Bruederheim Meteorite fell in 1960 ... Chondrules, spherical to subspherical minerals or mineral aggregates, form the most conspicuous textural features. The chondrules are in a fine to medium crystalline groundmass. This groundnass is seriate textured with the smallest grains being about 0.01 mm. in diameter. The larger grains of the groundmass are as much as 0.3 mm. in diameter. They are generally anhedral and have forms that range fron angular to subrounded."<ref name=Brown>{{ cite book |author=Harrison Brown, Bruce C. Murray |title=First Annual Report |publisher=National Aeronautics and Space Administration |location=Pasadena, California |date=January 31, 1961 |editor= |pages=41 |url=http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19660081661_1966081661.pdf |arxiv= |bibcode= |doi= |pmid= |isbn= |accessdate=2012-10-19 }}</ref> The Bruederheim meteorite is approximately 39.55 % SiO₂, 13.89 % FeO+Fe₂O₃ as FeO, and 24.69 % MgO by weight.<ref name=Brown/> Most meteorites that are recovered on Earth are chondrites: 86.2% of witnessed falls are chondrites,<ref name=Bischoff1995>{{ cite journal |title=Meteorites for the Sahara: Find locations, shock classification, degree of weathering and pairing |journal=Meteoritics |issn=0026-1114 |volume=30 |issue=1 |pages=113–122 |bibcode=1995Metic..30..113B |author=A. Bischoff, T. Geiger |year=1995 }}</ref> as are the overwhelming majority of meteorites that are found. There are currently over 27,000 chondrites in the world's collections. The largest individual stone ever recovered, weighing 1770&nbsp;kg, was part of the Jilin meteorite shower of 1976. Chondrite falls range from single stones to extraordinary showers consisting of thousands of individual stones, as occurred in the Holbrook fall of 1912, where an estimated 14,000 stones rained down on northern Arizona. The ordinary chondrites are thought to have originated from three parent asteroids, with the fragments making up the H chondrite, L chondrite and LL chondrite groups respectively.<ref name=Kring>David Kring (21 November 2013). "[https://www.youtube.com/watch?v=BNkS1uHUbq8&t=34m39s Asteroid Initiative Workshop Cosmic Explorations Speakers Session]". NASA (via YouTube). Retrieved 16 February 2019.</ref> A probable parent body of the H chondrites (comprising about 46% of the ordinary chondrites) is 6 Hebe, but its spectrum is dissimilar due to what is likely a metal impact melt component.<ref name=Gaffey>{{cite journal |bibcode=1998M&PS...33.1281G |first=M. J. |last=Gaffey |first2=S. L. |last2=Gilbert |title=Asteroid 6 Hebe: The probable parent body of the H-Type ordinary chondrites and the IIE iron meteorites |journal=Meteoritics & Planetary Science |volume=33 |pages=1281 |year=1998 |doi=10.1111/j.1945-5100.1998.tb01312.x |doi-access=free }}</ref> The ordinary chondrites comprise three mineralogically and chemically distinct groupings that differ in the amount of total iron, of iron metal and iron oxide in the silicates:<ref>{{Cite book|url=http://www.meteorite.fr/en/classification/ordinarychon.htm|title= Classification – Stony Meteorites – Ordinary tauch.Chondrites|website=www.meteorite.fr|accessdate=10 August 2017}}</ref> * The H chondrites have the '''H'''ighest total iron, high metal, but lower iron oxide (Fa) in the silicates * The L chondrites have '''L'''ower total iron, lower metal, but higher iron oxide (Fa) in the silicates * The LL chondrites have '''L'''ow total iron and '''L'''ow metal, but the highest iron oxide content (Fa) in the silicates. {{clear}} ==Carbonaceous chondrites== [[Image:Allende meteorite.jpg|thumb|right|250px|A slice of the Allende meteorite shows circular chondrules. Credit: [https://www.flickr.com/photos/33389938@N00 Shiny Things].{{tlx|free media}}]] A slice from the 4.5-billion-year-old Allende meteorite. This rock was formed along with the solar system. The C chondrites represent only a small proportion (4.6%)<ref name=Bischoff1995/> of meteorite falls. C chondrites contain a high proportion of carbon (up to 3%), which is in the form of graphite, carbonates and organic compounds, including amino acids, water and minerals that have been modified by the influence of water.<ref name=Buhler>BÜHLER, Springer-Verlag, 2013, {{ISBN|978-3-0348-6667-5}}, pp. 130.</ref> Some famous carbonaceous chondrites are: the Allende meteorite, Murchison meteorite, Orgueil meteorite, Ivuna meteorite, Murray meteorite, Tagish Lake meteorite, Sutter's Mill meteorite and Winchcombe meteorite. {{clear}} ==CB chondrites== [[Image:Gujba meteorite, bencubbinite (14785860604).jpg|thumb|right|250px|Gujba meteorite is a bencubbinite found in Nigeria, polished slice, 4.6&nbsp;x&nbsp;3.8&nbsp;cm, note the nickel-iron chondrules, which have been age-dated to 4.5627 billion years. Credit: .{{tlx|free media}}]] The group takes its name from the most representative member: [https://www.lpi.usra.edu/meteor/metbull.php?sea=Bencubbin&sfor=names&ants=&nwas=&falls=&valids=&stype=contains&lrec=50&map=ge&browse=&country=All&srt=name&categ=All&mblist=All&rect=&phot=&strewn=&snew=0&pnt=Normal%20table&code=5014 Bencubbin] (Australia). Although these chondrites contain over 50% nickel-iron metal, they are not classified as mesosiderites because their mineralogical and chemical properties are strongly associated with CR chondrites.<ref name=Meteoritefrcarb/> {{clear}} ==CH chondrites== "H" stands for "high metal" because CH chondrites may contain up to as much as 40% of metal.<ref name=Norton2002>{{cite book |title=The Cambridge Encyclopedia of Meteorites |last=Norton |first=O. Richard |date=2002 |publisher=Cambridge University Press |location=Cambridge |{{isbn|978-0-521-62143-4}} |page=139}}</ref> That makes them one of the most metal-rich of any of the chondrite groups, second only to the CB chondrites and some ungrouped chondrites such as NWA 12273. The first meteorite discovered was ALH 85085. Chemically, these chondrites are closely related to CR and CB groups. All specimens of this group belong only to petrologic types 2 or 3.<ref name=Meteoritefrcarb/> ==CI chondrites== This group was named after the Ivuna meteorite (Tanzania), have chemical compositions that are close to that measured in the solar photosphere (aside from gaseous elements, and elements such as lithium which are underrepresented in the Sun's photosphere by comparison to their abundance in CI&nbsp;chondrites). Six CI chondrites have been observed to fall: Ivuna, Orgueil, Alais, Tonk, Revelstoke, and Flensburg. ==CI1 chondrites== The Orgueil meteorite fell on May 14, 1864, a few minutes after 20:00 local time, near Orgueil in southern France. About 20 stones fell over an area of 5-10 square kilometres. ==CK chondrites== The group takes its name from Mighei chondrite (Ukraine), but the most famous member is the extensively studied Murchison meteorite meteorite. Many falls of this type have been observed and CM&nbsp;chondrites are known to contain a rich mix of complex organic compounds such as amino-acids and purine/pyrimidine nucleobases.<ref name=Meteoritefrcarb>"Carbonaceous chondrite" Meteorite.fr: All About Meteorites: Classification https://web.archive.org/web/20091012121808/http://www.meteorite.fr/en/classification/carbonaceous.htm 2009-10-12</ref><ref>{{Cite book | date = 28 April 2012 | title = Sutter's Mill Meteorite }}</ref><ref name=Pearce>{{cite journal|last1=Pearce|first1=Ben K. D.|last2=Pudritz|first2=Ralph E.|title=Seeding the Pregenetic Earth: Meteoritic Abundances of Nucleobases and Potential Reaction Pathways|journal=Astrophysical Journal|date=2015|volume=807|issue=1|page=85|doi=10.1088/0004-637X/807/1/85|arxiv = 1505.01465 |bibcode = 2015ApJ...807...85P |s2cid=93561811}}</ref> CM chondrite famous falls: * Murchison meteorite * Sutter's Mill meteorites * [https://www.sciencemag.org/news/2020/08/unusual-meteorite-more-valuable-gold-may-hold-building-blocks-life Aguas Zarcas meteorites]<ref>{{Cite book |title=Meteoritical Bulletin: Entry for Aguas Zarcas |url=https://www.lpi.usra.edu/meteor/metbull.php?sea=Aguas+Zarcas&sfor=names&ants=&nwas=&falls=&valids=&stype=&lrec=50&map=ge&browse=&country=All&srt=&categ=All&mblist=All&rect=&phot=&strewn=&snew=0&pnt=Normal%20table&code=69696 |accessdate=2020-08-21 |website=www.lpi.usra.edu }}</ref> * Jbilet Winselwan meteorites * Winchcombe meteorites ==CM chondrites== [[Image:Murchison crop.jpg|thumb|right|250px|Murchison meteorite is at the The National Museum of Natural History (Washington). Credit: [[c:user:Basilicofresco|Basilicofresco]].{{tlx|free media}}]] Amino acids in Ivuna and Orgueil were present at much lower concentrations than in CM chondrites (~30%), and that they had a distinct composition high in β-alanine, glycine, γ-Gamma-Aminobutyric acid (ABA), and beta-Aminobutyric acid (β-ABA) but low in 2-Aminoisobutyric acid|α-aminoisobutyric acid (AIB) and isovaline.<ref name=Ehrenfreund>{{cite journal |last=Ehrenfreund |first=Pascale |author2=Daniel P. Glavin |author3=Oliver Botta |author4=George Cooper |author5=Jeffrey L. Bada |year=2001 |title=Extraterrestrial amino acids in Orgueil and Ivuna: Tracing the parent body of CI type carbonaceous chondrites |journal=Proceedings of the National Academy of Sciences |volume=98 |issue=5 |pages=2138–2141 |doi=10.1073/pnas.051502898 |pmid=11226205 |pmc=30105|bibcode = 2001PNAS...98.2138E }}</ref> Most of the organic compound carbon in CI and CM carbonaceous chondrites is an insoluble complex material. That is similar to the description for kerogen. A kerogen-like material is also in the ALH84001 Martian meteorite (an achondrite). The CM meteorite Murchison meteorite has over 70 extraterrestrial amino acids and other compounds including carboxylic acids, hydroxy carboxylic acids, sulphonic and phosphonic acids, aliphatic, aromatic and polar hydrocarbons, fullerenes, heterocycles, carbonyl compounds, alcohols, amines and amides. {{clear}} ==CO chondrites== [[Image:Météorite Ornans, exposition Météorites, Muséum national d'histoire naturelle.jpg|thumb|right|250px|The Ornans meteorite observed in France in 1868. Credit: [[c:user:Eunostos|Eunostos]].{{tlx|free media}}]] The group takes its name from Ornans meteorite observed in France in 1868.<ref name=Ornans>{{cite book|title=Ornans|url=http://www.lpi.usra.edu/meteor/metbull.php?code=18030|publisher=Meteoritical Society|accessdate=4 January 2013}}</ref> The chondrule size is only about 0.15&nbsp;mm on average. They are all of petrologic type&nbsp;3. * 221 Eos – an asteroid from the asteroid belt and one of the likely parent bodies of the CO meteorites. Famous CO chondrite falls: * Ornans meteorites * [https://www.lpi.usra.edu/meteor/metbull.php?code=12229 Kainsaz meteorites] * [https://www.lpi.usra.edu/meteor/metbull.php?code=24215 Warrenton meteorites] * Moss meteorites Famous finds: * Dar al Gani 749 {{clear}} ==CR chondrites== The group takes its name from Cento Renazzo (Italy). The best parent body candidate is 2 Pallas.<ref name=Meteoritefrcarb/> Current usage of type 1 is simply to indicate meteorites that have experienced extensive aqueous alteration, to the point that most of their olivine and pyroxene have been altered to hydrous phases. This alteration took place at temperatures of 50 to 150 °C, so type 1 chondrites were warm, but not hot enough to experience thermal metamorphism. Chondrites have experienced extensive aqueous alteration, but still contain recognizable chondrules as well as primary, unaltered olivine and/or pyroxene. The fine-grained matrix is generally fully hydrated and minerals inside chondrules may show variable degrees of hydration. This alteration probably occurred at temperatures below 20 °C, and again, these meteorites are not thermally metamorphosed. Almost all CM and CR chondrites are petrologic type 2; with the exception of some ungrouped carbonaceous chondrites, no other chondrites are type 2. ==CV chondrites== [[Image:NWA 3118 meteorite.jpg|thumb|right|250px|NWA 3118 is shown. Credit: Mario Müller.{{tlx|free media}}]] CV chondrites observed falls: * Allende meteorites * [https://www.lpi.usra.edu/meteor/metbull.php?sea=Bali&sfor=names&ants=&nwas=&falls=&valids=&stype=contains&lrec=50&map=ge&browse=&country=All&srt=name&categ=All&mblist=All&rect=&phot=&strewn=&snew=0&pnt=Normal%20table&code=4928 Bali meteorites] *[https://www.lpi.usra.edu/meteor/metbull.php?code=30448 Bukhara meteorites] *[https://www.lpi.usra.edu/meteor/metbull.php?code=11206 Grosnaja meteorites] *[https://www.lpi.usra.edu/meteor/metbull.php?sea=Vigarano&sfor=names&ants=&nwas=&falls=&valids=&stype=contains&lrec=50&map=ge&browse=&country=All&srt=name&categ=All&mblist=All&rect=&phot=&strewn=&snew=0&pnt=Normal%20table&code=24174 Vigarano meteorites] {{clear}} ==CV3 chondrites== *[https://www.lpi.usra.edu/meteor/metbull.php?code=12218 Kaba meteorites] *[https://www.lpi.usra.edu/meteor/metbull.php?sea=Mokoia&sfor=names&ants=&nwas=&falls=&valids=&stype=contains&lrec=50&map=ge&browse=&country=All&srt=name&categ=All&mblist=All&rect=&phot=&strewn=&snew=0&pnt=Normal%20table&code=16713 Mokoia meteorites] ==C ungrouped== The most famous members: * Tagish Lake meteorites * Tarda meteorites ==Hypatia== [[Image:Hypatia stone sample.png|thumb|right|250px|The image shows a sample of the Hypatia stone. Credit: Romano Serra.{{tlx|fairuse}}]] Hypatia is a small stone found in Egypt in 1996, which may be the first known specimen of a comet nucleus on Earth, although defying physically-accepted models for hypervelocity processing of organic material.<ref>{{cite web|url=http://www.sci-news.com/space/science-libyan-desert-glass-diamond-bearing-pebble-evidence-comet-01446.html|title=Libyan desert glass: Diamond-Bearing Pebble Provides Evidence of Comet Striking Earth|work=sci-news.com, 8 October 2013}}</ref><ref name="ScD">{{cite web|url=https://www.sciencedaily.com/releases/2018/01/180109112437.htm|title=Extra-terrestrial Hypatia stone rattles solar system status quo|work=ScienceDaily.com, 9 January 2018}}</ref> Hypatia was discovered in December 1996 at {{coord|25|20|N|25|30|E}}, directly in proximity to a dark, slag-like glassy material that was interpreted to be a form of Libyan desert glass.<ref name=Kramers>{{cite journal |title=Unique chemistry of a diamond-bearing pebble from the Libyan Desert Glass strewnfield, SW Egypt: Evidence for a shocked comet fragment |doi=10.1016/j.epsl.2013.09.003 |bibcode=2013E&PSL.382...21K |volume=382 |journal=Earth and Planetary Science Letters |pages=21–31|date=2013 |last1=Kramers |first1=Jan D |last2=Andreoli |first2=Marco A.G |last3=Atanasova |first3=Maria |last4=Belyanin |first4=Georgy A |last5=Block |first5=David L |last6=Franklyn |first6=Chris |last7=Harris |first7=Chris |last8=Lekgoathi |first8=Mpho |last9=Montross |first9=Charles S |last10=Ntsoane |first10=Tshepo |last11=Pischedda |first11=Vittoria |last12=Segonyane |first12=Patience |last13=Viljoen |first13=K.S. (Fanus) |last14=Westraadt |first14=Johan E }}</ref> Although its status as an extraterrestrial rock is widely accepted, Hypatia is not officially classified as a true meteorite specimen by the Meteoritical Society due to its small size, where the original sample was cut apart and sent to multiple labs for study, reducing its original size of approximately 30 grams to about four grams.<ref name=Barakat>[https://drbarakataly.wordpress.com/2018/01/15/the-true-story-of-the-first-recorded-presolar-system-material-hypatia-stone/ See Barakat]: "The specimen is of a shiny grey-black colour and irregular shape. It measures roughly 3.5 x 3.2 x 2.1 cm and weights about 30 grams"; Pappas, Stephanie (January 18, 2018). [https://www.livescience.com/61409-extraterrestrial-hypatia-stone.html "Out-of-This-World Diamond-Studded Rock Just Got Even Weirder"]. ''Live Science''. Retrieved May 25, 2022.</ref> Hypatia may be a relict fragment of the hypothetical impacting body assumed to have produced the chemically-dissimilar Libyan desert glass.<ref name=Kramers/> If this association holds, Hypatia may have impacted Earth approximately 28 million years ago.<ref name=Collins>{{cite news |url=http://www.nzherald.co.nz/world/news/article.cfm?c_id=2&objectid=11974415 |title=Incredible diamond-studded 'alien' rock has minerals not found anywhere in our star system |last=Collins |first=Tim |date=2018-01-12 |work=NZ&nbsp;Herald |accessdate=2018-01-13|issn=1170-0777 }}</ref> Its unusual chemistry has prompted further speculation that Hypatia may predate the formation of the Solar System.<ref name="ScD" /> Compounds including polyaromatic hydrocarbons and silicon carbide associated with a previously-unknown nickel phosphide compound have been found, ratios of silicon to carbon are anti-correlated to terrestrial averages, or those of major planets like Mars or Venus, but some samples of interstellar dust overlap Hypatia distributions, although Hypatia's elemental chemistry also overlaps some terrestrial distributions.<ref>2018 Journal Geochimica et Cosmochimica Acta '''223''' 462. (Quotation from CERN Courier March 2018)</ref> "The stone has a bimodal matrix. Type 1 is essentially devoid of elements heavier than oxygen, whereas matrix 2 shows a unique, consistent element abundance pattern from aluminum to zinc, up to a sum of 4.5 wt%. Fe is dominant and Si markedly depleted, with a CI-chondrite normalized Si/Fe ratio of c. 0.1. Element abundance ratios to Fe show several correlations, with one group (Al, Si, K, Ca, Ti and Cu) being negatively correlated to another one (P, S and Ni)."<ref name=Kramers2022>{{ cite journal |author=Jan D. Kramers, Georgy A. Belyanin, Wojciech J. Przybyłowicz, Hartmut Winkler and Marco A.G. Andreoli |title=The chemistry of the extraterrestrial carbonaceous stone “Hypatia”: A perspective on dust heterogeneity in interstellar space |journal=Icarus |date=August 2022 |volume=282 |issue= |pages=115043 |url=https://www.sciencedirect.com/science/article/pii/S0019103522001555?via%3Dihub |arxiv= |bibcode= |doi=10.1016/j.icarus.2022.115043 |pmid= |accessdate=6 June 2022 }}</ref> The "amount of silicon in the Hypatia stone was extremely low — less than 1% of what would be expected for an object that formed in our solar system. Likewise, the levels of chromium, manganese, iron, sulfur, copper and vanadium were not typical of inner solar system material."<ref name=Waldek>{{ cite web |author=Stefanie Waldek |title=This tiny space rock might be the 1st physical evidence of a rare supernova |publisher=Space.com |location= |date=20 May 2022 |url=https://www.space.com/hypatia-stone-from-space-evidence-supernova?utm_source=SmartBrief |accessdate=6 June 2022 }}</ref> "We found a consistent pattern of trace element abundances that is completely different from anything in the solar system, primitive or evolved. Objects in the asteroid belt and meteors don't match this, either. So next, we looked outside the solar system."<ref name=KramersSpace>{{ cite web |author=Jan Kramers |title=This tiny space rock might be the 1st physical evidence of a rare supernova |publisher=Space.com |location= |date=20 May 2022 |url=https://www.space.com/hypatia-stone-from-space-evidence-supernova?utm_source=SmartBrief |accessdate=6 June 2022 }}</ref> The "composition of the Hypatia stone ruled out [interstellar bands of dust in the Milky Way, a red giant star and even a Type II supernova, which occurs when a massive star runs out of fuel, collapses, then explodes.]"<ref name=Waldek/> "In six of the 15 elements, proportions were between 10 and 100 times higher than the ranges predicted by theoretical models for supernovas of type Ia. These are the elements aluminum, phosphorus, chlorine, potassium, copper and zinc."<ref name=KramersSpace/> "Perhaps equally important, it shows that an individual anomalous 'parcel' of dust from outer space could actually be incorporated in the solar nebula that our solar system was formed from, without being fully mixed in. This goes against the conventional view that dust which our solar system was formed from was thoroughly mixed."<ref name=KramersSpace/> "The carbonaceous, diamondiferous stone named “Hypatia” represents a clear exception to the homogeneous chemistry of primitive solar system objects. The c. 30 g stone was found in 1996 by Dr. Aly Barakat of the Egyptian Mineral Resources Authority in the Libyan Desert Glass (LDG) area in southwest Egypt (Barakat, 2005). Its extraterrestrial origin was demonstrated via its 40Ar/36Ar ratio, which is much lower than that of Earth's atmosphere (Kramers ''et al.'', 2013)."<ref name=Kramers2022/> "He and Xe isotope compositions [...] strongly resemble those of the “Q” noble gas component hosted in the carbonaceous matter of chondrites (Ozima ''et al.'', 1998). Further, a significant {{chem|129|Xe}} excess was found in Hypatia (Avice ''et al.'', 2015). {{chem|129|Xe}} is the radiogenic daughter of extinct {{chem|129|I}} (half-life 1.57 × 10⁷ yrs.) and {{chem|129|Xe}} excesses are common in chondritic meteorites, where the I-Xe chronometer has been extensively used to constrain timescales of processes within the first 4 × 10⁷ yrs. of solar nebula history (Brazzle ''et al.'', 1999)."<ref name=Kramers2022/> "Deuteron nuclear reaction analysis (D-NRA) of the bulk matrix yielded atomic percentages of 74.5–77% C, 0.5–1.5% N and 22–25% O (Kramers ''et al.'', 2013). A petrographic study (Belyanin ''et al.'', 2018) demonstrated the ubiquitous presence of microdiamonds, thought to be shock-related, as well as one graphite- and multiple silicon carbide grains, the latter in association with a Ni phosphide compound not previously described, but no silicate minerals."<ref name=Kramers2022/> {{clear}} ==H chondrites== [[Image:Weston meteorite.jpg|thumb|right|250px|Weston meteorite is an H chondrite that fell in 1807. Credit: [https://www.flickr.com/photos/8435962@N06 Claire H.].{{tlx|free media}}]] The H type ordinary chondrites are the most common type of meteorite, accounting for approximately 40% of all those catalogued, 46% of the ordinary chondrites, and 44% of the chondrites.<ref>[http://internt.nhm.ac.uk/jdsml/research-curation/projects/metcat/metsPerGroup.dsml Natural History Museum, meteorite catalogue]</ref> A probable parent body for this group is the S-type asteroid 6 Hebe, with less likely candidates being 3 Juno and 7 Iris.<ref name=Gaffey1998>[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1998M%26PS...33.1281G&amp;db_key=AST&amp;data_type=HTML&amp;format=&amp;high=438c93072f28336 M. J. Gaffey & S. L. Gilbert ''Asteroid 6 Hebe: The probable parent body of the H-Type ordinary chondrites and the IIE iron meteorites''], Meteoritics & Planetary Science, Vol. 33, p. 1281 (1998).</ref> It is supposed that these meteorites arise from impacts onto small near-Earth asteroids broken off from 6 Hebe in the past, rather than originating from 6 Hebe directly. The '''H chondrites''' have very similar trace element abundances and Oxygen isotope ratios to the IIE iron meteorites, making it likely that they both originate from the same parent body. Their high iron abundance is about 25–31% by weight. Over half of this is present in metallic form, making these meteorites strongly magnetic despite the stony chondritic appearance. The most abundant minerals are bronzite]] (an orthopyroxene), and olivine. Characteristic is the fayalite (Fa) content of the olivine of 16 to 20&nbsp;mol%. They contain also 15–19% of nickel-iron metal and about 5% of troilite. The majority of these meteorites have been significantly metamorphosed, with over 40% being in petrologic class 5, most of the rest in classes 4 and 6. Only a few (about 2.5%) are of the largely unaltered petrologic class 3. {{clear}} ==H4 chondrites== [[Image:Meteorito Marília.jpg|thumb|right|250px|Marília is an H chondrite meteorite that fell to Earth on October 5, 1971, in Marília, São Paulo, Brazil. Credit: [[c:user:Gabisfunny|Gabisfunny]].{{tlx|free media}}]] It is classified as H4-ordinary chondrite.<ref>{{cite book|url=https://meteoritosbrasil.weebly.com/database/marlia|title=Marília|language=pt|publisher=Meteoritos Brasil|accessdate=July 24, 2018}}</ref> {{clear}} ==H6 chondrites== [[Image:Aarhus meteorites.png|thumb|right|250px|Aarhus meteorite pieces are from the find on 19 October 1951. Credit: Hanne Teglhus.{{tlx|fairuse}}]] The meteor split just before the otherwise undramatic impact and two pieces were recovered: Aarhus I (at 300g) and Aarhus II (at 420g), with Aarhus I found in the small woodland of Riis Skov, just a few minutes after impact.<ref name=Grady>{{cite book| first =Monica M |last = Grady | location = London |title=Catalogue of Meteorites|url= https://books.google.com/books?id=mkdHJR35Q_8C&pg=PA55 | accessdate =30 April 2014|date=31 August 2000|publisher= Natural History Museum, Cambridge University Press | {{ISBN|978-0-521-66303-8}}|page=55}}</ref><ref>StenoMusen 15. Pictures of the pieces.</ref> {{clear}} ==L chondrites== [[Image:NWA869Meteorite.jpg|thumb|right|250px|A 700 g individual is from the NWA 869 meteorite. Credit: [[c:user:Vesta|H. Raab]].{{tlx|free media}}]] The L type ordinary chondrites are the second most common group of meteorites, accounting for approximately 35% of all those catalogued, and 40% of the ordinary chondrites.<ref>[http://internt.nhm.ac.uk/jdsml/research-curation/projects/metcat/metsPerGroup.dsml Natural History Museum, meteorite catalogue]</ref> Their name comes from their relatively low iron abundance, with respect to the H chondrites, which are about 20–25% iron by weight. Characteristic is the fayalite content (Fa) in olivine of 21 to 25&nbsp;mol%. About 4–10% iron–nickel is found as a free metal, making these meteorites magnetic, but not as strongly as the H chondrites. "A zircon U-Pb date of 467.50±0.28 Ma from a distinct bed within the meteorite-bearing interval of southern Sweden that, combined with published cosmic-ray exposure ages of co-occurring meteoritic material, provides a precise age for the L chondrite breakup at 468.0±0.3 Ma."<ref name=Lindskog/> Many of the L chondrite meteors may have their origin in the Ordovician meteor event, radioisotope dated with uranium–lead at around 467.50±0.28 million years ago. Compared to other chondrites, a large proportion of the L chondrites have been heavily shocked, which is taken to imply that the parent body was catastrophically disrupted by a large impact. This impact has been dated via cosmic ray exposure at around 468.0±0.3 million years ago.<ref name=Lindskog>{{Cite journal|last=Lindskog|first=A. |last2=Costa|first2=M. M. |last3=Rasmussen|first3=C.M.Ø. |last4=Connelly|first4=J. N. |last5=Eriksson|first5=M. E. |date=2017-01-24|title=Refined Ordovician timescale reveals no link between asteroid breakup and biodiversification |journal=Nature Communications|volume=8|pages=14066 |doi=10.1038/ncomms14066| pmid=28117834|pmc=5286199 |issn=2041-1723 }}</ref><ref name=Schmitz>{{cite journal |last1=Schmitz |first1=Birger |date=2019-09-18 |title=An extraterrestrial trigger for the mid-Ordovician ice age: Dust from the break-up of the L-chondrite parent body |journal=Science Advances|volume=5 |issue=9 |pages= eaax4184|doi=10.1126/sciadv.aax4184 |pmid=31555741 |pmc=6750910 }}</ref> Earlier argon dating placed the event at around 470±6 million years ago.<ref name=Haack> H. Haack ''Meteorite, asteroidal, and theoretical constraints on the 500-Ma disruption of the L chondrite parent body'', Icarus, Vol. 119, p. 182 (1996).</ref><ref name=Korochantseva>Korochantseva [http://www.ingentaconnect.com/content/arizona/maps/2007/00000042/00000001/art00009 "L-chondrite asteroid breakup tied to Ordovician meteorite shower by multiple isochron 40Ar-39Ar dating"] Meteoritics & Planetary Science 42, 1, pp. 3-150, Jan. 2007.</ref> The most abundant minerals are olivine and hypersthene (an orthopyroxene), as well as iron–nickel alloy and troilite. Chromite, sodium-rich feldspar and calcium phosphates occur in minor amounts. The parent body/bodies for this group are not known, but plausible suggestions include 433 Eros and 8 Flora, or the Flora family as a whole. 433 Eros has been found to have a similar spectrum, while several pieces of circumstantial evidence for the Flora family exist: (1) the Flora family is thought to have formed about 1,000 to 500 million years ago; (2) the Flora family lies in a region of the asteroid belt that contributes strongly to the meteorite flux at Earth; (3) the Flora family consists of S-type asteroids, whose composition is similar to that of chondrite meteorites; and (4) the Flora family parent body was over {{convert|100|km}} in diameter. {{clear}} ==L1 chondrites== ==L2 chondrites== ==L3 chondrites== Julesburg meteorites are L3 chondrites. ==L4 chondrites== Kemer and Saratov meteorites are L4 chondrites. ==L5 chondrites== [[Image:Homestead meteorite.jpg|thumb|right|250px|20.4 gram partial slice of the historic witnessed fall "Homestead". Credit: [https://www.flickr.com/people/48082563@N08 Jon Taylor].{{tlx|free media}}]] This is a nice specimen with fusion crust along one edge. This slice was taken from a 450 gram fragment that resided in the American Museum of Natural History for over a century. The fall took place on Friday February 12, 1875 in the small town of Homestead, Iowa. At about 10:30 a brilliant fireball lit up the cold dark sky. The flash was followed by loud rumbling sounds as the meteoroid exploded over the snowy countryside. The first meteorite fragment was found two miles west of Homestead by a woman named Sarah Sherlock. The stone weighed seven pounds and six ounces. The largest mass weighed 74 lbs and was not discovered until spring. It along with a 48 pound mass were buried 2 feet in the ground. In total approximately 227kgs of this attractive L5 brecciated chondrite were discovered. {{clear}} ==L6 chondrites== [[Image:Berduc meteorite, full slice, reverse side.jpg|thumb|right|250px|Reverse side of the Berduc slice shows both the beautiful brecciation and fresh fusion crust of this veined L6 chondrite. Credit: [https://www.flickr.com/people/48082563@N08 Jon Taylor].{{tlx|free media}}]] Both the beautiful brecciation and fresh fusion crust of this veined L6 chondrite. 6.8 gram full slice. Date 9 August 2011, 08:27:57 {{clear}} ==L7 chondrites== L7 chondrites: PAT 91501 and LEW 88663 meteorites. ==LL chondrites== [[Image:Paragould meteorite sm.jpg|thumb|right|250px|The Paragould Meteorite is on display in Mullins Library at the University of Arkansas in Fayettville, Arkansas. Credit: [[w:user:The stuart|The stuart]].{{tlx|free media}}]] The LL chondrites are a group of stony meteorites, the least abundant group of the ordinary chondrites, accounting for about 10–11% of observed ordinary-chondrite falls and 8–9% of all meteorite falls. The composition of the Chelyabinsk meteorite is that of a LL chondrite meteorite. The material makeup of Itokawa, the asteroid visited by the Hayabusa spacecraft which landed on it and brought particles back to Earth also proved to be type LL chondrite. They contain 19–22% total iron and only 0.3–3% metallic iron. That means that most of the iron is present as iron oxide (FeO) in the silicates; olivine contains 26 to 32 mol% fayalite (Fa). The most abundant minerals are hypersthene (a pyroxene) and olivine. Other minerals include Fe–Ni, troilite (FeS), feldspar or feldspathic glass, chromite, and phosphates. LL chondrites contain the largest chondrules of the ordinary chondrite groups, averaging around 1 millimetre (0.039 in) diameter. The LL group includes many of the most primitive ordinary chondrites, including the well-studied Semarkona (type 3.0) chondrite. However, most LL chondrites have been thermally metamorphosed to petrologic types 5 and 6, meaning that their minerals are homogeneous in composition and chondrule borders are difficult to discern. ==Cryometeorites== [[Image:Hagelkorn mit Anlagerungsschichten.jpg|thumb|right|250px|A large hailstone (clear and white) with concentric rings is shown. Credit: [[commons:User:ERZ|ERZ]].]] [[Image:Small hail, fractured to show internal structure.jpg|thumb|right|250px|The image shows small hail that has been fractures to show internal structure. Credit: Erbe, Pooley: USDA, ARS, EMU.]] [[Image:Pital nevada.jpg|thumb|left|250px|On April 13, 2004, a blanket of hail fell during a storm in Cerro El Pital, El Salvador. Credit: [[commons:User:Wanakoo|Wanakoo]].]] [[Image:Bogota hailstorm.jpg|thumb|left|250px|The image captures a hailstorm in progress in Bogotá, D.C., Colombia, on March 3, 2006. Credit: [[w:User:Ju98 5|Ju98 5]].]] [[Image:Nssl0098 - Flickr - NOAA Photo Library.jpg|thumb|right|250px|This is a very large hailstone from the NOAA Photo Library. Credit: NOAA Legacy Photo; OAR/ERL/Wave Propagation Laboratory.]] [[Image:Wea02251 - Flickr - NOAA Photo Library.jpg|thumb|right|250px|This hailstone was four inches in diameter and weighed seven ounces. Credit: Archival Photography by Steve Nicklas, NOS, NGS.]] [[Image:Hailstone.jpg|thumb|right|250px|As of June 22, 2003, this is the largest hailstone ever recovered. Credit: NOAA.]] [[Image:Record hailstone Vivian, SD.jpg|thumb|right|250px|This is a record-setting hailstone that fell in Vivian, South Dakota on July 23, 2010. Credit: NWS Aberdeen, SD.]] [[Image:Graupel encasing a snow crystal.jpg|thumb|left|250px|Graupel is shown encasing an unseen snow crystal. Credit: Erbe, Pooley: USDA, ARS, EMU.]] [[Image:Snowflake 300um LTSEM, 13368.jpg|thumb|left|250px|Rime occurs on both ends of a columnar snow crystal. Credit: [[w:User:Brian0918|Brian0918]].]] [[Image:Sleet (ice pellets).jpg|thumb|left|250px|The image shows ice pellets aka sleet in North America, with a United States penny for scale. Credit: [[commons:User:Runningonbrains|Runningonbrains]].]] [[Image:Mwrime.JPG|thumb|right|250px|Rime ice is shown after deposition on a window. Credit: [[commons:User:Ws47|Ws47]].]] [[Image:Snow Clouds in Korea.jpg|thumb|right|250px|This image is a satellite photo of lake-effect snow bands near the Korean Peninsula. Credit: NASA.]] '''Hail''' is a form of solid [water] precipitation. It consists of balls or irregular lumps of ice, each of which is referred to as a '''hailstone'''.<ref name="webster">{{ cite book |title=Merriam-Webster definition of "hailstone" |publisher=Merriam-Webster |url=http://www.merriam-webster.com/dictionary/hailstone |accessdate=2013-01-23 }}</ref> Unlike graupel, which is made of rime, and ice pellets, which are smaller and translucent, hailstones – on Earth – consist mostly of water ice and measure between {{convert|5|and|200|mm|in}} in diameter. The METAR reporting code for hail {{convert|5|mm|in|abbr=on}} or greater is '''GR''', while smaller hailstones and graupel are coded '''GS'''. ... Hail has a diameter of {{convert|5|mm|in}} or more.<ref name="gloss">{{ cite book |url=http://amsglossary.allenpress.com/glossary/search?id=hail1 |title=Hail |date=2009 |accessdate=2009-07-15 |author=Glossary of Meteorology |publisher=American Meteorological Society }}</ref> Hailstones can grow to {{convert|15|cm|in|0}} and weigh more than {{convert|0.5|kg|lb|1}}.<ref name=NSSL2007>{{ cite book |url=http://www.photolib.noaa.gov/htmls/nssl0001.htm |title=Aggregate hailstone |author=National Severe Storms Laboratory |publisher=National Oceanic and Atmospheric Administration |date=2007-04-23 |accessdate=2009-07-15 }}</ref> Unlike ice pellets, hailstones are layered and can be irregular and clumped together. A cross-section through a large hailstone shows an onion-like structure. This means the hailstone is made of thick and translucent layers, alternating with layers that are thin, white and opaque. The image at left shows a blanket of hail precipitated on the ground at Cerro El Pital, El Savador. {{lang|es|"Cerro El Pital se encuentra a 12 kilómetros de La Palma, con una altura de 2730 msnm es el punto más alto del territorio Salvadoreño. Es una montaña en medio de un bosque nebuloso que suele tener una temperatura aproximada de 10 ºC. El 13 de abril de 2004, las temperaturas bajaron tanto que el cerro fue cubierto por una escarcha de hielo que causó conmoción entre los pobladores, atribuyendo el fenómeno a una supuesta "nevada"."}} The third image at right shows a hailstone that fell at Washington, D. C., on May 26, 1953, that was 4 in in diameter and weighed 7 oz. In the fourth image at right is the largest hailstone ever recovered in the United States as of June 22, 2003. This hailstone fell in Aurora, Nebraska. It has a 7-inch (17.8 cm) diameter and an approximate circumference of 18.75 inches.<ref name=Leslie>{{ cite book |author=John Leslie |title=Central Plains Storm Produced Largest Hailstone in U.S. History |publisher=NOAA Satellites and Information |location=Maryland |date=2008 |url=http://www.noaanews.noaa.gov/stories/s2008.htm |accessdate=2012-10-14 }}</ref> The next hailstone image is one, approximately 133 mm (5 1/4 inches) in diameter, that fell in Harper, Kansas on May 14, 2004. After 2003, another record-setting hailstone fell in Vivian, South Dakota, on July 23, 2010. Its diameter is 8 inches with a weight of 1 pound 15 ounces. Terminal velocity of hail, or the speed at which hail is falling when it strikes the ground, varies by the diameter of the hail stones. A hail stone of 1 cm (0.39 in) in diameter falls at a rate of 9 metres per second (20 mph), while stones the size of 8 centimetres (3.1 in) in diameter fall at a rate of 48 metres per second (110 mph). Hail stone velocity is dependent on the size of the stone, friction with air it is falling through, the motion of wind it is falling through, collisions with raindrops or other hail stones, and melting as the stones fall through a warmer atmosphere.<ref name=NSSL>{{ cite book |url=http://www.nssl.noaa.gov/primer/hail/hail_basics.html |title=Hail Basics |author=National Severe Storms Laboratory |publisher=National Oceanic and Atmospheric Administration |date=2006-11-15 |accessdate=2009-08-28 }}</ref> A '''megacryometeor''' is a very large chunk of ice which, despite sharing many textural, hydro-chemical and isotopic features detected in large hailstones, is formed under unusual atmospheric conditions which clearly differ from those of the cumulonimbus cloud scenario (i.e. clear-sky conditions). They are sometimes called huge hailstones, but do not need to form in thunderstorms. Jesus Martinez-Frias, a planetary geologist at the Center for Astrobiology in Madrid, pioneered research into megacryometeors in January 2000 after ice chunks weighing up to 6.6 pounds (3.0 kg) rained on Spain out of cloudless skies for ten days. '''Graupel''' ... also called '''soft hail''' or '''snow pellets''')<ref name=Webster>{{ cite book |url = http://www.merriam-webster.com/dictionary/graupel |title = Graupel - Definition, In: ''Merriam-Webster Dictionary'' |publisher = Merriam-Webster |accessdate = 15 Jan 2012 }}</ref> refers to precipitation that forms when supercooled droplets of water are collected and freeze on a falling snowflake, forming a {{convert|2|-|5|mm|in|3|abbr=on}} ball of rime. Strictly speaking, graupel is not the same as hail or ice pellets, although it is sometimes referred to as '''small hail'''. However, the World Meteorological Organization defines ''small hail'' as snow pellets encapsulated by ice, a precipitation halfway between graupel and hail.<ref name=Secretariat>{{ cite book |title=International Cloud Atlas |date=1975 |publisher=Secretariat of the World Meteorological Organization |location=Geneva |url=https://books.google.com/books?id=hkTEMgAACAAJ |ISBN=92-63-10407-7 }}</ref> The frozen droplets on the surface of rimed crystals are hard to resolve and the topography of a graupel particle is not easy to record with a light microscope because of the limited resolution and depth of field in the instrument. However, observations of snow crystals with a low-temperature scanning electron microscope (LT-SEM) clearly show cloud droplets measuring up to {{convert|50|µm|in|5|abbr=on}} on the surface of the crystals. The rime has been observed on all four basic forms of snow crystals, including plates, dendrites, columns and needles. As the riming process continues, the mass of frozen, accumulated cloud droplets obscures the identity of the original snow crystal, thereby giving rise to a graupel particle. Graupel commonly forms in high altitude climates and is both denser and more granular than ordinary snow, due to its rimed exterior. Macroscopically, graupel resembles small beads of polystyrene. The combination of density and low viscosity makes fresh layers of graupel unstable on slopes, and layers of {{convert|20|-|30|cm|in|abbr=on}} present a high risk of dangerous slab avalanches. In addition, thinner layers of graupel falling at low temperatures can act as ball bearings below subsequent falls of more naturally stable snow, rendering them also liable to avalanche.<ref>"[http://www.avalanche.org/~moonstone/snowpack/the%20relation%20of%20crystal%20riming%20to%20avalanche%20formation%20in%20new%20snow.htm The Relation of Crystal Riming to Avalanche Formation in New Snow]". ''Department of Atmospheric Sciences, University of Washington.''</ref> Graupel tends to compact and stabilise ("weld") approximately one or two days after falling, depending on the temperature and the properties of the graupel.<ref>[http://www.avalanche.org/~uac/encyclopedia/graupel.htm Graupel], www.avalanche.org.</ref> '''Ice pellets''' (also referred to as '''sleet''' by the United States National Weather Service<ref>{{ cite book |url=http://www.weather.gov/glossary/index.php?word=sleet |title= Sleet (glossary entry) |publisher= National Oceanic and Atmospheric Administration's National Weather Service |accessdate=2007-03-20 }}</ref>) are a form of precipitation consisting of small, translucent balls of ice. Ice pellets are usually smaller than hailstones<ref>{{ cite book |url=http://www.weather.gov/glossary/index.php?word=hail |title= Hail (glossary entry) |publisher= National Oceanic and Atmospheric Administration's National Weather Service |accessdate=2007-03-20 }}</ref> and are different from graupel, which is made of rime, or rain and snow mixed, which is soft. Ice pellets often bounce when they hit the ground, and generally do not freeze into a solid mass unless mixed with freezing rain. The METAR code for ice pellets is '''PL'''. '''Hard rime''' is a white ice that forms when the water droplets in fog freeze to the outer surfaces of objects. It is often seen on trees atop mountains and ridges in winter, when low-hanging clouds cause freezing fog. This fog freezes to the windward (wind-facing) side of tree branches, buildings, or any other solid objects, usually with high wind velocities and air temperatures between {{convert|−2|and|−8|°C|°F|1}}. '''Snow''' is precipitation in the form of flakes of crystalline water ice that fall from clouds. Since snow is composed of small ice particles, it is a granular material. It has an open and therefore soft structure, unless subjected to external pressure. Snowflakes come in a variety of sizes and shapes. Types that fall in the form of a ball due to melting and refreezing, rather than a flake, are known as hail, ice pellets or snow grains. {{clear}} ==Mercury== {{main|Hermiominerals}} [[Image:20130321 nwa 7325 ralew.jpg|thumb|right|250px|NWA 7325 is a unique meteorite. Credit: Stefan Ralew / sr-meteorites.de.]] "NWA 7325 is actually a group of 35 meteorite samples discovered in 2012 in Morocco. They are ancient, with Irving and his team dating the rocks to an age of about 4.56 billion years."<ref name=Kramer>{{ cite book |author=Miriam Kramer |title=Green Meteorite May Be from Mercury, a First |publisher=SPACE.com |location= |date=March 28, 2013 |url=http://www.space.com/20426-mercury-meteorite-discovery-messenger.html |accessdate=2013-03-31 }}</ref> "NWA 7325 has a lower magnetic intensity — the magnetism passed from a cosmic body's magnetic field into a rock — than any other rock yet found, Irving said. Data sent back from NASA's Messenger spacecraft currently in orbit around Mercury shows that the planet's low magnetism closely resembles that found in NWA 7325, Irving said."<ref name=Kramer/> "NWA 7325 has olivine in it that is insanely magnesium-rich. Iron and magnesium are two elements that are almost always found together in rocks; the ions they make have the same size and charge so they happily occupy the same positions in crystal lattices. It's weird to have a rock that is so dominantly magnesium-rich. Mercury's surface rocks are known (thanks to MESSENGER) to be unusually low in iron."<ref name=Lakdawalla/> "NWA 7325's oxygen isotope ratios do not match any known meteorites from any other planet-size body. In fact, they're not particularly similar to much of anything that we've measured oxygen isotope ratios for."<ref name=Lakdawalla/> "The ratios of Al/Si (0.224) and Mg/Si (0.332) plus the very low Fe content of NWA 7325 are consistent with the compositions of surface rocks on Mercury [6], but the Ca/Si ratio (0.582) is far too high. However, since NWA 7325 is evidently a plagioclase cumulate (and presumably excavated from depth), it may not match surface rocks on its parent body. The abundance of diopside rather than enstatite might be consistent with some earlier spectral observations of Mercury [7]."<ref name=Irving>{{ cite book |author=A. J. Irving |author2=S. M. Kuehner |author3=T. E. Bunch |author4=K. Ziegler |author5=G. Chen |author6=C. D. K. Herd |author7=R. M. Conrey |author8=S. Ralew |title=Ungrouped Mafic Achondrite Northwest Africa 7325: A Reduced, Iron-poor Cumulative Olivine Gabbro from a Differentiated Planetary Parent Body |publisher=Lunar and Planetary Science Conference |location= |date=March 20, 2013 |url=http://www.lpi.usra.edu/meetings/lpsc2013/pdf/2164.pdf |accessdate=2013-03-22 }}</ref> "[I]t's about 23 times harder to get a rock from Mercury to Earth than it is from Mars to Earth. Given that we've got more than 70 known Mars meteorites in our collections, that means we ought to have found 3 (give or take a couple) Mercury meteorites by now."<ref name=Lakdawalla>{{ cite book |author=Emily Lakdawalla |title=LPSC 2013: Do we have a meteorite from Mercury? |publisher=The Planetary Society |location= |date=March 21, 2013 |url=http://www.planetary.org/blogs/emily-lakdawalla/2013/03211549-lpsc-hermean-meteorite.html |accessdate=2013-03-22 }}</ref> Meteorites originating from Mercury can be called ''hermiometeorites''. {{clear}} ==Venus== {{main|Liquids/Liquid objects/Venus}} While there is little or no water on Venus, there is a phenomenon which is quite similar to snow. The Magellan probe imaged a highly reflective substance at the tops of Venus's highest mountain peaks which bore a strong resemblance to terrestrial snow. This substance arguably formed from a similar process to snow, albeit at a far higher temperature. Too volatile to condense on the surface, it rose in gas form to cooler higher elevations, where it then fell as precipitation. The identity of this substance is not known with certainty, but speculation has ranged from elemental tellurium to lead sulfide (galena).<ref name=Otten>{{ cite book |title='Heavy metal' snow on Venus is lead sulfide |author=Carolyn Jones Otten |publisher=Washington University in St Louis |url=http://news-info.wustl.edu/news/page/normal/633.html |date=2004 |accessdate=2007-08-21}}</ref> ==Earth== {{main|Rocks/Rocky objects/Earth|Geominerals}} [[Image:Barringer Meteor Crater, Arizona.jpg|thumb|left|250px|This is an aerial view of the Barringer Meteor Crater about 69 km east of Flagstaff, Arizona USA. Credit: D. Roddy, U.S. Geological Survey (USGS).]] [[Image:Meteor Crater - Arizona.jpg|thumb|right|250px|This is a Landsat image of the Barringer Meteor Crater from space. Credit: National Map Seamless Server, NASA Earth Observatory.]] [[Image:Canyon-diablo-meteorite.jpg|thumb|right|250px|This is an image of the {{w|Canyon Diablo (meteorite)|Canyon Diablo}} iron meteorite (IIIAB) 2,641 grams. Credit: Geoffrey Notkin, Aerolite Meteorites of Tucson, [[w:User:Geoking42|Geoking42]].]] [[Image:Meteor Crater 08 2010 151.JPG|thumb|right|250px|The Holsinger meteorite is the largest discovered fragment of the meteorite that created Meteor Crater and it is exhibited in the crater visitor center. Credit: [[commons:User:Mariordo|Mariordo]] Mario Roberto Duran Ortiz.]] In the image at left is an aerial view of the Barringer Meteor Crater about 69 km east of Flagstaff, Arizona USA. Although similar to the aerial view of the Soudan crater, the Barringer Meteor Crater appears angular at the farthest ends rather than round. '''Meteor Crater''' is a [[Meteorites|meteorite]] {{w|impact crater}} approximately 43 miles (69 km) east of {{w|Flagstaff, Arizona|Flagstaff}}, near {{w|Winslow, Arizona|Winslow}} in the northern {{w|Arizona}} desert of the {{w|United States}}. Because the US {{w|Department of the Interior}} Division of Names commonly recognizes names of natural features derived from the nearest post office, the feature acquired the name of "Meteor Crater" from the nearby post office named Meteor.<ref name=Barringer>[http://articles.adsabs.harvard.edu//full/1993Metic..28....9B/0000009.000.html J. P. Barringer's acceptance speech.] Meteoritics, volume 28, page 9 (1993). Retrieved on the SAO/NASA Astrophysics Data System</ref> The site was formerly known as the '''Canyon Diablo Crater''', and fragments of the meteorite are officially called the Canyon Diablo Meteorite. Scientists refer to the crater as '''Barringer Crater''' in honor of Daniel Barringer, who was first to suggest that it was produced by meteorite impact.<ref name=Grieve>Grieve, R.A.F. (1990) ''Impact Cratering on the Earth'', Scientific American '''262'''(4), 66–73.</ref> From space the crater appears almost like a square. The image at right has a resolution of 2 meters per pixel, and illumination is from the right. Layers of exposed limestone and sandstone are visible just beneath the crater rim, as are large stone blocks excavated by the impact. The Holsinger meteorite is the largest discovered fragment of the meteorite that created Meteor Crater and it is exhibited in the crater visitor center. The '''Canyon Diablo''' [[Meteorites|meteorite]] comprises many fragments of the {{w|asteroid}} that impacted at {{w|Barringer Crater}} ({{w|Meteor Crater}}), Arizona, USA. Meteorites have been found around the crater rim, and are named for nearby Canyon Diablo, which lies about three to four miles west of the crater. There are fragments in the collections of museums around the world including the {{w|Field Museum of Natural History}} in Chicago. The biggest fragment ever found is the {{w|Canyon Diablo (meteorite)|Holsinger Meteorite}}, weighing 639&nbsp;kg, now on display in the Meteor Crater Visitor Center on the rim of the crater. {{clear}} ==Moon== {{main|Liquids/Liquid objects/Moon|Selenominerals}} [[Image:Allan Hills 81005, lunar meteorite.jpg|thumb|right|250px|This image shows the lunar meteorite ''Allan Hills 81005''. Credit: NASA.]] [[Image:Lunar breccia Apollo sample 14321.jpg|left|thumb|300px|Lunar breccia Apollo sample 14321 formed somewhere between 4 and 4.1 billion years ago, about 12.4 miles beneath the Earth’s crust. Credit: David A. Kring/Center for Lunar Science and Exploration.{{tlx|fairuse}}]] '''Def.''' a meteorite that is known to have originated on the Moon" is called a '''lunar meteorite''', or '''selenometeorite'''. The meteorite called Allan Hills 81005 resembled some rocks brought back from the Moon by the Apollo program.<ref name=Marvin>{{ cite journal | doi = 10.1029/GL010i009p00775 | author = U. B. Marvin | year = 1983 | title = The discovery and initial characterization of Allan Hills 81005: The first lunar meteorite | url = | journal = Geophys. Res. Lett. | volume = 10 | issue = | pages = 775–8 | bibcode=1983GeoRL..10..775M }}</ref> Yamato 791197 is another lunar meteorite. About 134 lunar meteorites have been discovered so far (as of October, 2010), perhaps representing more than 50 separate meteorite falls (i.e., many of the stones are "paired" fragments of the same meteoroid). The total mass is more than 46&nbsp;kg. Lunar origin is established by comparing the mineralogy, the chemical composition, and the isotopic composition between meteorites and samples from the Moon collected by Apollo missions. Cosmic ray exposure history established with noble gas measurements have shown that all lunar meteorites were ejected from the Moon in the past 20 million years. Most left the Moon in the past 100,000 years. All six of the Apollo missions on which samples were collected landed in the central nearside of the Moon, an area that has subsequently been shown to be geochemically anomalous by the Lunar Prospector mission. In contrast, the numerous lunar meteorites are likely to be random samples of the Moon and consequently provide a more representative sampling of the lunar surface than the Apollo samples. Half the lunar meteorites, for example, likely sample material from the farside of the Moon. So far seifertite has only been found in Martian<ref name=Goresy>{{ cite journal |doi=10.1127/0935-1221/2008/0020-1812 |title=Seifertite, a dense orthorhombic polymorph of silica from the Martian meteorites Shergotty and Zagami |year=2008 |last1=Goresy|first1=Ahmed El|last2=Dera|first2=Przemyslaw|last3=Sharp|first3=Thomas G.|last4=Prewitt|first4=Charles T.|last5=Chen|first5=Ming|last6=Dubrovinsky|first6=Leonid|last7=Wopenka|first7=Brigitte|last8=Boctor|first8=Nabil Z.|last9=Hemley|first9=Russell J. |journal=European Journal of Mineralogy |volume=20 |issue=4 |pages=523 |url=http://www.schweizerbart.de/resources/downloads/paper_previews/58172.pdf }}</ref><ref name=Dera>{{ cite journal |author=Dera P |author2=Prewitt C T |author3=Boctor N Z |author4=Hemley R J |journal=American Mineralogist |volume=87 |year=2002 |page=1018 |title=Characterization of a high-pressure phase of silica from the Martian meteorite Shergotty |url=http://rruff.geo.arizona.edu/AMS/authors/Boctor%20N%20Z }}</ref> and lunar meteorites.<ref name=Aoudjehane>{{ cite journal |url=http://www.uair.arizona.edu/objectviewer?o=uadc%3A%2F%2Fazu_maps%2FVolume43%2FNumberSupplement%2Fea83b7e4-bcbb-44c2-af9e-a14b6d5ebdd4 |title=First evidence of high-pressure silica: stishovite and seifertite in lunar meteorite Northwest Africa 4734 |author=H. Chennaoui Aoudjehane |author2=A. Jambon |journal=Meteoritics & Planetary Science |volume=43 |issue=7, Supplement |page= A32 |year=2008 }}</ref> "A felsite clast in lunar breccia Apollo sample 14321 [arrowed in the image on the left], which has been interpreted as Imbrium ejecta, has petrographic and chemical features that are consistent with formation conditions commonly assigned to both lunar and terrestrial environments. A simple model of Imbrium impact ejecta [...] indicates a pre-impact depth of 30–70 km, i.e. near the base of the lunar crust. Results from Secondary Ion Mass Spectrometry trace element analyses indicate that zircon grains recovered from this clast have positive Ce/Ce^⁎ anomalies corresponding to an oxygen fugacity +2 to +4 log units higher than that of the lunar mantle, with crystallization temperatures of 771 ± 88 to 810 ± 37 °C (2σ) that are unusually low for lunar magmas. Additionally, Ti-in-quartz and zircon calculations indicate a pressure of crystallization of 6.9 ± 1.2 kbar, corresponding to a depth of crystallization of 167 ± 27 km on the Moon, contradicting ejecta modelling results. Such low-''T'', high-''f''O₂, and high-P have not been observed for any other lunar clasts, are not known to exist on the Moon, and are broadly similar to those found in terrestrial magmas."<ref name=Bellucci>{{ cite journal |author=J. J. Bellucci |author2=A. A. Nemchin |author3=M. Grange |author4=K. L. Robinson |author5=G. Collins |author6=M. J. Whitehouse |author7=J. F. Snape |author8=M. D. Norman |author9=D. A. Kring |title= |journal=Earth and Planetary Science Letters |date=15 March 2019 |volume=510 |issue= |pages=173-185 |url=https://www.sciencedirect.com/science/article/pii/S0012821X19300202 |arxiv= |bibcode= |doi=10.1016/j.epsl.2019.01.010 |pmid= |accessdate=28 January 2019 }}</ref> "The terrestrial-like redox conditions inferred for the parental magma of these zircon grains and other accessory minerals in the felsite contrasts with the presence of Fe-metal, bulk clast geochemistry, and the Pb isotope composition of K-feldspar grains within the clast, all of which are consistent with a lunar origin."<ref name=Bellucci/> The "felsite and its zircon crystallized on Earth at a modest depth of 19 ± 3 km in the continental crust where oxidizing, low-''T'', fluid-rich conditions are common. Subsequently, the clast was ejected from the Earth during a large impact, entrained in the lunar regolith as a terrestrial meteorite with the evidence of reducing conditions introduced during its incorporation into the Imbrium ejecta and host breccia."<ref name=Bellucci/> {{clear}} ==Mars== {{main|Areiominerals}} [[Image:NWA 2373 shergottite, 6mm piece.jpg|thumb|right|250px|This is a small sample from the NWA 2373 Meteorite. Credit: [http://www.flickr.com/people/47445767@N05 James St. John].]] Roughly three-quarters of all Martian meteorites (Areiometeorites) can be classified as shergottites. "[T]he most frequent type of rock (basaltic lithologies) among all known Martian meteorites is the basaltic shergottites."<ref name=Cowan>{{ cite book |author=DA Cowan |title=Speaker Abstracts |url=http://scholar.google.com/scholar?as_q=shergottite&num=100&btnG=Search+Scholar&as_epq=dominant+group&as_oq=&as_eq=&as_occt=any&as_sauthors=&as_publication=&as_ylo=&as_yhi=&as_sdt=1.&as_sdtp=on&as_sdtf=&as_sdts=3&hl=en |bibcode= |doi= |pmid= |accessdate=2011-08-07 }}</ref> "The dominant group of Martian meteorites, shergottites, are divided into two subgroups consisting of basalts and lherzolites.<ref name=Mikouchi>{{ cite journal |author=Takashi Mikouchi |author2=Masamichi Miyamoto |title=Lherzolitic Martian meteorites Allan Hills 77005, Lewis Cliff 88516 and Yamato-793605: Major and minor element zoning in pyroxene and plagioclase glass |journal=Antarctic Meteorite Research |month=March |year=2000 |volume=13 |issue=3 |pages=256-69 |url= |bibcode=2000AMR....13..256M |doi= |pmid= |accessdate=2011-08-07 }}</ref> Almost 100 rocks are known that demonstrably come from the Planet Mars. Meteorite researchers and collectors generally refer to the Martian rocks as the SNC meteorites - the shergottites, the nakhlites, and the chassignites. Most of these Martian rocks are shergottites. Shergottites are a group of Martian rocks named after the Shergotty Meteorite, the type example. The Shergotty Meteorite is a shergottite that was found and identified in 2004. The first image at right shows a small sample 6 mm from the NWA 2373 Meteorite (NWA = &quot;Northwest Africa&quot;). The light brown-colored material is the outer weathered surface of the rock. The greenish and black speckled surface shows the crystal &amp; mineral make-up of the rock itself. Mineral analysis performed by Theodore Bunch and James Wittke at Northern Arizona University has shown that NWA 2373 is composed principally of olivine, pigeonite &amp; augite pyroxene, plagioclase feldspar glass (maskelynite), chromite, Ti-magnetite, chlorapatite, and trace amounts of other minerals. It looks like an ultramafic rock, but it's apparently a basaltic shergottite (also regarded as a picritic shergottite). NWA 2373 is reportedly paired with the NWA 1068 Meteorite. Available isotopic dates on the NWA 1068 Meteorite show it formed 185 million years ago (late Amazonian, equivalent to Earth's Early Jurassic), and was ejected from the Martian surface about 2.2 million years ago (information based on cosmogenic isotope analysis). Very light snow is known to occur at high latitudes on Mars.<ref name=Minard>{{ cite book |url=http://news.nationalgeographic.com/news/2009/07/090702-snow-mars-phoenix.html |title="Diamond Dust" Snow Falls Nightly on Mars |author=Anne Minard |date=2009-07-02 |publisher=National Geographic News }}</ref> {{clear}} ==Asteroids== {{main|Rocks/Rocky objects/Asteroids|Astrominerals}} [[Image:Cumberland Falls meteorite.jpg|thumb|right|250px|This is an image of the Cumberland Falls meteorite which is considered to be an asteroidal achondrite. Credit: [http://flickr.com/people/8435962@N06 Claire H.].]] Aubrites are a group of meteorites that are primarily composed of the orthopyroxene enstatite, and are often called enstatite achondrites. Their igneous origin separates them from primitive enstatite achondrites and means they originated in an asteroid. Aubrites are typically light-colored, and with a brownish fusion crust. Most aubrites are heavily brecciated. Aubrites are primarily composed of large white crystals of the Fe-poor, Mg-rich orthopyroxene, or enstatite. Around this matrix, they have minor phases of olivine, nickel-iron metal, troilite, which indicate a magmatic formation under extremely reducing conditions. The severe brecciation of most aubrites attests to a violent history for their parent body. Since some aubrites contain chondritic xenoliths it is likely that the aubrite parent body collided with an asteroid of “F-chondritic” composition. Comparisons of aubrite spectra to the spectra of asteroids have revealed striking similarities between the aubrite group and the main belt Nysian asteroid family. A small member of this asteroid family, 3103 Eger, exhibits a near-Earth orbit, and is very likely the parent body of the aubrites. "The recent investigation of the orbital distribution of Centaurs (Emel’yanenko et al., 2005) showed that there are two dynamically distinct classes of Centaurs, a dominant group with semimajor axes a > 60 AU and a minority group with a < 60 AU."<ref name=Emelyanenko>{{ cite journal |author=V. V. Emel’yanenko |title=Structure and dynamics of the Centaur population: constraints on the origin of short-period comets |journal=Earth, Moon, and Planets |month=December |year=2005 |volume=97 |issue=3-4 |pages=341-51 |url=http://dccm.susu.ac.ru/acm2005.pdf |bibcode= |doi=10.1007/s11038-006-9095-5 |pmid= |accessdate=2011-10-06 }}</ref> "[T]he intrinsic number of such objects is roughly an order of magnitude greater than that for a<60 AU"<ref name=Emelyanenko/>. "From the dominant group, the asteroids evolve to intersect the Earth's orbit on a median time scale of about 60 Myr."<ref name=Michel>{{ cite journal |author=Patrick Michel |author2=Fabbio Migliorini |author3=Alessandro Morbidelli |author4=Vincenzo Zappalà |title=The Population of Mars-Crossers: Classification and Dynamical Evolution |journal=Icarus |month=June |year=2000 |volume=145 |issue=2 |pages=332-47 |url=http://www.obs-nice.fr/morby/papers/6358a.pdf |bibcode= |doi=10.1006/icar.2000.6358 |pmid= |accessdate=2011-10-06 }}</ref> "The MB group is the most numerous group of MCs. ... 50 % of the MB Mars-crossers [MCs] become ECs within 59.9 Myr and [this] contribution ... dominates the production of ECs"<ref name= Michel />. MB denotes the main belt of asteroids.<ref name= Michel /> EC denotes Earth-crossing.<ref name= Michel /> {{clear}} ==Diameters== [[Image:Beach Stones 2.jpg|thumb|right|250px|These are pebbles on a beach. Credit: [[commons:User:Slomox|Slomox]].]] [[Image:Boulder.JPG|thumb|right|250px|This image shows a rock apparently where it fell. Credit: [[commons:User:Sten Porse|Sten Porse]].]] '''Def.''' a particle classification system based on diameter is called the '''Wentworth scale'''. '''Def.''' a particle less than 1 micron in diameter is called a '''colloid'''. '''Def.''' a particle less than 3.9 microns in diameter is called a '''clay'''. '''Def.''' a particle from 3.9 to 62.5 microns in diameter called a '''silt'''. '''Def.''' a particle less than 62.5 microns in diameter is called a '''mud'''. '''Def.''' a particle from 62.5 microns to 2 mm in diameter is called a '''sand'''. '''Def.''' a particle from 2 to 64 mm in diameter is called a '''gravel'''. '''Def.''' a particle from 2 to 4 mm in diameter is called a '''granule'''. '''Def.''' a particle from 4 to 64 mm in diameter is called a '''pebble'''. '''Def.''' a particle from 64 to 256 mm in diameter is called a '''cobble'''. '''Def.''' a particle [or large piece of stone] greater than 256 mm in diameter that can theoretically be moved if enough force is applied is called a '''boulder'''. {{clear}} ==Natural sciences== [[Image:SEUtahStrat.JPG|thumb|right|250px|The picture shows an approximately angled slice through a small portion of the Earth's crust. It is from Glen Canyon National Recreation Area, Utah. Credit: [[w:User:Qfl247|Qfl247]].]] [[Image:Geology of Cyprus-Chalk.jpg|thumb|right|250px|This image is from a road cut through the Earth's crust on the island of Cyprus. Credit: [[w:User:MeanStreets|MeanStreets]].]] [[Image:Esquel pallasite partial slice.jpg|thumb|right|250px|This is a partial slice of the Esquel (meteorite) discovered in Esquel, Chubut Province, Argentina. Credit: [http://www.flickr.com/people/49698777@N02 M. Rehemtulla for the QUOI Media Group].]] [[Image:LvMS-Lvm.jpg|thumb|left|250px|The photomicrographs show of a sand grain held in an amorphous matrix, in plane-polarized light on top, cross-polarized light on bottom. Scale box in mm. Credit: [[w:User:Qfl247|Qfl247]].]] [[Image:Gabbro pmg ss 2006.jpg|thumb|left|250px|This is a photomicrograph of a thin section of gabbro. Credit: [[commons:User:Siim Sepp|Siim Sepp]].]] [[Image:CarmelOoids.jpg|thumb|250px|This photomicrograph is of a thin section of a limestone with ooids. The largest is approximately 1.2 mm in diameter. Credit: Photograph taken by Mark A. Wilson (Department of Geology, The College of Wooster).]] [[Image:820qtz.jpg|thumb|left|250px|This is a thin section with cross-polarized light through a sand-sized quartz grain of 0.13 mm diameter. Credit: Glen A. Izett, USGS.]] [[Image:Suvasvesi shocked quartz.jpg|thumb|right|250px|This is a thin section of a shocked quartz grain. Credit: Martin Schmieder.]] '''Def.''' the study of the origin, composition and structure of rock is called '''petrology'''. Each rock has a location and an environment. These are recorded. Sometimes a sequence of events is connectable to a rock in a location. '''Def.''' the scientific description and classification of rocks is called '''petrography'''. '''Def.''' a section formed by a plane cutting through an object, usually at right angles to an axis is called a '''cross section'''. '''Def.''' a laboratory preparation of a rock, mineral, soil, pottery, bones, or metal sample for use with a polarizing petrographic microscope, electron microscope and electron microprobe is called a '''thin section'''. At lower left is a thin section through a sand-sized quartz grain "from the USGS-NASA Langley core showing two well-developed, intersecting sets of shock lamellae produced by the late Eocene Chesapeake Bay bolide impact. This shocked quartz grain is from the upper part of the crater-fill deposits at a depth of 820.6 ft in the core. The corehole is located at the NASA Langley Research Center, Hampton, VA, near the southwestern margin of the Chesapeake Bay impact crater."<ref name=Izett>{{ cite book |author=Glen A. Izett |title=Shocked Quartz from the USGS -- NASA Langley Core |publisher=U. S. Geological Survey |location= |date=September 26, 2000 |url=http://geology.er.usgs.gov/eespteam/crater/shockquartz.html |accessdate=2012-10-23 }}</ref> "Very high pressures produced by strong shock waves cause dislocations in the crystal structure of quartz grains along preferred orientations. These dislocations appear as sets of parallel lamellae in the quartz when viewed with a petrographic microscope. Bolide impacts are the only natural process known to produce shock lamellae in quartz grains."<ref name=Izett/> Lower right shows another thin section in plane polarized light of a shocked quartz grain with two sets of decorated planar deformation features (PDFs) surrounded by a cryptocrystalline matrix from the Suvasvesi South impact structure, Finland. In a specimen of shocked quartz, stishovite can be separated from quartz by applying hydrogen fluoride (HF); unlike quartz, stishovite will not react.<ref name=Fleischer>{{ cite journal |author=Michael Fleischer |year=1962 |title=New mineral names |journal=American Mineralogist |volume=47 |issue=2 |pages=172–4 |publisher=Mineralogical Society of America |doi= |url=http://rruff.info/uploads/AM47_805.pdf |accessdate= |format=PDF }}</ref> Minute amounts of stishovite has been found within diamonds<ref name=Wirth>{{ cite journal |doi=10.1016/j.epsl.2007.04.041 |title=Inclusions of nanocrystalline hydrous aluminium silicate "Phase Egg" in superdeep diamonds from Juina (Mato Grosso State, Brazil) |year=2007 |author=R Wirth |author2=C. Vollmer |author3=F. Brenker |author4=S. Matsyuk |author5=F. Kaminsky |journal=Earth and Planetary Science Letters |volume=259 |pages=384 |bibcode=2007E&PSL.259..384W |issue=3–4 }}</ref>. The major evidence for a volcanic origin for tektites "includes: close analogy between shaped tektites and small volcanic bombs, and between layered tektites and lava or tuff-lava flows or huge bombs; analogy between flanged tektites and volcanic bombs ablated by gasjets: long-time, multistage formation of some tektites that corresponds to wide variations in their radiometric ages; well-ordered long compositional trends (series) typical of magmatic differentiation; different compositional tektite families (subseries) comparable to different stages (phases) of the volcanic process."<ref name=Izokh>{{ cite journal |author=EP Izokh |title=Origin of tektites: an alternative to terrestrial impact theory |journal=Chemie der Erde : Beitrage zur Chemischen Mineralogie, Petrographie und Geologie |month= |year=1996 |volume=56 |issue= |pages=458-74 |url=http://ukpmc.ac.uk/abstract/MED/11541098 |arxiv= |bibcode= |doi= |pmid=11541098 |accessdate=2012-10-23 }}</ref> "As with the North American microtektite-bearing cores, all the Australasian microtektite-bearing cores containing coesite and shocked quartz also contained volcanic ash, which complicated the search."<ref name=Glass>{{ cite journal |author=B. P. Glass |author2=Jiquan Wu |title=Coesite and shocked quartz discovered in the, Australasian and North American, microtektite layers |journal=Geology |month=May |year=1993 |volume=21 |issue=5 |pages=435-8 |url=http://geology.geoscienceworld.org/content/21/5/435.short |arxiv= |bibcode= |doi=10.1130/0091-7613(1993)021<0435:CASQDI>2.3.CO;2 |pmid= |accessdate=2012-10-23 }}</ref> {{clear}} ==Hypotheses== {{main|Hypotheses}} # Passage through the Earth's magnetic field and natural electric field of magnetic meteors may cause deflection and a slowing down during flight such that collision with the Earth does not create a crater. ==See also== {{div col|colwidth=20em}} * [[Radiation astronomy/Alloys|Alloys]] * [[Rocks/Rocky objects/Ceres|Ceres]] * [[Radiation astronomy/Craters|Crater astronomy]] * [[Liquids/Liquid objects/Meteorites|Liquid meteorites]] * [[Rocks/Rocky objects/Mercury|Mercury]] * [[Radiation/Meteors|Meteor astronomy]] * [[Rocks/Meteorites/Laboratory|Meteorite laboratory]] * [[Liquids/Liquid objects/Moon|Moon]] * [[Planetary science]] * [[Liquids/Liquid objects/Venus|Venus]] {{Div col end}} ==References== {{reflist|2}} ==Further reading== * {{ cite book | url = http://books.google.com/?id=QDU7AAAAIAAJ&pg=PA152 | page =152 | title = Planet earth: cosmology, geology, and the evolution of life and environment | author = Cesare Emiliani | publisher = Cambridge University Press | date = 1992 | isbn = 978-0-521-40949-0 | chapter = Meteorites }} ==External links== * [http://www.iau.org/ International Astronomical Union] * [http://nedwww.ipac.caltech.edu/ NASA/IPAC Extragalactic Database - NED] * [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center] * [http://www.osti.gov/ Office of Scientific & Technical Information] * [http://www.ncbi.nlm.nih.gov/pccompound PubChem Public Chemical Database] * [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System] * [http://www.scirus.com/srsapp/advanced/index.jsp?q1= Scirus for scientific information only advanced search] * [http://cas.sdss.org/astrodr6/en/tools/quicklook/quickobj.asp SDSS Quick Look tool: SkyServer] * [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database] * [http://simbad.harvard.edu/simbad/ SIMBAD Web interface, Harvard alternate] * [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA.] * [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl Universal coordinate converter] <!-- footer templates --> {{tlx|Radiation astronomy resources}}{{tlx|Chemistry resources}}{{tlx|Geology resources}}{{tlx|History of science resources}}{{Principles of radiation astronomy}}{{Sisterlinks|Meteorites}} <!-- categories --> [[Category:Geology resources]] [[Category:History of Sciences/Resources]] [[Category:Radiation astronomy/Resources]] [[Category:Rocks resources]] [[Category:Meteorites]] j5q3he4vpb4clrsh3n0psx760a835r0 0 139027 2690295 1659483 2024-12-04T19:11:41Z 69.85.234.247 /* Diagnosis */ 2690295 wikitext text/x-wiki This page is written for those taking the ASE A4 test * Always mark the placement of any part before removal ==Steering== Always disable the [[../Air Bag/]] system before any change to the steering column. [[../Steering Columns/]] should be locked into place to preserve timing and prevent overextending the clock-spring. Replace [[../Bushings/]] when they no longer hold the part in position. Replace [[../Bearings/]] when they have any visible play, or have a turning torque outside of specification. Adjustments can affect front wheel toe ===Diagnosis=== Bump steer can be caused by * Loose idler arm * Loose rack mountings * Loose or damaged mounting Steering wheel free play can be caused by *Worn couplings *worn u joint *Loose mounting Steering problems are sometimes subframe alignment problems ====Manual Steering Gears==== Also known as worm gears Noises, binding, looseness, hard steering point to bearing and lubrication problems. ====Rack and Pinion Steering Gear==== ====Other parts==== Air bag Steering wheel Clock spring Worm gear Idler arm Rack and pinion Tie rods ==Power Steering Units== Power steering uses hydraulic pressure to apply force to the worm gear or rack === Diagnosis === Weeping, or a slight fluid leak requires repair or replacement * inspect power steering fluid * Remove and replace power steering pump belt, tensioner, pulley * diagnose power steering pump problems * Remove and replace power steering pump * power steering pressure testio * Remove and replace power steering hydraulics * Steering gear worm bearing preload and sector lash * Remove and replace steering gear seals and gaskets [[File:RecirculatingBall.png|thumb]] * Remove and replace rack and pinion unit * Remove and replace rack and pinion steering gear bellows/boots. * flush, fill, bleed power steering hydraulics * Diagnose, Remove and replace variable-assist steering systems ;[[w:Rack and pinion|Rack and Pinion Steering]]:can provide assistance by applying pressure to one side ==Steering Linkage== * Inspect and adjust steering linkage geometry [[File:Ackermann.svg|thumb]] ====Pitman arm==== Pitman arm is only bent in collisions, and wears on other parts. These are pressed on, and are difficult to separate from the splines. ====Center link==== If bent, will affect front wheel toe ====Idler arm==== * Remove and Replace tie rods * Remove and Replace steering linkage damper Remove and Replace ==Suspension== [[File:Suspension.jpg|thumb]] <!-- [[File:Automotive suspension comparison.svg|thumb]] --> ==Front Suspension== ==Rear Suspension== ==Alignment== ==Frame== ;[[Wikipedia:Frame (vehicle)|Frame]] ===Springs=== You should understand spring frequency, even if you never measure the frequency of the spring. Most spring are made of a chrome-vanadium [[alloy]]. Very tough stuff. ;[[w:Leaf spring|Leaf spring]] ;[[w:Coil spring|Coil spring]] ;[[w:Torsion bar suspension|Torsion bars]] ;[[w:Constant-velocity joint|Constant-velocity joint]]:Usable in torque transfer ===Suspension=== ;[[w:Kingpin (automotive part)|Kingpin]] ;[[w:MacPherson strut|MacPherson Struts]] ;[[w:Short long arms suspension|Short/Long Arm]] == ASE Task List== ==Glossary== ;Aspect ratio ;Bead:The lip of the tire that seals the rim flange ;Belt:Material under the tread ;Center section AKA Spider:Center of a steel wheel ;Conicity ;DOT tire code ;E-metric tire ;High flotation tire ;Inner liner ;Load index ;Lug nuts ;Major splice ;Offset ;Ply steer ;Rim width ;Run-flat ;Sidewall ;Speed rating ;TPC ;[[w:Tread|Tread]] ;[[w:Unsprung mass|Unsprung Weight]]:Weight below the spring in a vehicle ;[[w:UTQG|Uniform Tire Quality Grading System (UTQG)]]:Tires sold in the USA are rated in Tread wear, Traction, and Temperature Resistance as per [[w:National Highway Traffic Safety Administration|NHTSA]]. ;[[w:Wear bar|Wear bar]]s:raised features located at the bottom of the tread grooves that indicate the tire has reached its wear limit [[Automotive Diagnostics]] [[Category:Automotive engineering]] [[Category:Automotive Technology|{{SUBPAGENAME}}]] ajnz3twqm02wx2o1mujhxb5fe5pcxi7 0 139384 2690272 2690076 2024-12-04T15:00:05Z Young1lim 21186 /* Adder */ 2690272 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20211108.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20221130.pdf|A]]|| || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20241204.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] htrnfjitpwlblz8q1wlzuexpnn3pzav 1 158983 2690344 2545827 2024-12-05T05:58:03Z Piotrus 571 /* Inconsistent capitalization of titles */ new section 2690344 wikitext text/x-wiki {{WikiJournal_discussions|Tubal Pregnancy with embryo (crop2).jpg}} {{Archive box| *[[/2014-2019|2014–2019]] *[[Talk:WikiJournal of Medicine/Open tasks and discussions|Open tasks and discussions]] Discussions may also take place at the <br>'''[https://groups.google.com/forum/#!forum/{{WikiJXyz}}/join public mailing list]''' }}[[Category:WikiJournal of Medicine]] == SHERPA/RoMEO == I've submitted to the details for WikiJMed to SHERPA/RoMEO via the [http://sherpa.ac.uk/forms/new-journal.php journal submission form]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:50, 2 May 2019 (UTC) :Great! [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 15:04, 10 June 2019 (UTC) == Consensus Report on Reproducibility and Replicability in Science (National Academies of Sciences, Engineering, and Medicine) == National Academies of Sciences, Engineering. ''Reproducibility and Replicability in Science'', 2019. https://doi.org/10.17226/25303. Below is a brief summary from the [https://www.psychologicalscience.org/ Association for Psychological Science] (APS). The National Academies pre-publication full report is available as a [https://cart.nap.edu/cart/cart.php?list=fs&action=buy%20it&record_id=25303&isbn=0-309-48613-0 print book], [https://www.nap.edu/download/25303 PDF], or to [https://www.nap.edu/read/25303 read online]. === Brief Summary === The National Academies of Sciences, Engineering, and Medicine (NASEM) has released a consensus report on reproducibility and replicability in science. The report defines key terms, examines the state of reproducibility and replicability in science, and reviews current activities aimed at strengthening the reliability of the scientific enterprise. Reproducibility and Replicability in Science, funded by the National Science Foundation and the Alfred P. Sloan Foundation, concludes a thorough process that spanned more than a year. The report was authored by a multidisciplinary committee including APS William James Fellow Timothy Wilson (University of Virginia) and APS Fellow Wendy Wood (University of Southern California). Recognizing that different fields use the same terms in different ways, the report established clear definitions of reproducibility and replicability. The report defines reproducibility as “achieving consistent results using the same input data, computational steps, methods, code, and conditions of analysis as prior studies—known as computational reproducibility within some fields.” Replicability is defined as “obtaining consistent results across studies that are aimed at answering the same scientific question but have obtained independent data.” The report also assesses the current state of reproducibility and replicability in science. “There is no crisis, but also no time for complacency,” said the chair of the committee, physician Harvey Fineberg, in an event marking the public release of the report. The committee concludes that efforts are needed to strengthen both reproducibility and replicability in science, recognizing that these aspects are important but not always easy to attain. Given that replicability of individual studies can vary, the report notes, integrating multiple channels of evidence from a variety of studies is essential to understanding the reliability of scientific knowledge. The study also provides suggestions for how reproducibility and replication can be improved. The report makes a variety of recommendations for scientists and researchers in presenting their research findings, suggesting that they: * Convey clear information about computational methods and data products that support published reports * Provide accurate and appropriate characterization of relevant uncertainties when they report research findings * Provide a complete description of how a reported result was reached * Avoid overstating the implications of research findings and exercise caution in their review of research-related press releases * The report also includes recommendations for universities, science funders, journalists, policymakers, and other stakeholders; it also discusses how concerns about reproducibility and replicability might have the potential to affect how the public views the scientific enterprise. To read the new National Academies report ''Reproducibility and Replicability in Science'', [https://www.nap.edu/catalog/25303/reproducibility-and-replicability-in-science click here]. [[User:Markworthen|<span style="color:#539; font-family:copperplate gothic">&nbsp; - Mark D Worthen PsyD</span>]] [[User talk:Markworthen|<span style="color:#64B; font-family:times new roman">(talk)</span>]] 14:58, 10 May 2019 (UTC) == BASE == The journal is [https://www.base-search.net/Record/7016ef9358ef46e3836d87b198f000d7bbeb7b97c280f6456b99567e5b4e44e9/ now indexed in BASE] via DOAJ. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 06:05, 7 June 2019 (UTC) :Great! [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 15:04, 10 June 2019 (UTC) == Board member re-elections == As per the [[WikiJournal of Medicine/Bylaws#Section 5. Duration of Term|bylaws of WikiJMed]] "Editorial Board Members shall serve four-year terms. There is no limit to the number of terms any individual Editorial Board Member may serve." In the [[WikiJournal of Medicine/Editors|editorial board of WikiJMed]], there are 2 members at the end of their terms: [[User:CFCF|Carl Fredrik Sjöland]] and [[User:Taketa|Mike Nicolaije]]. If you would like to extend your terms, we've previously simply used the same system as applications. I suggest doing this at [[Talk:WikiJournal_of_Medicine/Editors]] as the logical location (e.g. [https://en.wikiversity.org/w/index.php?title=Talk:WikiJournal_of_Medicine/Editors&action=edit&section=new&preload=WikiJournal_of_Medicine%2FEditorial_board%2FApplication&summary=Editorial+board+application using this link]) [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:24, 14 June 2019 (UTC) :Dear T.Shafee and all, :thank you for the notice. I am not applying for a new term. I enjoyed my time on the board, with as a highlight Wikimania 2016. For the moment I would like to work on other wikiprojects. :All the best, [[User:Taketa|Taketa]] ([[User talk:Taketa|discuss]] • [[Special:Contributions/Taketa|contribs]]) 14:48, 14 June 2019 (UTC) == Dyslexia article == I had some concerns about the [[WikiJournal Preprints/Dyslexia|Dyslexia article]]. The authors responded promptly, politely, and professionally to the [[Talk:WikiJournal Preprints/Dyslexia|concerns I (and others) posted]]. I can't remember the precise context, but a couple of months ago I had planned to review the article mainly for [[w:copy editing|copy editing]], but also to make sure statements were adequately supported by their cited references. Unfortunately, time constraints resulted in my failure to follow through on that commitment. At this point I don't think it's fair to the authors to drag out the review any longer. Thus, if the editors for the article—Eric Youngstrom and Jitendra Kumar Sinha—decide the article is "good to go", i.e., ready to move to [[WikiJournal User Group/Potential upcoming articles|Stages 6 and 7]], then I support whatever decision the editors make. Part of my reasoning is that if I or anyone else discovers problems with grammar, syntax, etc. (or citations not supporting a statement) then we can judiciously edit [[w:Dyslexia|the Wikipedia article]]. Plus, it's a [[w:good article|good article]] per Wikipedia standards, and it has received extensive review. ''(I also posted what I write here to the listserv.)'' Thanks! Mark [[User:Markworthen|<span style="color:#539; font-family:copperplate gothic">&nbsp; - Mark D Worthen PsyD</span>]] [[User talk:Markworthen|<span style="color:#64B; font-family:times new roman">(talk)</span>]] 21:27, 6 August 2019 (UTC) :Note that [[User:Eyoungstrom]] stated on the mailing list that they will be doing the final proofread in the next 2 weeks. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:11, 25 August 2019 (UTC) == Bylaws == Hi. Just wanted to point out the following confusing sections - I think there are a few words missing (my suggested additions are underlined): * Article III Section 1 *: "{{highlight|(a) The}} voting procedures in ARTICLE IV apply to:" - I suggest removing the "(a)", since this isn't part of the list, but merely introducing the list *: "(e) Amendment of these bylaws as specified in ARTICLE {{highlight|IX<u>.</u>}}" - all the other entries have periods at the end * Article III Section 2 *: The (a)(b)(c)... suggest that voters meet ''one of'' the listed qualifications, but (g) says "Not an individual voting for herself/himself" - this suggests that anyone may vote, but only those that meet a different qualification can vote for themselves. I suggest explicitly stating that these are "or" qualifications, but that (g) is ''in addition to'' the other requirements * Article VIII Section 2 *: "The property {{highlight|of <u>Wiki.J.Med.</u> is}} irrevocably dedicated to charitable purposes and no part of the funds allotted by WikiJournal {{highlight|to <u>Wiki.J.Med.</u> shall}} ever inure to the benefit of any Editorial Board Member or to the benefit of any private individual other than compensation in a reasonable amount to its contractors for services rendered. * Article VIII Section 3 *: Upon the dissolution or winding-up of Wiki.J.Med., the resultant assets remaining after payment, or provision for payment, of all debts and liabilities {{highlight|of <u>Wiki.J.Med.</u> shall}} be distributed to WikiJournal. If this is not possible, the resultant assets shall be distributed to Wikimedia Foundation. * Article VIII Section 4 *: "No loans shall be contracted on behalf of {{highlight|the <u>Wiki.J.Med.</u> and}} no evidence of indebtedness shall be issued in its name unless authorized by a resolution of the Editorial Board." Thanks, --[[User:DannyS712|DannyS712]] ([[User talk:DannyS712|discuss]] • [[Special:Contributions/DannyS712|contribs]]) 00:43, 7 August 2019 (UTC) Thank you, [[User:DannyS712|DannyS712]], for pointing these out! I'm not sure they warrant a vote on a bylaws change right now, but I've added them to [[WikiJournal User Group/Bylaws/Proposed changes]], so that they will be accounted for in the next update. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 10:34, 1 February 2020 (UTC) ==PMC aplication== {{cot}} {{:WikiJournal_of_Medicine/Applications/PubMed_Central}} {{cob}} :Greetings [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]. If the next application to PMC is accepted, will all of the articles published in previous years also be indexed in PubMed/PMC? Thanks. [[User:Biosthmors|Biosthmors]] ([[User talk:Biosthmors|discuss]] • [[Special:Contributions/Biosthmors|contribs]]) 14:59, 7 March 2021 (UTC) ::@[[User:Biosthmors|Biosthmors]]. I believe that we submit back-issues up to two years to them ([https://www.ncbi.nlm.nih.gov/pmc/about/guidelines/#backcontent relevant policy]). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 03:22, 8 March 2021 (UTC) == Twitter share button code == This is not a major problem, but when someone can get to it ... The Twitter share button code for the recent (and very good!) Hepatitis D article has the old Twitter handle (@WiJouMed) in the code. That just needs to be changed to @WikiJMed. Thanks! [[User:Markworthen|<span style="color:#539; font-family:copperplate gothic">&nbsp; - Mark D Worthen PsyD</span>]] [[User talk:Markworthen|<span style="color:#64B; font-family:times new roman">(talk)</span>]] 14:04, 30 March 2020 (UTC) :Done! Luckily an easy fix to {{tlx|share}}. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 00:49, 31 March 2020 (UTC) == SCOPUS == WikiJMed to be indexed in SCOPUS. You can see the application and process [[WikiJournal of Medicine/Applications/SCOPUS|here]]. Also announced variously on [https://twitter.com/WikiJMed/status/1273595131975909377 twitter] and [https://www.facebook.com/WikiJMed/photos/a.2448190831929392/3102687373146398 FB]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 01:34, 20 June 2020 (UTC) :''WikiLMed'' is now indexed in Scopus with an [https://www.scopus.com/sourceid/21101024226 ID 21101024226]. It will be good to link on the journal home page. [[User:Chhandama|Chhandama]] ([[User talk:Chhandama|discuss]] • [[Special:Contributions/Chhandama|contribs]]) 05:05, 20 October 2021 (UTC) ::@[[User:Chhandama|Chhandama]]: Good suggestion. I also took the opportunity to do a few additional layout updates [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 05:44, 23 October 2021 (UTC) == Add "Quality prose" to Author guidelines == I highly recommend that we add a new "Quality prose" section to the Author guidelines (for both Research and Review articles). Here is a draft of such a section: === Quality prose === Manuscripts submitted to the WikiJMed should exhibit clear, correct, concise, comprehensible, and consistent writing. Articles should "say what they mean and mean what they say."<ref>{{Cite journal|date=2021-01-08|title=Wikipedia:WikiProject Guild of Copy Editors|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:WikiProject_Guild_of_Copy_Editors&oldid=999038911|journal=Wikipedia|language=en}}</ref> As the ''Publication Manual of the American Psychological Association'' states: <blockquote>The main objective of scholarly writing is clear communication, which can be achieved by presenting ideas in an orderly and concise manner. ... Precise, clear word choice and sentence structure also contribute to the creation of a substantive, impactful work.<ref>''Publication Manual of the American Psychological Association'', 7th ed., (Washington, D.C.: American Psychological Association, 2020), 111.</ref></blockquote> Although WikiJMed editors will carefully review manuscripts for quality prose, we do not provide a copy editing service. In other words, submit a manuscript only after you ''know'' that your article exhibits pithy prose.<ref>''Webster's Third New International Dictionary of the English Language, Unabridged'', ed. Philip B. Gove (Springfield, MA: G. & C. Merriam, 1961, 1993, periodically updated as Merriam-Webster Unabridged), s.v. "[https://unabridged.merriam-webster.com/unabridged/pithy pithy]", ("pithy ''adjective'' ... 2 : a : containing much meaning and substance in a terse concentrated form : brief and to the point : full of significance : meaty").</ref> How do you know that your prose passes muster? Ask one or two colleagues known for writing well to review your manuscript. Also seriously consider hiring a professional copy editor to review your manuscript and offer recommendations.<ref>Search results for "professional copy editor": [[google:professional+copy+editor|Google]] | [https://www.bing.com/search?q=professional+copy+editor Bing] | [https://duckduckgo.com/?q=professional+copy+editor DuckDuckGo]</ref> Here are some recommended writing resources to help you write articles that make a difference. === WRITING RESOURCES === ==== Writing resources: Books ==== Garner, Bryan A. ''Garner's Modern English Usage''. 4th ed. New York: Oxford University Press, 2016. Stein, Sol. ''Stein on Writing''. New York: St. Martin's Press, 1995. Strunk, William Jr., and E. B. White. ''The Elements of Style''. 4th ed. New York: Longman, 1999. Zinsser, William. ''On Writing Well''. 7th ed., rev.. New York: Harper Collins, 2006. ==== Writing resources: Online writing labs ==== Purdue University. ''Purdue Online Writing Lab'' ("Purdue OWL"). https://owl.purdue.edu/owl/purdue_owl.html University of North Carolina at Chapel Hill. "Tips & Tools." ''The Writing Center''. https://writingcenter.unc.edu/tips-and-tools/ ==== Writing resources: Wikipedia ==== '''[[w:WP:COPYEDIT|Basic copyediting]]''' '''[[w:WP:Writing better articles#Use clear, precise and accurate terms|Use clear, precise and accurate terms]]''' '''[[w:WP:REFERS|Use of "refers to"]]''' and related phrases such as "relates to". ====Writing resources: Dictionaries==== ===== Dictionaries: Free online ===== ''American Heritage Dictionary of the English Language Online''. https://ahdictionary.com/ . COMMENT: The best for pithy definitions. ''Merriam-Webster.com Dictionary''. https://www.merriam-webster.com/ . COMMENT: Solid, reliable definitions. ''Oxford Languages'' via Google. Search Google for the word or, if you do not see a definition right away, search for the word + "definition". COMMENT: Fast & reliable. Not as comprehensive as Merriam-Webster. Not as concise as American Heritage. ===== Dictionaries: Subscription-based online ===== ''Oxford English Dictionary'' (OED Online). https://www.oed.com/ . COMMENT: The best for etymology; eloquent. ''Webster's Third New International Dictionary of the English Language, Unabridged'', ed. Philip B. Gove (Springfield, MA: G. & C. Merriam, 1961, 1993, periodically updated as ''Merriam-Webster Unabridged''), https://unabridged.merriam-webster.com/unabridged/ . COMMENT: Exquisitely written and comprehensive. ===== Dictionaries: Print books ===== ''American Heritage Dictionary of the English Language''. 5th ed., rev. Boston: Houghton Mifflin Harcourt, 2018. (Usually marketed as "50th Anniversary edition.") ''Webster's Third New International Dictionary of the English Language, Unabridged''. Edited by Philip B. Gove. Springfield, MA: Merriam-Webster, 1961, rev. 1993. ===== Dictionaries: General comment ===== There are other good dictionaries. Find two or three you prefer by comparing definitions and related material over time. === References === {{reflist-talk}} Thank you for considering my recommendation. [[User:Markworthen|<span style="color:#539; font-family:copperplate gothic">&nbsp; - Mark D Worthen PsyD</span>]] [[User talk:Markworthen|<span style="color:#64B; font-family:times new roman">(talk)</span>]] 07:47, 7 March 2021 (UTC) :I agree with adding a quality prose / readability guideline to [[WikiJournal_of_Medicine/Publishing#General_guidelines]] (or as its own section). I also think it's worth including something similar for the sister journals, since it's pretty broadly relevant, so I'll also format up something to add to the central [[WikiJournal_User_Group/Publishing]] page. The specific resources might be collapsed, linked out to, or footnoted so as not to make it too long, but make sure the material is available. I'm also a big fan of [https://www.americanscientist.org/blog/the-long-view/the-science-of-scientific-writing Gopen & Swan's 'The Science of Scientific Writing'], which I was introduced to when writing my thesis. We ideally want these sorts of issues dealt with by the authors earlier in the process than later. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 10:38, 11 March 2021 (UTC) ::I agree with everything you wrote. :0) [[User:Markworthen|<span style="color:#539; font-family:copperplate gothic">&nbsp; - Mark D Worthen PsyD</span>]] [[User talk:Markworthen|<span style="color:#64B; font-family:times new roman">(talk)</span>]] 17:00, 7 April 2021 (UTC) == Format of abstracts == Is it allowed to change the format of [[WikiJournal of Medicine/Does the packaging of health information affect the assessment of its reliability? A randomized controlled trial protocol]] or [[WikiJournal of Medicine/Viewer interaction with YouTube videos about hysterectomy recovery]] to the format of [[WikiJournal of Medicine/Comparison between the Lund-Browder chart and the BurnCase 3D® for consistency in estimating total body surface area burned]], because of the missing visual effect on [[WikiJournal of Medicine|the front page]] (missing ":")? [[User:Habitator terrae|Habitator terrae]] ([[User talk:Habitator terrae|discuss]] • [[Special:Contributions/Habitator terrae|contribs]]) 21:15, 23 June 2021 (UTC) :@[[User:Habitator terrae|Habitator terrae]] Ah, I see what you mean. When the front page strips out line returns to save space, the abstract section indicators become unclear. I think you're right that those two should be formatted to add colons (and indeed that should be the standard format going forward). Since it doesn't change the meaning of the content, it's fine to make the change directly. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 03:37, 24 June 2021 (UTC) == Wikipedia integrated == What does it mean Wikipedia integrated? I don't see any explanation, what does it mean or how it works. --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:15, 2 August 2021 (UTC) :There are a few aspects to it. Firstly, some of the [[WikiJournal of Medicine/Publishing#Publication formats|main publication formats]] are directly converted to Wikipedia pages ([[WikiJournal of Medicine/Epidemiology of the Hepatitis D virus|example]]), and some even from [[WikiJournal of Medicine/What are Systematic Reviews?|Wikipedia pages]]. Additionally, some articles have their images integrated into Wikipedia ([[WikiJournal of Medicine/Cell disassembly during apoptosis|example]]). There's some more in-depth info and history in [[c:File:2020_WikiJournal_overview_and_comparison_(Open_Publishing_Fest)_recording.webm|this presentation from 2020]]. But good point that it's be worth linking out to more information. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 23:50, 2 August 2021 (UTC) I see, thanks for the explanation. --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:41, 6 September 2021 (UTC) == Impact factor == So it is said, that the Journal was not added to the Web of Science yet. Is this per request process or do they do it automatically or how does it work? --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 6 September 2021 (UTC) :Getting into Web of Science remains our focus. It is a manual submission process so we will try again later. It is also important to note that there are [[:w:Impact factor#Criticism|multiple, well-known criticism]] over what impact factor stands for and what are its gaps. Moreover, different engines arrive at different results. For example, Google Scholar counts every citation including student thesis, conference abstracts and government reports (which means that the impact factor may be inflated). ResearchGate lets you upload conference poster, which can serve as a way self-cite your own publications and inflate your personal impact factor. On the other hand, Web of Science tend to underestimate impact factor because it excludes things like peer-reviewed book chapters. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:39, 29 March 2022 (UTC) == Recruiting technical editors == We are hiring new [[WikiJournal User Group/Technical editors|technical editors]] for the journals. Please see [https://www.linkedin.com/posts/andrewcleung_technical-editor-job-poster-activity-6912636772371828736-LteF this job posting for details.] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:24, 29 March 2022 (UTC) :The application period is now closed, and we are assessing existing applications. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 19:33, 22 May 2022 (UTC) == Rubriq does not work since 2017 == Three is a sentence on the Wikijournal of Medicine [[WikiJournal_of_Medicine/Editorial_guidelines#Finding_peer_reviewers|Editorial guidelines]] page "As a last possibility, authors may pay for a peer review to be performed by Rubriq (with a request to abide by the journal's peer review guidelines)." There was a link to a Wikipedia article on Rubriq: [https://en.wikipedia.org/wiki/Scholarly_peer_review#Rubriq Rubriq] I have figured out that Rubriq does not work since 2017. I have edited the Wikipedia page but not the page on the Editorial guidelines of Wikijournal of Medicine. I recommend to replace the link to Rubriq to a link to another service that works (if any) or remove this advise altogether. --[[User:Maxim Masiutin|Maxim Masiutin]] ([[User talk:Maxim Masiutin|discuss]] • [[Special:Contributions/Maxim Masiutin|contribs]]) 21:24, 2 May 2022 (UTC) :When trying to reach Rubriq, I was redirected to [https://www.researchsquare.com/publishers/editorial-services Research Square]. I'm not sure it's interchangeable though, so I've simply removed the Rubriq option from the editorial guidelines. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 19:39, 21 August 2022 (UTC) == The links to PubMed Central (PMC) are generating incorrectly == Hello, {{ping|Evolution and evolvability}} The links to PubMed Central (PMC) are generating incorrectly from the template ''cite journal|pmc=...'' in the WikiJournal of Medicine, for example, click the links to "PMC" at the references section at https://en.wikiversity.org/wiki/WikiJournal_Preprints/Androgen_backdoor_pathway In contrast, the links from the same template on Wikipedia are generated correctly, see https://en.wikipedia.org/wiki/Androgen_backdoor_pathway [[User:Maxim Masiutin|Maxim Masiutin]] ([[User talk:Maxim Masiutin|discuss]] • [[Special:Contributions/Maxim Masiutin|contribs]]) 11:03, 19 June 2022 (UTC) == Proposal to introduce "Inactivity removal policy" to the bylaws == There is an ongoing discussion to propose introducing an inactivity removal policy for editorial board members. Full details [[Talk:WikiJournal User Group#Proposal to introduce "Inactivity removal policy" to the bylaws|can be viewed here]]. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:24, 12 September 2022 (UTC) == Double DOI to same article == I came across [[WikiJournal of Medicine/The Kivu Ebola Epidemic]], which has two DOIs (10.15347/WJM/2021.005 and 10.15347/WJM/2022.001) both pointing to it. I suspect the 2021.005 is wrong, since the article was accepted in April 2022. Not quite sure how to delete an DOI. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 05:09, 21 September 2022 (UTC) : Hi {{u|OhanaUnited}}. DOIs can't be deleted but they can be aliased, which designates one of them as primary DOI and the other as the secondary DOI. Then, we just go ahead and display the primary one. I believe this process is fairly straightforward as outlined [https://www.crossref.org/documentation/reports/conflict-report/#00243 here] for anyone with access to the Crossref depositor credentials. There appears to be one other example of this, with 10.15347/wjm/2015.001 and 10.15347/wjm/2014.013 both pointing to the same article. —[[User:Bobamnertiopsis|Collin]] (Bobamnertiopsis)^{[[User talk:Bobamnertiopsis|t]] [[Special:Contributions/Bobamnertiopsis|c]]} 12:52, 21 September 2022 (UTC) ::Fixed both. Thanks @[[User:Bobamnertiopsis|Bobamnertiopsis]] for spotting the second pair of DOI conflict. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 00:59, 22 September 2022 (UTC) == Backlog at [[WikiJournal of Medicine/Potential upcoming articles]] == I fear that due to Athikhun Suwannakhan and/or possibly other editors becoming inactive, a number of articles (including mine, see [[Talk:WikiJournal_Preprints/Where_experts_and_amateurs_meet:_the_ideological_hobby_of_medical_volunteering_on_Wikipedia#Thank you for revising your submission|context here]])) have been stalled. Can someone take them over? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 15:12, 8 June 2023 (UTC) :@[[User:OhanaUnited|OhanaUnited]] Ping... is anyone able to look into this? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 14:16, 15 June 2023 (UTC) ::{{re|Piotrus}} Athikhun wouldn't be inactive, as I co-presented alongside with him in-person at the [[meta:EduWiki Conference 2023/Program/WikiJournal|EduWiki Conference]]. {{ping|Athikhun.suw}}, do you have a status update for [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia|this submission]]? It appears to be stalled at the final decision stage by the editorial board. I do know that the handling editor is heading to China for 3 weeks and will be unresponsive to emails. As for other stalled submissions, [[WikiJournal Preprints/Next Generation Junctional Tourniquet: A Case Report in a Non-injured Subject|one]] of them was my fault and I did a follow-up in March but didn't hear back from the author (and didn't mark the status in the "Notes" field). I send a last follow-up reminder email and gave the author a 10-day deadline to respond. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 17:09, 16 June 2023 (UTC) :::@[[User:OhanaUnited|OhanaUnited]] @[[User:Athikhun.suw|Athikhun.suw]] An update would be appreciated. We are now half a year at the stage the article, as far as I understand (given the positive reviews) should be simply published? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 04:57, 9 July 2023 (UTC) ::::@[[User:Rwatson1955|Rwatson1955]], can you take a look please? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:33, 10 July 2023 (UTC) :::::{{re|Piotrus}} FYI, your article [https://en.wikiversity.org/w/index.php?title=WikiJournal_User_Group/Technical_editors/tasks&diff=prev&oldid=2541107 has been accepted]. The backend is processing the updates to reflect the acceptance status. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 17:48, 29 July 2023 (UTC) ::::::@[[User:OhanaUnited|OhanaUnited]] Thank you for the update! [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 02:34, 30 July 2023 (UTC) == Publication Schedule? == Are there any editors that can describe the publication schedule? I ask because it has been a few months since an article of mine has been finalized: https://en.wikiversity.org/wiki/WikiJournal_of_Medicine/Alternative_androgens_pathways It doesn't show up on the front page or in the "upcoming articles" list, making it difficult for users to find. I would imagine it would make sense to have a rolling front page for publications. Things look rather dead for a journal from the front page, and there is at least this article that has been ready to go for months. It would make more sense to publish articles to the front page faster, attracting more attention and hopefully higher quality submissions etc. [[User:Maneesh|Maneesh]] ([[User talk:Maneesh|discuss]] • [[Special:Contributions/Maneesh|contribs]]) 16:02, 11 August 2023 (UTC) == Inconsistent capitalization of titles == Just noting that it appears we don't have a preference for this, which results in the past article title looking a bit disjointed. Maybe time to chose one format and stick to it? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:58, 5 December 2024 (UTC) h3xwet7myi2vjzxkaxlp3nybdck6vui 0 163408 2690308 2093999 2024-12-04T20:42:55Z RockTransport 2992610 2690308 wikitext text/x-wiki *'''Learning communities''': [[Learning communities/Inspiring parents|Inspiring parents]], [[Learning communities/System development|system development]] * '''Website''': http://scratch.mit.edu *'''Wikipedia''': [[Wikipedia: Scratch (programming language)]] == What is Scratch? == * Scratch is a 'programming' platform for children. Children can use Scratch to make games, interactive stories and animations. * Scratch is developed by [[Wikipedia: Massachusetts Institute of Technology|MIT]], one of the most prestigious universities in the world. == About this learning project == In 2014 learning community 'system development' started a learning project for teachers and parents to learn more about Scratch and how it enables young children to improve their computing skills. This learning project is block-based, which makes it suitable for younger audiences and introduces the basics of coding. == Index == * [[/Alphabetical index/]] * [[/Log/]] * [[/To-do/]] * [[/Wikibook/]] [[Category:Visual programming]] 2e13bixlzlh0ksgelo8reh6w62tiif9 1 170731 2690309 2687815 2024-12-04T20:56:06Z Mikael Häggström 12130 /* Editorial board application of Luis Rafael Moscote-Salazar */ No response 2690309 wikitext text/x-wiki <noinclude> {{WikiJournal editorial application top |archive box = {{Archive box|[[/Archive 2015-2017]] <br>[[/Archive 2018]] <br>[[/Archive 2019]] <br>[[/Archive 2020]] <br>[[/Archive 2021]] <br>[[/Archive 2022]] <br>[[/Archive 2023]] }} }} </noinclude> ==Associate editor application of James Bibey== {{WikiJournal editor application submitted | position =Associate editor | name =James Bibey | qualifications =2nd Year Medical Student | link = | areas_of_expertise =General medicine (basic anatomy, physiology, biochemistry, statistics, ethics) | professional_experience =Maths lecturing, anatomical prosection preparation. | publishing_experience =N/A | open_experience =Significant editing history on English Wikipedia (primarily medicine and anatomy), Wikimedia Commons, and Wikidata under username "Bibeyjj". | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:Bibeyjj|Bibeyjj]] ([[User talk:Bibeyjj|discuss]] • [[Special:Contributions/Bibeyjj|contribs]]) 19:23, 24 September 2021 (UTC) }} *{{Support}}. The applicant is [https://en.wikipedia.org/wiki/Special:Contributions/Bibeyjj active in related topics in Wikipedia], and I think we can really need the help for our journal too. [[User:Bibeyjj|Bibeyjj]], I hope you are still interested in this position. I'm sorry for the late response to your application, as you see we are quite busy with the everyday matters of the project. If elected, would you be willing to help out for instance in [[WikiJournal_of_Medicine/Editorial_guidelines#Arranging_peer_review|finding peer reviewers]] for article submissions to the journal? [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 19:10, 22 May 2022 (UTC) *{{Support}}. Mikael summarises the reasons well above and I agree [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 06:56, 23 May 2022 (UTC) *They seem a good candidate for assoc editor status, and it would be useful experience for them as well as helpful skills for us. We've been a bit stalled on applications over the last year, so it will be good to get organised again. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:59, 11 June 2022 (UTC) :I agree. I made [https://en.wikipedia.org/wiki/User_talk:Bibeyjj#Associate_editor_application an entry on the user's talk page] whether he's still interested. If so, I think we can go ahead and approve this application. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 15:52, 16 June 2022 (UTC) ::I haven't heard back from the wiki talk page, so I sent an email through the wiki system as well. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 03:13, 2 August 2022 (UTC) *{{Support}}. [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:52, 2 November 2022 (UTC) ==Editorial board application of Luis Rafael Moscote-Salazar== {{WikiJournal editor application submitted | journal =WikiJournal of Medicine | position =Editorial board | name =Luis Rafael Moscote-Salazar | qualifications =MD | link =https://neuroclani.org/ | areas_of_expertise =Neurosurgery, Neurotrauma, Stroke, Neurointervention, Neurocritical Care, Neurointervention, Evidence Based Medicine | professional_experience =Neurosurgeon graduated from the University of Cartagena, Founder of the Colombian Clinical Research Group in Neurocritical Care and Co-founder of the Latinamerican Council of Neurocritical Care (CLaNi). Research communication, research leadership, mentorship. | publishing_experience = Experience with publishing in peer-reviewed journals (see Google Scholar profile. Peer-reviewer for several international journals in Neurosurgery and Medicine (see Publons profile). | open_experience = | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:NeuroQuimbaya|NeuroQuimbaya]] ([[User talk:NeuroQuimbaya|discuss]]) 2022-07-12 }} :No details were provided, so I have contacted the applicant to request that they add more information. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 01:27, 7 July 2022 (UTC) ::I've updated the application above with the new replacement information that they sent through. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 03:55, 13 July 2022 (UTC) :::'''Pending more specific presentation'''. I'm tending towards support, as the applicant seems active in research and publishing according to orcid ([https://orcid.org/0000-0002-4180-6962]). I think it is appropriate to let him join us. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 03:12, 2 August 2022 (UTC) ::::[[User:NeuroQuimbaya|Dr. Moscote-Salazar]], some questions that have been raised are: ::::*Would you be able to provide a webpage or other presentation about yourself? The link provided (https://neuroclani.org/) directs to a more general website. ::::*Could you provide one or two sentences of what motivates you to join WikiJMed? ::::*Would you be willing to begin contributing as an associate editor? The tasks can be described here: [[WikiJournal_of_Medicine/Associate_editors]]. We feel we can really need some help with peer review coordination, and you'll have the opportunity to later become promoted to editorial board membership. ::::[[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 02:11, 11 August 2022 (UTC) We need to establish if this applicant has specific Wikipedia experience [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:52, 3 May 2023 (UTC) *Given the lack of open experience, I think the applicant should be considered for associate editor position. {{re|Rwatson1955}} and {{re|Mikael Häggström}} Did this individual provide more information since our latest follow-up in May? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:01, 4 July 2023 (UTC) ::I have not received any response from the talk page entry... [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 20:56, 4 December 2024 (UTC) ==Editorial board application of Helmar Bornemann-Cimenti== {{WikiJournal editor application submitted | position =Editorial board | name =Helmar Bornemann-Cimenti | qualifications =MD DMedSc MSc (interdisciplinary Pain Medicine) MBA (health care management) | link =https://forschung.medunigraz.at/fodok/suchen.person_uebersicht?sprache_in=en&menue_id_in=101&id_in=2001978 | areas_of_expertise =Pain medicine, Anesthesiology | professional_experience =Deputy Head and Chair of the Department of Anesthesiology and Intensive Care Medicine, Medical University of Graz | publishing_experience =Editor in 2 scientific journals ("BMC Anesthesiology", "Pain and Therapy"), Guest-editor in "Life" | open_experience =Wikipedia author since 2006 | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:Bornhelm|Bornhelm]] ([[User talk:Bornhelm|discuss]] • [[Special:Contributions/Bornhelm|contribs]]) 09:00, 4 February 2023 (UTC) }} *Being followed up by [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:50, 3 May 2023 (UTC) **Note: The applicant mentioned that he used to edit under the account [https://de.wikipedia.org/wiki/Benutzer:Borne User:Borne], to whicch he lost hte password and subsequently edited anonymously before signing up for [[user:Bornehelm]] (which is why that account doesn't show edit history back to 2006). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:05, 19 June 2023 (UTC) *'''Support''' - Though the application itself is slim, I think he is actually quite a good candidate. The work done under the old [https://de.wikipedia.org/wiki/Benutzer:Borne User:Borne] account was relevant and expertise in anesthesiology and pain medicine would be of use to the board. His work with ''BMC Anesthesiology'' is particular useful, since BMC is one of the most established OA publishing groups. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:30, 19 June 2023 (UTC) *'''Support''' likewise, having cleared up the confusion over his Wikipedia page, I support this one. [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 02:01, 20 June 2023 (UTC) *'''Support''' Seems like a suitable candidate. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 01:58, 4 July 2023 (UTC) ==Editorial board application of Alfred Amendolara== {{WikiJournal editor application submitted | position =Editorial board | name =Alfred Amendolara | qualifications =MS, DO (3rd year student, expected graduation 2025) | link =https://orcid.org/0000-0001-9696-8961 | areas_of_expertise =neuroscience/neurology, machine learning and AI, epidemiology | professional_experience =My research interests are fairly diverse and span from designing and implementing machine learning models to investigating the neuronal pathways responsible for central pattern generation. I initially worked as a graduate research assistant at Fortune Lab at New Jersey Institute of Technology. During this time I completed my thesis which modeled influenza trends using cutting edge machine learning tools. I went on to work as a research associate at Severi Lab, also at New Jersey Institute of Technology for several years. There I investigated zebra fish motor circuitry using both behavioral experiments and, more recently, computational modeling. I continue to be affiliated with NJIT, although no longer in a paid position. I am currently a 3rd year medical student. I spent the first year in the Addiction Lab at Brigham Young University performing electrophysiological experiments investigating the neural pathways responsible for alcohol dependence. I am currently working in the Payne Lab at Noorda College of Osteopathic Medicine as a graduate research assistant. Here I lead a number of projects including protein modeling of KCC2 channels in the brain (in order to investigate its role in addiction behaviors) as well as several in progress systematic reviews and meta-analyses. Additionally I am involved in an on-going clinical trail investigating mechanical nerve stimulation for the treatment of migraines. Over the past year I have actively regularly in reviewing for a number of journals. | publishing_experience =Founding co-Editor-in-chief of Intermountain Journal of Translational Medicine. I currently serve as the co-editor-in-chief of Intermountain Journal of Translational Medicine, a newly formed peer-reviewed journal published by Noorda College of Osteopathic Medicine aimed at early career researchers in the United States and beyond. We hope to complete our first issue by December of this year. | open_experience =I have edited Wikipedia pages in the past as a none-registered user. | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:Alfred Amendolara|Alfred Amendolara]] ([[User talk:Alfred Amendolara|discuss]] • [[Special:Contributions/Alfred Amendolara|contribs]]) 05:12, 3 August 2023 (UTC) }} *I am not sure that there is sufficient editorial experience with mainstream journals/publishers or with Wikipedia to support [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 10:34, 12 January 2024 (UTC) *'''Decline''' While the Intermountain Journal of Translational Medicine is open access, it is far too new and published too few papers for me to comment on its quality. The author made no contributions to any wiki projects since the application to demonstrate their continued interest in the open movement or publishing. In the applicant's ORCID profile, several Open Science Framework items were misclassified as "journal articles". This suggests that applicant is confused between research registry and publications. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:38, 14 October 2024 (UTC) ==Editorial board application of Md. Tanzir Islam== {{WikiJournal editor application submitted | position =Editorial board | name =Md. Tanzir Islam | qualifications =MBBS, MD (Phase A - Nephrology) | link = | areas_of_expertise =Medical Education, Public Health, Clinical Research, Nephrology | professional_experience =As a dedicated physician and an ardent advocate for clinical research, I possess a profound foundation in medical education, public health, clinical research, and nephrology. My medical odyssey commenced at Bangabandhu Sheikh Mujib Medical College (BSMMC), further refined by an enriching internship at Dhaka Medical College (DMC). My passion lies in diminishing the gap between healthcare providers and patients through strategic knowledge dissemination and active participation in diverse educational platforms. Professional profiles: LinkedIn (https://www.linkedin.com/in/tanzir-islam-britto-629277129), ORCID (https://orcid.org/0009-0009-3936-055X), Cureus (https://cureus.com/users/515538-tanzir-islam-britto), Loop (https://loop.frontiersin.org/people/2410099/). | publishing_experience =I have written several medical articles and books, contributing significantly to fields such as pediatric oncology and nephrology. My works include "A Systematic Review on Childhood Non-Hodgkin Lymphoma: An Overlooked Entity," and "A Systematic Review of Pediatric Dialysis in Asia: Unveiling Demographic Trends, Clinical Representation, and Outcomes," both of which have been instrumental in highlighting critical areas in pediatric healthcare and advocating for advanced research and improved patient outcomes. | open_experience =Honored as a CUREUS Laureate and an esteemed peer reviewer, my endeavors underscore the paramount importance of excellence in medical research and scholarly publication. My scholarly contributions span across nephrology, endocrinology, and diabetes, epitomizing my unwavering commitment to propelling the frontiers of medical science and education. | policy_confirm =I hereby affirm my commitment to uphold the principles and policies of the WikiJournal of Medicine, ensuring integrity, transparency, and the advancement of medical knowledge. Md. Tanzir Islam [[User:Vespercasper|Vespercasper]] ([[User talk:Vespercasper|discuss]] • [[Special:Contributions/Vespercasper|contribs]]) 19:46, 24 January 2024 (UTC) }} * This applicant lacks sufficient editorial experience to be recommended [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 07:22, 2 February 2024 (UTC) * '''Decline''' The applicant lacks sufficient level of publication and editorial experience at this moment. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:42, 14 October 2024 (UTC) ==Associate editor application of Tesleemah Taye Abdulkareem== {{WikiJournal editor application submitted | position =Associate editor | name =Tesleemah Taye Abdulkareem | qualifications =Doctor of Optometry | link = | areas_of_expertise =Optometry | professional_experience = Founder Mira Sight Foundation (2022- Present), Extern optometrist in 2024 for 6 months at University of Ilorin Teaching Hospital, Optometrist Assistant at University of Medicine Teaching Hospital and Apple Eye clinic in 2022 | publishing_experience =The Pattern of Intraocular Pressure in Myopia: Students of University of Ilorin as a Case Study 2023 Seminar Presentation: Patient case study at University of Ilorin Teaching hospital Taiwo. E. A, Abdulkareem. T. T, Fajemisin. E. “The Nutraceutical Potential of Carrots Carotenoids in Chronic Eyes Defects (Ceds): A Review” Ssrn Electronic Journal, July 12, 2021. Kindly find the links below: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4939216 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3885012 | open_experience =I am a board member of Wiki Project Medicine who has improved upon a good number of health articles and translated more into the Yoruba Language. Also, in 2023, I was the project lead of Wikimedia Awareness in Akure and one of the core organizers of the Wikiclimate Campus Tour Nigeria Project. To perfect my open organizing skill, I am a Certified Organiser for the organizer lab and experienced Wikimedia projects editor with over 300+ articles across English Wikipedia, Yoruba Wikipedia, Wikiquotes and Wikivoyage. Between 2022-2023, I was a training Associate with Free Knowledge Africa | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:Tesleemah|Tesleemah]] ([[User talk:Tesleemah|discuss]] • [[Special:Contributions/Tesleemah|contribs]]) 13:53, 17 October 2024 (UTC) }} :@[[User:Tesleemah|Tesleemah]] Thank you for your application and my apologies for the delay in responding to it. Do you have any peer-reviewed publications (e.g. journal articles or book chapters)? The SSM papers you linked are preprints, which are not peer-reviewed publications. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:37, 6 November 2024 (UTC) ::Ohh I was told the first one was published but guess it's just Preprint (I was a participatory author about 4 years ago and it was first time writing a research woek) and I am yet to publish the second one [[User:Tesleemah|Tesleemah]] ([[User talk:Tesleemah|discuss]] • [[Special:Contributions/Tesleemah|contribs]]) 03:33, 14 November 2024 (UTC) :::@[[User:Tesleemah|Tesleemah]] Given your experience, would you be open to consider applying for associate editor position as opposed to editorial board position? For editorial board, we're looking for individuals with peer-reviewed publications. Associate editor positions require lesser experience. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:45, 14 November 2024 (UTC) ::::Alright that's fine [[User:Tesleemah|Tesleemah]] ([[User talk:Tesleemah|discuss]] • [[Special:Contributions/Tesleemah|contribs]]) 19:28, 14 November 2024 (UTC) ::::: Thanks. I have changed your application to associate editor. I am in '''support''' of your associate editor application. Please wait for other editorial board members to review and comment on your application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:39, 18 November 2024 (UTC) ==Associate editor application of Truong Gia Hy Do== {{WikiJournal editor application submitted | position =Associate editor | name =Truong Gia Hy Do | qualifications =BS (Genetics) PHD (1st year student) | link =https://medicine.umich.edu/dept/cdb/hy-do | areas_of_expertise =Biomedical sciences, biology, genetics | professional_experience =Graduate research assistant, tutoring | publishing_experience =One co-authorship publication (doi: 10.3389/fcell.2024.1460669), another co-authored paper submitted for publication | open_experience =I am an active editor on Vietnamese Wikipedia (primarily translation, have published 160 articles) | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:Dotruonggiahy12|Dotruonggiahy12]] ([[User talk:Dotruonggiahy12|discuss]] • [[Special:Contributions/Dotruonggiahy12|contribs]]) 21:01, 16 November 2024 (UTC) }} :* '''Support''' An editor who has good mix of open and publishing experience who understands the wiki environment. His professional experience will grow over time. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:43, 18 November 2024 (UTC) c7lr91pzjgn1zgb3zosv93d3q8ptc0p 0 171005 2690265 2690089 2024-12-04T14:41:39Z Young1lim 21186 /* Geometric Series Examples */ 2690265 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20241204.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|C.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] o52dyud99iyboqm8yiqv2wun4y1gp4m 2690275 2690265 2024-12-04T15:05:33Z Young1lim 21186 /* Geometric Series Examples */ 2690275 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20241204-1.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|C.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] f5s8c5r4v6u6rnmekwyvltzoqc9t5cz 0 212733 2690352 2690062 2024-12-05T11:05:43Z Young1lim 21186 /* Using Libraries */ 2690352 wikitext text/x-wiki ==''' Part I '''== <!----------------------------------------------------------------------> === Introduction === * Overview * Memory * Number <!----------------------------------------------------------------------> === Python for C programmers === * Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]]) * Statement Level ([[Media:CProg.Statement.1A.20230509.pdf |pdf]]) * Output with print * Formatted output * File IO <!----------------------------------------------------------------------> === Using Libraries === * Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]]) * Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]]) * Packages ([[Media:Python.Work2.Package.1A.20241203.pdf |pdf]]) * Libraries ([[Media:Python.Work2.Library.1A.20241109.pdf |pdf]]) * Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Repetition === * Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]]) * Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling a Big Work === * Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]]) * Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]]) * Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Series of Data === * Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]]) * Lists ([[Media:Python.Series3.List.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]]) * Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]]) * Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Various Kinds of Data === * Types * Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]]) * Files ([[Media:Python.Data4.File.1.A.pdf |pdf]]) <!----------------------------------------------------------------------> === Class and Objects === * Classes & Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]]) * Inheritance <!----------------------------------------------------------------------> </br> == Python in Numerical Analysis == </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering] b53d0h14wr0t1kwyb9nb19ofmxkwokl 2690354 2690352 2024-12-05T11:09:20Z Young1lim 21186 /* Using Libraries */ 2690354 wikitext text/x-wiki ==''' Part I '''== <!----------------------------------------------------------------------> === Introduction === * Overview * Memory * Number <!----------------------------------------------------------------------> === Python for C programmers === * Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]]) * Statement Level ([[Media:CProg.Statement.1A.20230509.pdf |pdf]]) * Output with print * Formatted output * File IO <!----------------------------------------------------------------------> === Using Libraries === * Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]]) * Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]]) * Packages ([[Media:Python.Work2.Package.1A.20241204.pdf |pdf]]) * Libraries ([[Media:Python.Work2.Library.1A.20241109.pdf |pdf]]) * Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Repetition === * Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]]) * Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling a Big Work === * Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]]) * Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]]) * Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Series of Data === * Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]]) * Lists ([[Media:Python.Series3.List.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]]) * Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]]) * Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Various Kinds of Data === * Types * Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]]) * Files ([[Media:Python.Data4.File.1.A.pdf |pdf]]) <!----------------------------------------------------------------------> === Class and Objects === * Classes & Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]]) * Inheritance <!----------------------------------------------------------------------> </br> == Python in Numerical Analysis == </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering] nw8og1ffdl6nfm8kbl3tf4yoqzwaup3 2690358 2690354 2024-12-05T11:20:04Z Young1lim 21186 /* Using Libraries */ 2690358 wikitext text/x-wiki ==''' Part I '''== <!----------------------------------------------------------------------> === Introduction === * Overview * Memory * Number <!----------------------------------------------------------------------> === Python for C programmers === * Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]]) * Statement Level ([[Media:CProg.Statement.1A.20230509.pdf |pdf]]) * Output with print * Formatted output * File IO <!----------------------------------------------------------------------> === Using Libraries === * Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]]) * Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]]) * Packages ([[Media:Python.Work2.Package.1A.20241205.pdf |pdf]]) * Libraries ([[Media:Python.Work2.Library.1A.20241109.pdf |pdf]]) * Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Repetition === * Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]]) * Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling a Big Work === * Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]]) * Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]]) * Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Series of Data === * Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]]) * Lists ([[Media:Python.Series3.List.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]]) * Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]]) * Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Various Kinds of Data === * Types * Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]]) * Files ([[Media:Python.Data4.File.1.A.pdf |pdf]]) <!----------------------------------------------------------------------> === Class and Objects === * Classes & Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]]) * Inheritance <!----------------------------------------------------------------------> </br> == Python in Numerical Analysis == </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering] 73ykkoqxhdkm00a4mogy0hzbnzaos6o 0 228630 2690263 2479567 2024-12-04T13:36:01Z 144.121.121.194 /* Fluid Friction */ 2690263 wikitext text/x-wiki We all slow down our vehicles whenever needed by applying brakes. Do you know why a vehicle slows down when brakes are applied? Not only vehicles any object moving on the surface of another object slows down and stops without any external force acting on it because of "friction". Before going to types of friction, lets know about friction. According to law of physics any object in the world can't be friction-less. == Friction == [[File:Friction diagram.png|thumb|294x294px|Forces on an object in motion]] Forces and motion of object. [[Friction]] is a force that opposes the motion of two contacting surfaces. === Laws of Friction === 1. Friction depends on the hardness or roughness of the contacting surfaces.Fluid friction depends on the viscosity (the thickness) of the fluid... 2. Friction is directly proportion to the normal force pressing the contacting forces together... 3. Friction does not depend on the area of the contacting surface... 4. In the case of sliding, friction is reduced at very high relative speeds. However, in the case of fluid friction, friction increases with increase in relative speed of movement. To stop an object in motion, a force must act on it in the opposite direction of motion. The force that opposes the motion of the object is called the frictional force. Look at the diagram. At first the block is at rest, then the pushing force keeps the block moving. As the block slides over the surface, the frictional force acts on it in the opposite direction. A unit of friction is a Newton, as forces are measured using Newtons. Friction generally depends on weight of the object and nature of the surface between the moving object and supporting surface. == Types of Friction == Different types of motion of the object gives rise to different types of friction. Generally, there are 4 types of friction. They are static friction, sliding friction, rolling friction, and fluid friction. The next sections will explore these forces and when they are applied. === Static Friction === Static friction exists between a stationary object and the surface on which it is resting. It prevents an object from moving against the surface. Example: Static friction prevents an object like a book from falling off the desk, even if the desk is slightly tilted. It helps us pick up an object without it slipping through our fingers. When we want to move an object first we must overcome the static friction acting between the object and the surface on which the object is resting. [[File:Blank book on a table.jpg|center|thumb|A stationary book on surface]] === Sliding Friction === Sliding friction occurs between objects as they slide against each other. When sliding friction is acting there must be another force existing to keep the body moving. Example: When a man is pushing an object on a rough surface the force acting is called "sliding friction". === Air resistance === Here on Earth we tend to take air resistance (aka.“drag”) for granted. We just assume that when we throw a ball, launch an aircraft, deorbit a spacecraft, or fire a bullet from a gun, that the act of it traveling through our atmosphere will naturally slow it down. But what is the reason for this? Just how is air able to slow an object down, whether it is in free-fall or in flight? Air friction is experienced by the objects moving through the open air. Air friction acts between the object and the air through which it is moving. It is also called drag. This force depends upon the object's shape, material, speed with which it is moving and the viscosity of the fluid. Viscosity is the measure of the resistance of the air to flow and it differs from one density to another. Example: It slow downs the motion of airplane flying in the air, here the engine of the airplane helps the plane to overcome the fluid friction and move forward. [[File:Jet2 aeroplane landing at EDI.jpg|center|thumb|264x264px|A plane in the air]] There is another type of friction (a special case) === Limiting Friction === Limiting friction is the maximum opposing force that comes into play when one body is just at the verge of moving over the surface of the other body jyy2qdedbkkf2d3na4rvzatiw0nd70m 1 228878 2690341 2687817 2024-12-05T05:53:41Z Piotrus 571 /* Editorial board application of Laura G. Campo */ 2690341 wikitext text/x-wiki <noinclude> {{WikiJournal editorial application top |archive box = {{Archive box|[[/Archive 2017]] <br>[[/Archive 2018]] <br>[[/Archive 2019]] <br>[[/Archive 2020]] <br>[[/Archive 2022]] <br>[[/Archive 2023]] }} }} </noinclude> ==Editorial board application of Hernan Perez Molano== {{WikiJournal editor application submitted | position =Editorial board | name =Hernan Perez Molano | qualifications =PHD in Political science, Master in Ethnomusicology | link =https://es.linkedin.com/in/hernan-p%C3%A9rez-molano-918252a1 | areas_of_expertise =Peacebuilding, social innovation, political science, ethnomusicology | professional_experience =Doctor of Political Science, Administration, and International Relations, from the Complutense University of Madrid (Spain), trained in ethnographic, sociological, and anthropological techniques (Master's in Musicology, specializing in Ethnomusicology) at the Sorbonne University (France). His research, entitled "Obstacles and Resistances in the Construction of Alternative Peace: Comparative Ethnographies of the Reintegration of Former Combatants in Colinas, Guaviare, and Icononzo, Tolima," describes the construction of peace at the local level from the perspective of local social innovation ecosystems, based on a multi-sited ethnography (2019-2023). :Coordinator of the Social Innovation Program (2015-2020) at the Research and Extension Office of the National University of Colombia, Bogotá campus. He has experience in supporting academia in formulating and implementing social innovation projects, utilizing participatory methodologies, design thinking, and fostering creative capacity in the context of community youth processes, as well as in communication and culture for peacebuilding. He was a former member of the formulating team, facilitator, and coordinator of the Innovation Laboratory for Peace (Trust for the Americas - National University of Colombia), and the Spaces of Re-cognition for Peace project of the Academic Vice-Rectory of the National University of Colombia. | publishing_experience = | open_experience =Official for the Education program of Wikimedia Colombia | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:HerPerezM|HerPerezM]] ([[User talk:HerPerezM|discuss]] • [[Special:Contributions/HerPerezM|contribs]]) 21:42, 20 July 2023 (UTC) }} * I approached him at EduWiki Conference to discuss WikiJournal and potential collaboration. I fully support his application to join the editorial board. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:47, 21 July 2023 (UTC) * [[File:Symbol support vote.svg|14px]]I support this application for editor. [[User:Smvital|<b><span style="color: #0000FF;">Smvital</span></b>]][[User talk:Smvital|<b><span style="color: green;">^{Talk page}</span></b>]] 10:46, 1 August 2023 (UTC) * '''support''' - It's also a support from me. Very useful professional bacckground, and experience with Wikimedia Colombia's educaction programme is definitely a bonus. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 10:45, 28 August 2023 (UTC) * I support this application. I agree; his area of study and experience will make him very suitable. [[User:Fransplace|<b><span style="color: #0000FF;">Fransplace</span></b>]][[User talk:Fransplace|<b><span style="color: green;">^{Talk page}</span></b>]] 20:01, 10 September 2023 (UTC) *'''Support''', of course. Hopefully, you'll have more time than I to help (I sadly overestimated my amount of time for this year...). --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 08:05, 13 September 2023 (UTC) * '''support''' [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:18, 13 September 2023 (UTC) * '''support''' - a very welcome addition to the WikiJ Hum Team --[[User:Mstefan|Mstefan]] ([[User talk:Mstefan|discuss]] • [[Special:Contributions/Mstefan|contribs]]) 12:48, 13 September 2023 (UTC) '''Result: Accepted into the editorial board.''' : [[WikiJournal User Group/Editorial guidelines#Adding editorial board members|Next steps]] (add <code>DONE</code> or <code><nowiki>{{Done}}</nowiki></code> after someone has performed the task): # [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) {{clickable button 2|Onboarding email template|url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member}} # Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template # Add their name and start data to the [d:{{WJQboard|default=Q75674277}} relevant editorial board] on wikidata # Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members # Welcome them at the {{#if:|wjm|WJM}}board mailing list so that they are informed # Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 06:31, 6 November 2023 (UTC) ==Editorial board application of Lihao Gan== {{WikiJournal editor application submitted | position =Editorial board | name =Lihao Gan | qualifications =PHD.Professor | link =https://faculty.ecnu.edu.cn/_s11/glh_en/main.psp | areas_of_expertise =Epistemology，Communication Studies，Media Discourse Analysis，Rhetoric | professional_experience =Gan Lihao (born October 1977) is a professor and doctoral supervisor at East China Normal University. He is a distinguished talent of the Pujiang Talent Program in Shanghai. He has also served as a visiting scholar in the Department of Linguistics at the University of California, Berkeley. Additionally, he holds the position of Deputy Director at the National Discourse Ecology Research Center and serves as an executive member of the Chinese Rhetoric Society, a council member of the Shanghai Language Society, and a committee member of the Audiovisual Communication branch of the Chinese Association for the History of Journalism and Communication. | publishing_experience =Gan Lihao is known for his pioneering contributions to the fields of "Life Rhetoric" and "Behavioral Dramatism Theory." His research primarily revolves around human communication discourse, aiming to promote individual growth, harmonious family dynamics, intercommunication among domestic communities, and international dialogues within the context of the human community's shared destiny and peaceful development. He focuses on three main research directions: family education discourse analysis based on empathetic rhetoric, discourse research on national governance rooted in speech acts, and global knowledge discourse analysis centered around digital communities. Gan Lihao has authored several significant works, including "Contrastive Structures Under the Influence of Spatial Dynamics," "Communication Rhetoric: Theory, Methods, and Case Studies," "Reshaping China's National Image and Wikipedia Knowledge Discourse Research," and "Political Science on Wikipedia" (in progress). | open_experience =wikipedia editor,wikipedia researcher | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:Ganlihao|Ganlihao]] ([[User talk:Ganlihao|discuss]] • [[Special:Contributions/Ganlihao|contribs]]) 06:30, 4 September 2023 (UTC) }} * This editor approached us at the Wikimania Singapore event and we discussed how we need experts in humanities to contribute and assist with reviewing the backlogged submissions. He expressed an interest after seeing our poster at Wikimania. He led a team of researchers from China to investigate and publish research articles about Wikipedia. As such, his professional, publishing and open experiences are quite extensive. Since he primarily publishes in Chinese language, I suggested that he initially apply for associate editor position to familiarize himself with publishing and communicating in English to gain confidence in this area. I fully {{support}} his application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:52, 7 September 2023 (UTC) * I support this application and agree an associate editor position will be best to begin with. [[User:Fransplace|<b><span style="color: #0000FF;">Fransplace</span></b>]][[User talk:Fransplace|<b><span style="color: green;">^{Talk page}</span></b>]] 20:05, 10 September 2023 (UTC) *'''Support''', of course. Hopefully, you'll have more time than I to help (I sadly overestimated my amount of time for this year...). --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 08:06, 13 September 2023 (UTC) * '''support''' [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:19, 13 September 2023 (UTC) * '''support''' Gan Lihao coming on as an associate editor, but we should also decide on a clear idea of what the process would be (timeline/criteria) to move them (or any other associate editor in a similar situation) to full editor --[[User:Mstefan|Mstefan]] ([[User talk:Mstefan|discuss]] • [[Special:Contributions/Mstefan|contribs]]) 12:52, 13 September 2023 (UTC) *:Good point. I think we will "cross that bridge" and evaluate once we see the [[WikiJournal of Humanities/Potential upcoming articles|backlog submissions]] getting chipped away by the newly recruited editors and associate editor. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:11, 18 September 2023 (UTC) '''Result: Accepted into the editorial board.''' : [[WikiJournal User Group/Editorial guidelines#Adding editorial board members|Next steps]] (add <code>DONE</code> or <code><nowiki>{{Done}}</nowiki></code> after someone has performed the task): # [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) {{clickable button 2|Onboarding email template|url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member}} # Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template # Add their name and start data to the [d:{{WJQboard|default=Q75674277}} relevant editorial board] on wikidata # Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members # Welcome them at the {{#if:|wjm|WJM}}board mailing list so that they are informed # Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 06:31, 6 November 2023 (UTC) ==Editorial board application of Laura G. Campo== {{WikiJournal editor application submitted | position =Editorial board | name =Laura G. Campo | qualifications =Bachelor Degree in Literature, Especialized in Edition | link =https://www.linkedin.com/in/laura-giselle-campo-sepulveda/ | areas_of_expertise =Literature, Education, Humanities | professional_experience =Literary analyst specializing in text editing. My career has been focused on the editing and proofreading of technical and literary documents. I also have experience accompanying research projects on journalism, literature, art and cultural articles. | publishing_experience =Journal editorial coordinator, Editorial assistant, Content creator,Copyeditor, Proofreader. | open_experience =Currently I coordinate the editorial production of the Universidad Pedagogica Nacional's (Colombia) scientistic journals | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:LaGCampo|LaGCampo]] ([[User talk:LaGCampo|discuss]] • [[Special:Contributions/LaGCampo|contribs]]) 13:39, 31 October 2023 (UTC) }} * I met Laura while presenting WikiJournal during Open Access week in Colombia. I '''support''' her application given her expertise in journal administration. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 06:29, 6 November 2023 (UTC) * I support this application. [[User:Fransplace|<b><span style="color: #0000FF;">Fransplace</span></b>]][[User talk:Fransplace|<b><span style="color: green;">^{Talk page}</span></b>]] 21:27, 10 January 2024 (UTC) * Laura is highly qualified, I support this application.[[User:Jacknunn|Jacknunn]] ([[User talk:Jacknunn|discuss]] • [[Special:Contributions/Jacknunn|contribs]]) 10:13, 31 January 2024 (UTC) * I support, looks like an ideal addition [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 07:20, 2 February 2024 (UTC) * Sure, particularly given OhanaUnited met them in person. --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:53, 5 December 2024 (UTC) ==Associate editor application of Taofeeq Idowu ABDULKAREEM== {{WikiJournal editor application submitted | position =Associate editor | name = Taofeeq Idowu ABDULKAREEM | qualifications = B.A History and International Studies; Member of Historical Society of Nigeria; Founder and Writer for Taofeeq’s Exposure | link = https://www.linkedin.com/in/taofeeq-idowu-abdulkareem-mhsn-b3479a1b2 | areas_of_expertise = History and International Studies | professional_experience = His professional experience can be found in Research, Content writing and Proofreading. He has made series of research in different historical events among which were titled " 'The Great Wall of China', 'The first Nigeria’s National Anthem', 'India’s great voyage to the Mars' " among others. He made a pioneer work on a topic he used for his undergraduate project research titled "Change and Continuity in Sociopolitical Role of Women in Owo, 1900-1970". This significant work was a culmination of historical research and historical analysis which would be used for further reference in the subject matter. He was appointed as the Project Coordinator for the Undergraduate Project Research because of his resourcefulness in research and editing. During the period, he coordinated over 30 co-supervises and helped a lot of them with the research and also editing. This makes the Supervisor work much more easier. As a member of University of Ilorin Model United Nations, he has made numerous research on International happenings and International relations | publishing_experience = He is a content writer, content editor, researcher, proofreader. He was a member of the Editorial team of the 2023 Journal of the National Association of Ondo State Students, University of Ilorin, Ilorin, Nigeria; He was the Assistant Director of Research and Editorial of the Alternative Dispute Resolution, University of Ilorin, Ilorin, Nigeria; He was an astute writer and editor for Union of Campus Journalists, University of Ilorin, Ilorin, Nigeria. He provided proofreading assistance for his Long Essay Undergraduate research Supervisor, thereby successfully proofread over 20 undergraduate Project Researches suitable for publication. His experience can also be found in helping editing articles that are suitable and professional for publish | open_experience = He is having over 3 years of experience in Wikimedia. He is keen interested individual in open source as he is more interested in people accessing information. He was the Vice President, Training and Development for Wikimedia Fan Club, University of Ilorin where he trained a lot of members on editing on Wikipedia and various other Sibling projects. He led Wikimedia Awareness in Ogbomosho Project where series of people were trained. He had also co-facilitated series of Projects among which are Wikimedia Promotion in Akure, Wikimedia Promotion in Lead City University, Wiki and Health Articles in Nigeria among other projects | policy_confirm = I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 09:05, 11 September 2024 (UTC) }} * {{ping|Taofeeq Abdulkareem}} Sorry for the delay, I recently found time to review your application. You definitely have sufficient level of professional and open experience (as demonstrated in your contribution activities on wiki). I would like to know more about your publishing experience. Can you tell me more, such as providing links to your published works? Do you have a list of your publications? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:46, 14 October 2024 (UTC) *:@[[User:OhanaUnited|OhanaUnited]] Thanks for the review and kind comments. *:Kindly find attached below the list of Publications: *:# Change and Continuity in Socio-political Role of Women in Owo, 1900-1970 *:# The Great Wall of China *:# The First Nigeria's National Anthem *:# India's great voyage to the Mars *:# 60 Years Journey of Nigeria's Independence *:Links to the Publications respectively: *:* https://drive.google.com/file/d/16c8WDHbArhFit9-p8isLMJ9CzgKklzBp/view?usp=drivesdk *:* https://taofeeqexposure.wordpress.com/2020/07/09/the-great-wall-of-china/ *:* https://taofeeqexposure.wordpress.com/2020/07/11/the-first-nigeria-national-anthem/ *:* https://taofeeqexposure.wordpress.com/2020/08/16/indiathe-pride-of-asia-the-great-journey-to-mars/ *:* https://taofeeqexposure.wordpress.com/2020/10/01/60-years-journey-of-nigerias-independence/ *:[[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 12:09, 16 October 2024 (UTC) *::@[[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] Thank you. Blog posts are not what I considered as publishing experience. Other than the undergraduate thesis, do you have any examples of publishing in a peer-reviewed journal article or book chapter? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 16:20, 24 October 2024 (UTC) *:::Thank you for your prompt response. I appreciate your feedback and understand your concerns regarding my publishing experience. While my publication record in peer-reviewed journals may be limited, I would like to highlight my research experience in significant aspects of humanities, including [cultural studies, historical analysis, among others aspects]. Although blog posts may not be traditional publications, they demonstrate my ability to make research and communicate complex ideas to diverse audiences. *:::Beyond publishing, I've developed valuable skills through Undergraduate thesis research, Editing and proofreading for others, Research assistance in humanities topics. *:::I bring strong research foundation in humanities, excellent writing, editing, and proofreading skills, ability to communicate complex ideas engagingly, experience working with diverse authors and topics, passion for promoting high-quality humanities research. I am eager to leverage these skills to support Wikimedia Journal's mission. I understand the importance of peer-reviewed publications and commit to further developing my expertise. *:::I would appreciate consideration of my application, recognizing the diverse experiences and skills I bring. Thank you for your time, and I look forward to your response. [[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 09:40, 27 October 2024 (UTC) *::::I am '''support'''ive of your associate editor application, contingent on mentorship from board members, to help you gain experience around the publishing area. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:54, 14 November 2024 (UTC) *:::::Thank you for your prompt and warm response. I am thrilled to join the team and contribute to the Humanities journal. As a passionate, ambitious, and evolving individual, I am committed to continuous learning, growth, and development. *:::::I would greatly appreciate mentorship from the board members to enhance my publishing knowledge and skills. I am eager to apply these skills in my role and contribute meaningfully to the team's growth and success. *:::::I look forward to the next steps and onboarding process, I am delighted to be part of this team and make a positive impact. [[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 20:25, 14 November 2024 (UTC) *::::::Please wait for other editorial board members to review and comment on your application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:44, 18 November 2024 (UTC) ==Associate editor application of Sideeq Abubakar Galadima== {{WikiJournal editor application submitted | position =Associate editor | name =Sideeq Abubakar Galadima | qualifications =B.A. History and International Studies | link = | areas_of_expertise =History, Diplomacy, Planning and Management | professional_experience =His professional experience is deeply rooted in his academic background in History and International Studies, which has familiarized him with the intricacies of objective research, writing, and reportage. His expertise in these areas was further strengthened by his active engagement in news and report writing as a member of the Union of Campus Journalists during his undergraduate studies. Additionally, his experience as a Wikimedia editor has honed his proofreading skills. As an event planner, he has developed exceptional attention to detail, which has become an integral part of his skillset. Notably, his pioneering research work, titled "Colonialism and the Continuity of Ilorin Cultural Heritage, 1900-1960," demonstrates his ability to conduct in-depth historical analysis and research. This work will undoubtedly serve as a valuable reference for future studies in related fields, such as cultural diplomacy. | publishing_experience =He's a researcher, news and reports writer, content editor, proofreader | open_experience =He possesses over three years of experience in Wikimedia, driven by a strong interest in open-source initiatives. Notably, he served as the Special Duties Officer for the Wikimedia Fan Club at the University of Ilorin, where he played a pivotal role in facilitating and training sessions on Wikipedia and its sister projects, as well as co-facilitating workshops, including "Wiki and Health Articles in Nigeria" and "Wikimedia Awareness in Ogbomosho". Through these endeavors, He demonstrated his expertise in promoting open-source knowledge sharing and community engagement. His experience and commitment to Wikimedia's mission have equipped him with a unique skill set, poised to contribute to future initiatives. | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 17:54, 11 September 2024 (UTC) }} * I really appreciate Sideeq's Wikipedia contributions to topics in Africa. It sounds like the highest degree earned is B.A., and no journal editor experience? I think normally we expect a PhD and some academic journal experience. Also it would be good to have a link to the ""Colonialism and the Continuity of Ilorin Cultural Heritage, 1900-1960", which I wasn't able to find. [[User:Aoholcombe|Aoholcombe]] ([[User talk:Aoholcombe|discuss]] • [[Special:Contributions/Aoholcombe|contribs]]) 23:25, 2 October 2024 (UTC) *:I agree with your comment. I wasn't able to find this applicant's published work list and I am hesitant with professional experience even for applying as an associate editor position. While the applicant has some experience with open access, the activity was sporadic. However, I think it may be beneficial to have additional volunteers to support this journal that deals with the administrative side of things and less reliant on professional and publishing experiences' side of the journal. @[[User:Albakry028|Albakry028]], in case you didn't see the previous comment, can you provide us with more information? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:55, 14 October 2024 (UTC) *:Thank you for acknowledging my contributions to African topics on Wikipedia. I appreciate your recognition of my efforts. Regarding your inquiries, I would like to clarify that my highest educational attainment is a Bachelor of Arts degree. Nevertheless, my editorial expertise has enabled me to assist colleagues with their research projects, leveraging my skills in research and academic writing. I understand and respect the standard expectations associated with academic roles. However, I was entrusted with this responsibility due to my demonstrated expertise. Regarding my research work, I am pleased to share the link to my project: "Colonialism and the Continuity of Ilorin Cultural Heritage, 1900-1960." https://drive.google.com/file/d/1bxysalU-AT7JakWfJCFxeWqwpFCz_C7s/view?usp=drivesdk @[[User:Aoholcombe|Aoholcombe]] @[[User:OhanaUnited|OhanaUnited]] [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 13:50, 16 October 2024 (UTC) :@[[User:Albakry028|Albakry028]] Thanks very much for providing your writing example. Do you have any publishing experience? We are looking for something beyond undergraduate thesis (for example, peer-reviewed journal article or book chapters). I am trained as a scientist and therefore will need more information to assess an applicant's suitability in applying for a humanities position. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 16:18, 24 October 2024 (UTC) :Although my publishing experience is limited to my undergraduate thesis, I'm confident in my potential. I bring transferable skills: research expertise, writing proficiency, adaptability, analytical thinking and effective communication. I'm eager to apply research methodology perspectives to humanities contexts, quickly learn and adapt. I'm poised to contribute innovatively through interdisciplinary research, engaging teaching methods and collaborative projects. I appreciate your consideration of potential over conventional metrics. [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 13:38, 25 October 2024 (UTC) ::I am happy to '''support''' your associate editor application, contingent on board members' availability, to mentor you to gain experience around the publishing area. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:55, 14 November 2024 (UTC) :::Thank you for your kind and supportive message. I am thrilled to join the team and grateful for the opportunity to work alongside experienced board members. I am eager to benefit from their mentorship and expertise, which will undoubtedly enhance my skills and knowledge in the publishing field. :::As a dedicated and passionate individual, I am committed to contributing to the humanities journal and supporting its growth. I am excited to embark on this journey and engage in meaningful discussions as a team member. :::I look forward to the next steps and onboarding process. [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 20:44, 14 November 2024 (UTC) ::::Please wait for other editorial board members to review and comment on your application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:38, 18 November 2024 (UTC) gsn1z74ojr89saueufrypupni62ljku 2690342 2690341 2024-12-05T05:54:29Z Piotrus 571 /* Associate editor application of Taofeeq Idowu ABDULKAREEM */ 2690342 wikitext text/x-wiki <noinclude> {{WikiJournal editorial application top |archive box = {{Archive box|[[/Archive 2017]] <br>[[/Archive 2018]] <br>[[/Archive 2019]] <br>[[/Archive 2020]] <br>[[/Archive 2022]] <br>[[/Archive 2023]] }} }} </noinclude> ==Editorial board application of Hernan Perez Molano== {{WikiJournal editor application submitted | position =Editorial board | name =Hernan Perez Molano | qualifications =PHD in Political science, Master in Ethnomusicology | link =https://es.linkedin.com/in/hernan-p%C3%A9rez-molano-918252a1 | areas_of_expertise =Peacebuilding, social innovation, political science, ethnomusicology | professional_experience =Doctor of Political Science, Administration, and International Relations, from the Complutense University of Madrid (Spain), trained in ethnographic, sociological, and anthropological techniques (Master's in Musicology, specializing in Ethnomusicology) at the Sorbonne University (France). His research, entitled "Obstacles and Resistances in the Construction of Alternative Peace: Comparative Ethnographies of the Reintegration of Former Combatants in Colinas, Guaviare, and Icononzo, Tolima," describes the construction of peace at the local level from the perspective of local social innovation ecosystems, based on a multi-sited ethnography (2019-2023). :Coordinator of the Social Innovation Program (2015-2020) at the Research and Extension Office of the National University of Colombia, Bogotá campus. He has experience in supporting academia in formulating and implementing social innovation projects, utilizing participatory methodologies, design thinking, and fostering creative capacity in the context of community youth processes, as well as in communication and culture for peacebuilding. He was a former member of the formulating team, facilitator, and coordinator of the Innovation Laboratory for Peace (Trust for the Americas - National University of Colombia), and the Spaces of Re-cognition for Peace project of the Academic Vice-Rectory of the National University of Colombia. | publishing_experience = | open_experience =Official for the Education program of Wikimedia Colombia | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:HerPerezM|HerPerezM]] ([[User talk:HerPerezM|discuss]] • [[Special:Contributions/HerPerezM|contribs]]) 21:42, 20 July 2023 (UTC) }} * I approached him at EduWiki Conference to discuss WikiJournal and potential collaboration. I fully support his application to join the editorial board. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:47, 21 July 2023 (UTC) * [[File:Symbol support vote.svg|14px]]I support this application for editor. [[User:Smvital|<b><span style="color: #0000FF;">Smvital</span></b>]][[User talk:Smvital|<b><span style="color: green;">^{Talk page}</span></b>]] 10:46, 1 August 2023 (UTC) * '''support''' - It's also a support from me. Very useful professional bacckground, and experience with Wikimedia Colombia's educaction programme is definitely a bonus. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 10:45, 28 August 2023 (UTC) * I support this application. I agree; his area of study and experience will make him very suitable. [[User:Fransplace|<b><span style="color: #0000FF;">Fransplace</span></b>]][[User talk:Fransplace|<b><span style="color: green;">^{Talk page}</span></b>]] 20:01, 10 September 2023 (UTC) *'''Support''', of course. Hopefully, you'll have more time than I to help (I sadly overestimated my amount of time for this year...). --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 08:05, 13 September 2023 (UTC) * '''support''' [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:18, 13 September 2023 (UTC) * '''support''' - a very welcome addition to the WikiJ Hum Team --[[User:Mstefan|Mstefan]] ([[User talk:Mstefan|discuss]] • [[Special:Contributions/Mstefan|contribs]]) 12:48, 13 September 2023 (UTC) '''Result: Accepted into the editorial board.''' : [[WikiJournal User Group/Editorial guidelines#Adding editorial board members|Next steps]] (add <code>DONE</code> or <code><nowiki>{{Done}}</nowiki></code> after someone has performed the task): # [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) {{clickable button 2|Onboarding email template|url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member}} # Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template # Add their name and start data to the [d:{{WJQboard|default=Q75674277}} relevant editorial board] on wikidata # Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members # Welcome them at the {{#if:|wjm|WJM}}board mailing list so that they are informed # Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 06:31, 6 November 2023 (UTC) ==Editorial board application of Lihao Gan== {{WikiJournal editor application submitted | position =Editorial board | name =Lihao Gan | qualifications =PHD.Professor | link =https://faculty.ecnu.edu.cn/_s11/glh_en/main.psp | areas_of_expertise =Epistemology，Communication Studies，Media Discourse Analysis，Rhetoric | professional_experience =Gan Lihao (born October 1977) is a professor and doctoral supervisor at East China Normal University. He is a distinguished talent of the Pujiang Talent Program in Shanghai. He has also served as a visiting scholar in the Department of Linguistics at the University of California, Berkeley. Additionally, he holds the position of Deputy Director at the National Discourse Ecology Research Center and serves as an executive member of the Chinese Rhetoric Society, a council member of the Shanghai Language Society, and a committee member of the Audiovisual Communication branch of the Chinese Association for the History of Journalism and Communication. | publishing_experience =Gan Lihao is known for his pioneering contributions to the fields of "Life Rhetoric" and "Behavioral Dramatism Theory." His research primarily revolves around human communication discourse, aiming to promote individual growth, harmonious family dynamics, intercommunication among domestic communities, and international dialogues within the context of the human community's shared destiny and peaceful development. He focuses on three main research directions: family education discourse analysis based on empathetic rhetoric, discourse research on national governance rooted in speech acts, and global knowledge discourse analysis centered around digital communities. Gan Lihao has authored several significant works, including "Contrastive Structures Under the Influence of Spatial Dynamics," "Communication Rhetoric: Theory, Methods, and Case Studies," "Reshaping China's National Image and Wikipedia Knowledge Discourse Research," and "Political Science on Wikipedia" (in progress). | open_experience =wikipedia editor,wikipedia researcher | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:Ganlihao|Ganlihao]] ([[User talk:Ganlihao|discuss]] • [[Special:Contributions/Ganlihao|contribs]]) 06:30, 4 September 2023 (UTC) }} * This editor approached us at the Wikimania Singapore event and we discussed how we need experts in humanities to contribute and assist with reviewing the backlogged submissions. He expressed an interest after seeing our poster at Wikimania. He led a team of researchers from China to investigate and publish research articles about Wikipedia. As such, his professional, publishing and open experiences are quite extensive. Since he primarily publishes in Chinese language, I suggested that he initially apply for associate editor position to familiarize himself with publishing and communicating in English to gain confidence in this area. I fully {{support}} his application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:52, 7 September 2023 (UTC) * I support this application and agree an associate editor position will be best to begin with. [[User:Fransplace|<b><span style="color: #0000FF;">Fransplace</span></b>]][[User talk:Fransplace|<b><span style="color: green;">^{Talk page}</span></b>]] 20:05, 10 September 2023 (UTC) *'''Support''', of course. Hopefully, you'll have more time than I to help (I sadly overestimated my amount of time for this year...). --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 08:06, 13 September 2023 (UTC) * '''support''' [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:19, 13 September 2023 (UTC) * '''support''' Gan Lihao coming on as an associate editor, but we should also decide on a clear idea of what the process would be (timeline/criteria) to move them (or any other associate editor in a similar situation) to full editor --[[User:Mstefan|Mstefan]] ([[User talk:Mstefan|discuss]] • [[Special:Contributions/Mstefan|contribs]]) 12:52, 13 September 2023 (UTC) *:Good point. I think we will "cross that bridge" and evaluate once we see the [[WikiJournal of Humanities/Potential upcoming articles|backlog submissions]] getting chipped away by the newly recruited editors and associate editor. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:11, 18 September 2023 (UTC) '''Result: Accepted into the editorial board.''' : [[WikiJournal User Group/Editorial guidelines#Adding editorial board members|Next steps]] (add <code>DONE</code> or <code><nowiki>{{Done}}</nowiki></code> after someone has performed the task): # [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) {{clickable button 2|Onboarding email template|url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member}} # Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template # Add their name and start data to the [d:{{WJQboard|default=Q75674277}} relevant editorial board] on wikidata # Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members # Welcome them at the {{#if:|wjm|WJM}}board mailing list so that they are informed # Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 06:31, 6 November 2023 (UTC) ==Editorial board application of Laura G. Campo== {{WikiJournal editor application submitted | position =Editorial board | name =Laura G. Campo | qualifications =Bachelor Degree in Literature, Especialized in Edition | link =https://www.linkedin.com/in/laura-giselle-campo-sepulveda/ | areas_of_expertise =Literature, Education, Humanities | professional_experience =Literary analyst specializing in text editing. My career has been focused on the editing and proofreading of technical and literary documents. I also have experience accompanying research projects on journalism, literature, art and cultural articles. | publishing_experience =Journal editorial coordinator, Editorial assistant, Content creator,Copyeditor, Proofreader. | open_experience =Currently I coordinate the editorial production of the Universidad Pedagogica Nacional's (Colombia) scientistic journals | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:LaGCampo|LaGCampo]] ([[User talk:LaGCampo|discuss]] • [[Special:Contributions/LaGCampo|contribs]]) 13:39, 31 October 2023 (UTC) }} * I met Laura while presenting WikiJournal during Open Access week in Colombia. I '''support''' her application given her expertise in journal administration. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 06:29, 6 November 2023 (UTC) * I support this application. [[User:Fransplace|<b><span style="color: #0000FF;">Fransplace</span></b>]][[User talk:Fransplace|<b><span style="color: green;">^{Talk page}</span></b>]] 21:27, 10 January 2024 (UTC) * Laura is highly qualified, I support this application.[[User:Jacknunn|Jacknunn]] ([[User talk:Jacknunn|discuss]] • [[Special:Contributions/Jacknunn|contribs]]) 10:13, 31 January 2024 (UTC) * I support, looks like an ideal addition [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 07:20, 2 February 2024 (UTC) * Sure, particularly given OhanaUnited met them in person. --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:53, 5 December 2024 (UTC) ==Associate editor application of Taofeeq Idowu ABDULKAREEM== {{WikiJournal editor application submitted | position =Associate editor | name = Taofeeq Idowu ABDULKAREEM | qualifications = B.A History and International Studies; Member of Historical Society of Nigeria; Founder and Writer for Taofeeq’s Exposure | link = https://www.linkedin.com/in/taofeeq-idowu-abdulkareem-mhsn-b3479a1b2 | areas_of_expertise = History and International Studies | professional_experience = His professional experience can be found in Research, Content writing and Proofreading. He has made series of research in different historical events among which were titled " 'The Great Wall of China', 'The first Nigeria’s National Anthem', 'India’s great voyage to the Mars' " among others. He made a pioneer work on a topic he used for his undergraduate project research titled "Change and Continuity in Sociopolitical Role of Women in Owo, 1900-1970". This significant work was a culmination of historical research and historical analysis which would be used for further reference in the subject matter. He was appointed as the Project Coordinator for the Undergraduate Project Research because of his resourcefulness in research and editing. During the period, he coordinated over 30 co-supervises and helped a lot of them with the research and also editing. This makes the Supervisor work much more easier. As a member of University of Ilorin Model United Nations, he has made numerous research on International happenings and International relations | publishing_experience = He is a content writer, content editor, researcher, proofreader. He was a member of the Editorial team of the 2023 Journal of the National Association of Ondo State Students, University of Ilorin, Ilorin, Nigeria; He was the Assistant Director of Research and Editorial of the Alternative Dispute Resolution, University of Ilorin, Ilorin, Nigeria; He was an astute writer and editor for Union of Campus Journalists, University of Ilorin, Ilorin, Nigeria. He provided proofreading assistance for his Long Essay Undergraduate research Supervisor, thereby successfully proofread over 20 undergraduate Project Researches suitable for publication. His experience can also be found in helping editing articles that are suitable and professional for publish | open_experience = He is having over 3 years of experience in Wikimedia. He is keen interested individual in open source as he is more interested in people accessing information. He was the Vice President, Training and Development for Wikimedia Fan Club, University of Ilorin where he trained a lot of members on editing on Wikipedia and various other Sibling projects. He led Wikimedia Awareness in Ogbomosho Project where series of people were trained. He had also co-facilitated series of Projects among which are Wikimedia Promotion in Akure, Wikimedia Promotion in Lead City University, Wiki and Health Articles in Nigeria among other projects | policy_confirm = I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 09:05, 11 September 2024 (UTC) }} * {{ping|Taofeeq Abdulkareem}} Sorry for the delay, I recently found time to review your application. You definitely have sufficient level of professional and open experience (as demonstrated in your contribution activities on wiki). I would like to know more about your publishing experience. Can you tell me more, such as providing links to your published works? Do you have a list of your publications? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:46, 14 October 2024 (UTC) *:@[[User:OhanaUnited|OhanaUnited]] Thanks for the review and kind comments. *:Kindly find attached below the list of Publications: *:# Change and Continuity in Socio-political Role of Women in Owo, 1900-1970 *:# The Great Wall of China *:# The First Nigeria's National Anthem *:# India's great voyage to the Mars *:# 60 Years Journey of Nigeria's Independence *:Links to the Publications respectively: *:* https://drive.google.com/file/d/16c8WDHbArhFit9-p8isLMJ9CzgKklzBp/view?usp=drivesdk *:* https://taofeeqexposure.wordpress.com/2020/07/09/the-great-wall-of-china/ *:* https://taofeeqexposure.wordpress.com/2020/07/11/the-first-nigeria-national-anthem/ *:* https://taofeeqexposure.wordpress.com/2020/08/16/indiathe-pride-of-asia-the-great-journey-to-mars/ *:* https://taofeeqexposure.wordpress.com/2020/10/01/60-years-journey-of-nigerias-independence/ *:[[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 12:09, 16 October 2024 (UTC) *::@[[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] Thank you. Blog posts are not what I considered as publishing experience. Other than the undergraduate thesis, do you have any examples of publishing in a peer-reviewed journal article or book chapter? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 16:20, 24 October 2024 (UTC) *:::Thank you for your prompt response. I appreciate your feedback and understand your concerns regarding my publishing experience. While my publication record in peer-reviewed journals may be limited, I would like to highlight my research experience in significant aspects of humanities, including [cultural studies, historical analysis, among others aspects]. Although blog posts may not be traditional publications, they demonstrate my ability to make research and communicate complex ideas to diverse audiences. *:::Beyond publishing, I've developed valuable skills through Undergraduate thesis research, Editing and proofreading for others, Research assistance in humanities topics. *:::I bring strong research foundation in humanities, excellent writing, editing, and proofreading skills, ability to communicate complex ideas engagingly, experience working with diverse authors and topics, passion for promoting high-quality humanities research. I am eager to leverage these skills to support Wikimedia Journal's mission. I understand the importance of peer-reviewed publications and commit to further developing my expertise. *:::I would appreciate consideration of my application, recognizing the diverse experiences and skills I bring. Thank you for your time, and I look forward to your response. [[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 09:40, 27 October 2024 (UTC) *::::I am '''support'''ive of your associate editor application, contingent on mentorship from board members, to help you gain experience around the publishing area. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:54, 14 November 2024 (UTC) *:::::Thank you for your prompt and warm response. I am thrilled to join the team and contribute to the Humanities journal. As a passionate, ambitious, and evolving individual, I am committed to continuous learning, growth, and development. *:::::I would greatly appreciate mentorship from the board members to enhance my publishing knowledge and skills. I am eager to apply these skills in my role and contribute meaningfully to the team's growth and success. *:::::I look forward to the next steps and onboarding process, I am delighted to be part of this team and make a positive impact. [[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 20:25, 14 November 2024 (UTC) *::::::Please wait for other editorial board members to review and comment on your application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:44, 18 November 2024 (UTC) *'''Support'''. Having read the above, welcome aboard. --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:54, 5 December 2024 (UTC) ==Associate editor application of Sideeq Abubakar Galadima== {{WikiJournal editor application submitted | position =Associate editor | name =Sideeq Abubakar Galadima | qualifications =B.A. History and International Studies | link = | areas_of_expertise =History, Diplomacy, Planning and Management | professional_experience =His professional experience is deeply rooted in his academic background in History and International Studies, which has familiarized him with the intricacies of objective research, writing, and reportage. His expertise in these areas was further strengthened by his active engagement in news and report writing as a member of the Union of Campus Journalists during his undergraduate studies. Additionally, his experience as a Wikimedia editor has honed his proofreading skills. As an event planner, he has developed exceptional attention to detail, which has become an integral part of his skillset. Notably, his pioneering research work, titled "Colonialism and the Continuity of Ilorin Cultural Heritage, 1900-1960," demonstrates his ability to conduct in-depth historical analysis and research. This work will undoubtedly serve as a valuable reference for future studies in related fields, such as cultural diplomacy. | publishing_experience =He's a researcher, news and reports writer, content editor, proofreader | open_experience =He possesses over three years of experience in Wikimedia, driven by a strong interest in open-source initiatives. Notably, he served as the Special Duties Officer for the Wikimedia Fan Club at the University of Ilorin, where he played a pivotal role in facilitating and training sessions on Wikipedia and its sister projects, as well as co-facilitating workshops, including "Wiki and Health Articles in Nigeria" and "Wikimedia Awareness in Ogbomosho". Through these endeavors, He demonstrated his expertise in promoting open-source knowledge sharing and community engagement. His experience and commitment to Wikimedia's mission have equipped him with a unique skill set, poised to contribute to future initiatives. | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 17:54, 11 September 2024 (UTC) }} * I really appreciate Sideeq's Wikipedia contributions to topics in Africa. It sounds like the highest degree earned is B.A., and no journal editor experience? I think normally we expect a PhD and some academic journal experience. Also it would be good to have a link to the ""Colonialism and the Continuity of Ilorin Cultural Heritage, 1900-1960", which I wasn't able to find. [[User:Aoholcombe|Aoholcombe]] ([[User talk:Aoholcombe|discuss]] • [[Special:Contributions/Aoholcombe|contribs]]) 23:25, 2 October 2024 (UTC) *:I agree with your comment. I wasn't able to find this applicant's published work list and I am hesitant with professional experience even for applying as an associate editor position. While the applicant has some experience with open access, the activity was sporadic. However, I think it may be beneficial to have additional volunteers to support this journal that deals with the administrative side of things and less reliant on professional and publishing experiences' side of the journal. @[[User:Albakry028|Albakry028]], in case you didn't see the previous comment, can you provide us with more information? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:55, 14 October 2024 (UTC) *:Thank you for acknowledging my contributions to African topics on Wikipedia. I appreciate your recognition of my efforts. Regarding your inquiries, I would like to clarify that my highest educational attainment is a Bachelor of Arts degree. Nevertheless, my editorial expertise has enabled me to assist colleagues with their research projects, leveraging my skills in research and academic writing. I understand and respect the standard expectations associated with academic roles. However, I was entrusted with this responsibility due to my demonstrated expertise. Regarding my research work, I am pleased to share the link to my project: "Colonialism and the Continuity of Ilorin Cultural Heritage, 1900-1960." https://drive.google.com/file/d/1bxysalU-AT7JakWfJCFxeWqwpFCz_C7s/view?usp=drivesdk @[[User:Aoholcombe|Aoholcombe]] @[[User:OhanaUnited|OhanaUnited]] [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 13:50, 16 October 2024 (UTC) :@[[User:Albakry028|Albakry028]] Thanks very much for providing your writing example. Do you have any publishing experience? We are looking for something beyond undergraduate thesis (for example, peer-reviewed journal article or book chapters). I am trained as a scientist and therefore will need more information to assess an applicant's suitability in applying for a humanities position. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 16:18, 24 October 2024 (UTC) :Although my publishing experience is limited to my undergraduate thesis, I'm confident in my potential. I bring transferable skills: research expertise, writing proficiency, adaptability, analytical thinking and effective communication. I'm eager to apply research methodology perspectives to humanities contexts, quickly learn and adapt. I'm poised to contribute innovatively through interdisciplinary research, engaging teaching methods and collaborative projects. I appreciate your consideration of potential over conventional metrics. [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 13:38, 25 October 2024 (UTC) ::I am happy to '''support''' your associate editor application, contingent on board members' availability, to mentor you to gain experience around the publishing area. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:55, 14 November 2024 (UTC) :::Thank you for your kind and supportive message. I am thrilled to join the team and grateful for the opportunity to work alongside experienced board members. I am eager to benefit from their mentorship and expertise, which will undoubtedly enhance my skills and knowledge in the publishing field. :::As a dedicated and passionate individual, I am committed to contributing to the humanities journal and supporting its growth. I am excited to embark on this journey and engage in meaningful discussions as a team member. :::I look forward to the next steps and onboarding process. [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 20:44, 14 November 2024 (UTC) ::::Please wait for other editorial board members to review and comment on your application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:38, 18 November 2024 (UTC) cxb8jltxumy897my2d66ty1wyrouxng 2690343 2690342 2024-12-05T05:56:04Z Piotrus 571 /* Associate editor application of Sideeq Abubakar Galadima */ 2690343 wikitext text/x-wiki <noinclude> {{WikiJournal editorial application top |archive box = {{Archive box|[[/Archive 2017]] <br>[[/Archive 2018]] <br>[[/Archive 2019]] <br>[[/Archive 2020]] <br>[[/Archive 2022]] <br>[[/Archive 2023]] }} }} </noinclude> ==Editorial board application of Hernan Perez Molano== {{WikiJournal editor application submitted | position =Editorial board | name =Hernan Perez Molano | qualifications =PHD in Political science, Master in Ethnomusicology | link =https://es.linkedin.com/in/hernan-p%C3%A9rez-molano-918252a1 | areas_of_expertise =Peacebuilding, social innovation, political science, ethnomusicology | professional_experience =Doctor of Political Science, Administration, and International Relations, from the Complutense University of Madrid (Spain), trained in ethnographic, sociological, and anthropological techniques (Master's in Musicology, specializing in Ethnomusicology) at the Sorbonne University (France). His research, entitled "Obstacles and Resistances in the Construction of Alternative Peace: Comparative Ethnographies of the Reintegration of Former Combatants in Colinas, Guaviare, and Icononzo, Tolima," describes the construction of peace at the local level from the perspective of local social innovation ecosystems, based on a multi-sited ethnography (2019-2023). :Coordinator of the Social Innovation Program (2015-2020) at the Research and Extension Office of the National University of Colombia, Bogotá campus. He has experience in supporting academia in formulating and implementing social innovation projects, utilizing participatory methodologies, design thinking, and fostering creative capacity in the context of community youth processes, as well as in communication and culture for peacebuilding. He was a former member of the formulating team, facilitator, and coordinator of the Innovation Laboratory for Peace (Trust for the Americas - National University of Colombia), and the Spaces of Re-cognition for Peace project of the Academic Vice-Rectory of the National University of Colombia. | publishing_experience = | open_experience =Official for the Education program of Wikimedia Colombia | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:HerPerezM|HerPerezM]] ([[User talk:HerPerezM|discuss]] • [[Special:Contributions/HerPerezM|contribs]]) 21:42, 20 July 2023 (UTC) }} * I approached him at EduWiki Conference to discuss WikiJournal and potential collaboration. I fully support his application to join the editorial board. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:47, 21 July 2023 (UTC) * [[File:Symbol support vote.svg|14px]]I support this application for editor. [[User:Smvital|<b><span style="color: #0000FF;">Smvital</span></b>]][[User talk:Smvital|<b><span style="color: green;">^{Talk page}</span></b>]] 10:46, 1 August 2023 (UTC) * '''support''' - It's also a support from me. Very useful professional bacckground, and experience with Wikimedia Colombia's educaction programme is definitely a bonus. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 10:45, 28 August 2023 (UTC) * I support this application. I agree; his area of study and experience will make him very suitable. [[User:Fransplace|<b><span style="color: #0000FF;">Fransplace</span></b>]][[User talk:Fransplace|<b><span style="color: green;">^{Talk page}</span></b>]] 20:01, 10 September 2023 (UTC) *'''Support''', of course. Hopefully, you'll have more time than I to help (I sadly overestimated my amount of time for this year...). --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 08:05, 13 September 2023 (UTC) * '''support''' [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:18, 13 September 2023 (UTC) * '''support''' - a very welcome addition to the WikiJ Hum Team --[[User:Mstefan|Mstefan]] ([[User talk:Mstefan|discuss]] • [[Special:Contributions/Mstefan|contribs]]) 12:48, 13 September 2023 (UTC) '''Result: Accepted into the editorial board.''' : [[WikiJournal User Group/Editorial guidelines#Adding editorial board members|Next steps]] (add <code>DONE</code> or <code><nowiki>{{Done}}</nowiki></code> after someone has performed the task): # [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) {{clickable button 2|Onboarding email template|url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member}} # Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template # Add their name and start data to the [d:{{WJQboard|default=Q75674277}} relevant editorial board] on wikidata # Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members # Welcome them at the {{#if:|wjm|WJM}}board mailing list so that they are informed # Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 06:31, 6 November 2023 (UTC) ==Editorial board application of Lihao Gan== {{WikiJournal editor application submitted | position =Editorial board | name =Lihao Gan | qualifications =PHD.Professor | link =https://faculty.ecnu.edu.cn/_s11/glh_en/main.psp | areas_of_expertise =Epistemology，Communication Studies，Media Discourse Analysis，Rhetoric | professional_experience =Gan Lihao (born October 1977) is a professor and doctoral supervisor at East China Normal University. He is a distinguished talent of the Pujiang Talent Program in Shanghai. He has also served as a visiting scholar in the Department of Linguistics at the University of California, Berkeley. Additionally, he holds the position of Deputy Director at the National Discourse Ecology Research Center and serves as an executive member of the Chinese Rhetoric Society, a council member of the Shanghai Language Society, and a committee member of the Audiovisual Communication branch of the Chinese Association for the History of Journalism and Communication. | publishing_experience =Gan Lihao is known for his pioneering contributions to the fields of "Life Rhetoric" and "Behavioral Dramatism Theory." His research primarily revolves around human communication discourse, aiming to promote individual growth, harmonious family dynamics, intercommunication among domestic communities, and international dialogues within the context of the human community's shared destiny and peaceful development. He focuses on three main research directions: family education discourse analysis based on empathetic rhetoric, discourse research on national governance rooted in speech acts, and global knowledge discourse analysis centered around digital communities. Gan Lihao has authored several significant works, including "Contrastive Structures Under the Influence of Spatial Dynamics," "Communication Rhetoric: Theory, Methods, and Case Studies," "Reshaping China's National Image and Wikipedia Knowledge Discourse Research," and "Political Science on Wikipedia" (in progress). | open_experience =wikipedia editor,wikipedia researcher | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:Ganlihao|Ganlihao]] ([[User talk:Ganlihao|discuss]] • [[Special:Contributions/Ganlihao|contribs]]) 06:30, 4 September 2023 (UTC) }} * This editor approached us at the Wikimania Singapore event and we discussed how we need experts in humanities to contribute and assist with reviewing the backlogged submissions. He expressed an interest after seeing our poster at Wikimania. He led a team of researchers from China to investigate and publish research articles about Wikipedia. As such, his professional, publishing and open experiences are quite extensive. Since he primarily publishes in Chinese language, I suggested that he initially apply for associate editor position to familiarize himself with publishing and communicating in English to gain confidence in this area. I fully {{support}} his application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:52, 7 September 2023 (UTC) * I support this application and agree an associate editor position will be best to begin with. [[User:Fransplace|<b><span style="color: #0000FF;">Fransplace</span></b>]][[User talk:Fransplace|<b><span style="color: green;">^{Talk page}</span></b>]] 20:05, 10 September 2023 (UTC) *'''Support''', of course. Hopefully, you'll have more time than I to help (I sadly overestimated my amount of time for this year...). --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 08:06, 13 September 2023 (UTC) * '''support''' [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:19, 13 September 2023 (UTC) * '''support''' Gan Lihao coming on as an associate editor, but we should also decide on a clear idea of what the process would be (timeline/criteria) to move them (or any other associate editor in a similar situation) to full editor --[[User:Mstefan|Mstefan]] ([[User talk:Mstefan|discuss]] • [[Special:Contributions/Mstefan|contribs]]) 12:52, 13 September 2023 (UTC) *:Good point. I think we will "cross that bridge" and evaluate once we see the [[WikiJournal of Humanities/Potential upcoming articles|backlog submissions]] getting chipped away by the newly recruited editors and associate editor. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:11, 18 September 2023 (UTC) '''Result: Accepted into the editorial board.''' : [[WikiJournal User Group/Editorial guidelines#Adding editorial board members|Next steps]] (add <code>DONE</code> or <code><nowiki>{{Done}}</nowiki></code> after someone has performed the task): # [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) {{clickable button 2|Onboarding email template|url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member}} # Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template # Add their name and start data to the [d:{{WJQboard|default=Q75674277}} relevant editorial board] on wikidata # Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members # Welcome them at the {{#if:|wjm|WJM}}board mailing list so that they are informed # Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 06:31, 6 November 2023 (UTC) ==Editorial board application of Laura G. Campo== {{WikiJournal editor application submitted | position =Editorial board | name =Laura G. Campo | qualifications =Bachelor Degree in Literature, Especialized in Edition | link =https://www.linkedin.com/in/laura-giselle-campo-sepulveda/ | areas_of_expertise =Literature, Education, Humanities | professional_experience =Literary analyst specializing in text editing. My career has been focused on the editing and proofreading of technical and literary documents. I also have experience accompanying research projects on journalism, literature, art and cultural articles. | publishing_experience =Journal editorial coordinator, Editorial assistant, Content creator,Copyeditor, Proofreader. | open_experience =Currently I coordinate the editorial production of the Universidad Pedagogica Nacional's (Colombia) scientistic journals | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:LaGCampo|LaGCampo]] ([[User talk:LaGCampo|discuss]] • [[Special:Contributions/LaGCampo|contribs]]) 13:39, 31 October 2023 (UTC) }} * I met Laura while presenting WikiJournal during Open Access week in Colombia. I '''support''' her application given her expertise in journal administration. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 06:29, 6 November 2023 (UTC) * I support this application. [[User:Fransplace|<b><span style="color: #0000FF;">Fransplace</span></b>]][[User talk:Fransplace|<b><span style="color: green;">^{Talk page}</span></b>]] 21:27, 10 January 2024 (UTC) * Laura is highly qualified, I support this application.[[User:Jacknunn|Jacknunn]] ([[User talk:Jacknunn|discuss]] • [[Special:Contributions/Jacknunn|contribs]]) 10:13, 31 January 2024 (UTC) * I support, looks like an ideal addition [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 07:20, 2 February 2024 (UTC) * Sure, particularly given OhanaUnited met them in person. --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:53, 5 December 2024 (UTC) ==Associate editor application of Taofeeq Idowu ABDULKAREEM== {{WikiJournal editor application submitted | position =Associate editor | name = Taofeeq Idowu ABDULKAREEM | qualifications = B.A History and International Studies; Member of Historical Society of Nigeria; Founder and Writer for Taofeeq’s Exposure | link = https://www.linkedin.com/in/taofeeq-idowu-abdulkareem-mhsn-b3479a1b2 | areas_of_expertise = History and International Studies | professional_experience = His professional experience can be found in Research, Content writing and Proofreading. He has made series of research in different historical events among which were titled " 'The Great Wall of China', 'The first Nigeria’s National Anthem', 'India’s great voyage to the Mars' " among others. He made a pioneer work on a topic he used for his undergraduate project research titled "Change and Continuity in Sociopolitical Role of Women in Owo, 1900-1970". This significant work was a culmination of historical research and historical analysis which would be used for further reference in the subject matter. He was appointed as the Project Coordinator for the Undergraduate Project Research because of his resourcefulness in research and editing. During the period, he coordinated over 30 co-supervises and helped a lot of them with the research and also editing. This makes the Supervisor work much more easier. As a member of University of Ilorin Model United Nations, he has made numerous research on International happenings and International relations | publishing_experience = He is a content writer, content editor, researcher, proofreader. He was a member of the Editorial team of the 2023 Journal of the National Association of Ondo State Students, University of Ilorin, Ilorin, Nigeria; He was the Assistant Director of Research and Editorial of the Alternative Dispute Resolution, University of Ilorin, Ilorin, Nigeria; He was an astute writer and editor for Union of Campus Journalists, University of Ilorin, Ilorin, Nigeria. He provided proofreading assistance for his Long Essay Undergraduate research Supervisor, thereby successfully proofread over 20 undergraduate Project Researches suitable for publication. His experience can also be found in helping editing articles that are suitable and professional for publish | open_experience = He is having over 3 years of experience in Wikimedia. He is keen interested individual in open source as he is more interested in people accessing information. He was the Vice President, Training and Development for Wikimedia Fan Club, University of Ilorin where he trained a lot of members on editing on Wikipedia and various other Sibling projects. He led Wikimedia Awareness in Ogbomosho Project where series of people were trained. He had also co-facilitated series of Projects among which are Wikimedia Promotion in Akure, Wikimedia Promotion in Lead City University, Wiki and Health Articles in Nigeria among other projects | policy_confirm = I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 09:05, 11 September 2024 (UTC) }} * {{ping|Taofeeq Abdulkareem}} Sorry for the delay, I recently found time to review your application. You definitely have sufficient level of professional and open experience (as demonstrated in your contribution activities on wiki). I would like to know more about your publishing experience. Can you tell me more, such as providing links to your published works? Do you have a list of your publications? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:46, 14 October 2024 (UTC) *:@[[User:OhanaUnited|OhanaUnited]] Thanks for the review and kind comments. *:Kindly find attached below the list of Publications: *:# Change and Continuity in Socio-political Role of Women in Owo, 1900-1970 *:# The Great Wall of China *:# The First Nigeria's National Anthem *:# India's great voyage to the Mars *:# 60 Years Journey of Nigeria's Independence *:Links to the Publications respectively: *:* https://drive.google.com/file/d/16c8WDHbArhFit9-p8isLMJ9CzgKklzBp/view?usp=drivesdk *:* https://taofeeqexposure.wordpress.com/2020/07/09/the-great-wall-of-china/ *:* https://taofeeqexposure.wordpress.com/2020/07/11/the-first-nigeria-national-anthem/ *:* https://taofeeqexposure.wordpress.com/2020/08/16/indiathe-pride-of-asia-the-great-journey-to-mars/ *:* https://taofeeqexposure.wordpress.com/2020/10/01/60-years-journey-of-nigerias-independence/ *:[[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 12:09, 16 October 2024 (UTC) *::@[[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] Thank you. Blog posts are not what I considered as publishing experience. Other than the undergraduate thesis, do you have any examples of publishing in a peer-reviewed journal article or book chapter? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 16:20, 24 October 2024 (UTC) *:::Thank you for your prompt response. I appreciate your feedback and understand your concerns regarding my publishing experience. While my publication record in peer-reviewed journals may be limited, I would like to highlight my research experience in significant aspects of humanities, including [cultural studies, historical analysis, among others aspects]. Although blog posts may not be traditional publications, they demonstrate my ability to make research and communicate complex ideas to diverse audiences. *:::Beyond publishing, I've developed valuable skills through Undergraduate thesis research, Editing and proofreading for others, Research assistance in humanities topics. *:::I bring strong research foundation in humanities, excellent writing, editing, and proofreading skills, ability to communicate complex ideas engagingly, experience working with diverse authors and topics, passion for promoting high-quality humanities research. I am eager to leverage these skills to support Wikimedia Journal's mission. I understand the importance of peer-reviewed publications and commit to further developing my expertise. *:::I would appreciate consideration of my application, recognizing the diverse experiences and skills I bring. Thank you for your time, and I look forward to your response. [[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 09:40, 27 October 2024 (UTC) *::::I am '''support'''ive of your associate editor application, contingent on mentorship from board members, to help you gain experience around the publishing area. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:54, 14 November 2024 (UTC) *:::::Thank you for your prompt and warm response. I am thrilled to join the team and contribute to the Humanities journal. As a passionate, ambitious, and evolving individual, I am committed to continuous learning, growth, and development. *:::::I would greatly appreciate mentorship from the board members to enhance my publishing knowledge and skills. I am eager to apply these skills in my role and contribute meaningfully to the team's growth and success. *:::::I look forward to the next steps and onboarding process, I am delighted to be part of this team and make a positive impact. [[User:Taofeeq Abdulkareem|Taofeeq Abdulkareem]] ([[User talk:Taofeeq Abdulkareem|discuss]] • [[Special:Contributions/Taofeeq Abdulkareem|contribs]]) 20:25, 14 November 2024 (UTC) *::::::Please wait for other editorial board members to review and comment on your application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:44, 18 November 2024 (UTC) *'''Support'''. Having read the above, welcome aboard. --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:54, 5 December 2024 (UTC) ==Associate editor application of Sideeq Abubakar Galadima== {{WikiJournal editor application submitted | position =Associate editor | name =Sideeq Abubakar Galadima | qualifications =B.A. History and International Studies | link = | areas_of_expertise =History, Diplomacy, Planning and Management | professional_experience =His professional experience is deeply rooted in his academic background in History and International Studies, which has familiarized him with the intricacies of objective research, writing, and reportage. His expertise in these areas was further strengthened by his active engagement in news and report writing as a member of the Union of Campus Journalists during his undergraduate studies. Additionally, his experience as a Wikimedia editor has honed his proofreading skills. As an event planner, he has developed exceptional attention to detail, which has become an integral part of his skillset. Notably, his pioneering research work, titled "Colonialism and the Continuity of Ilorin Cultural Heritage, 1900-1960," demonstrates his ability to conduct in-depth historical analysis and research. This work will undoubtedly serve as a valuable reference for future studies in related fields, such as cultural diplomacy. | publishing_experience =He's a researcher, news and reports writer, content editor, proofreader | open_experience =He possesses over three years of experience in Wikimedia, driven by a strong interest in open-source initiatives. Notably, he served as the Special Duties Officer for the Wikimedia Fan Club at the University of Ilorin, where he played a pivotal role in facilitating and training sessions on Wikipedia and its sister projects, as well as co-facilitating workshops, including "Wiki and Health Articles in Nigeria" and "Wikimedia Awareness in Ogbomosho". Through these endeavors, He demonstrated his expertise in promoting open-source knowledge sharing and community engagement. His experience and commitment to Wikimedia's mission have equipped him with a unique skill set, poised to contribute to future initiatives. | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 17:54, 11 September 2024 (UTC) }} * I really appreciate Sideeq's Wikipedia contributions to topics in Africa. It sounds like the highest degree earned is B.A., and no journal editor experience? I think normally we expect a PhD and some academic journal experience. Also it would be good to have a link to the ""Colonialism and the Continuity of Ilorin Cultural Heritage, 1900-1960", which I wasn't able to find. [[User:Aoholcombe|Aoholcombe]] ([[User talk:Aoholcombe|discuss]] • [[Special:Contributions/Aoholcombe|contribs]]) 23:25, 2 October 2024 (UTC) *:I agree with your comment. I wasn't able to find this applicant's published work list and I am hesitant with professional experience even for applying as an associate editor position. While the applicant has some experience with open access, the activity was sporadic. However, I think it may be beneficial to have additional volunteers to support this journal that deals with the administrative side of things and less reliant on professional and publishing experiences' side of the journal. @[[User:Albakry028|Albakry028]], in case you didn't see the previous comment, can you provide us with more information? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:55, 14 October 2024 (UTC) *:Thank you for acknowledging my contributions to African topics on Wikipedia. I appreciate your recognition of my efforts. Regarding your inquiries, I would like to clarify that my highest educational attainment is a Bachelor of Arts degree. Nevertheless, my editorial expertise has enabled me to assist colleagues with their research projects, leveraging my skills in research and academic writing. I understand and respect the standard expectations associated with academic roles. However, I was entrusted with this responsibility due to my demonstrated expertise. Regarding my research work, I am pleased to share the link to my project: "Colonialism and the Continuity of Ilorin Cultural Heritage, 1900-1960." https://drive.google.com/file/d/1bxysalU-AT7JakWfJCFxeWqwpFCz_C7s/view?usp=drivesdk @[[User:Aoholcombe|Aoholcombe]] @[[User:OhanaUnited|OhanaUnited]] [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 13:50, 16 October 2024 (UTC) :@[[User:Albakry028|Albakry028]] Thanks very much for providing your writing example. Do you have any publishing experience? We are looking for something beyond undergraduate thesis (for example, peer-reviewed journal article or book chapters). I am trained as a scientist and therefore will need more information to assess an applicant's suitability in applying for a humanities position. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 16:18, 24 October 2024 (UTC) :Although my publishing experience is limited to my undergraduate thesis, I'm confident in my potential. I bring transferable skills: research expertise, writing proficiency, adaptability, analytical thinking and effective communication. I'm eager to apply research methodology perspectives to humanities contexts, quickly learn and adapt. I'm poised to contribute innovatively through interdisciplinary research, engaging teaching methods and collaborative projects. I appreciate your consideration of potential over conventional metrics. [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 13:38, 25 October 2024 (UTC) ::I am happy to '''support''' your associate editor application, contingent on board members' availability, to mentor you to gain experience around the publishing area. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:55, 14 November 2024 (UTC) :::Thank you for your kind and supportive message. I am thrilled to join the team and grateful for the opportunity to work alongside experienced board members. I am eager to benefit from their mentorship and expertise, which will undoubtedly enhance my skills and knowledge in the publishing field. :::As a dedicated and passionate individual, I am committed to contributing to the humanities journal and supporting its growth. I am excited to embark on this journey and engage in meaningful discussions as a team member. :::I look forward to the next steps and onboarding process. [[User:Albakry028|Albakry028]] ([[User talk:Albakry028|discuss]] • [[Special:Contributions/Albakry028|contribs]]) 20:44, 14 November 2024 (UTC) ::::Please wait for other editorial board members to review and comment on your application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:38, 18 November 2024 (UTC) *'''Support'''. Having read the above, welcome aboard. --[[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:56, 5 December 2024 (UTC) 8whhm233v8tdljg6lof4uhbvzhkw87g 2 234481 2690282 2460482 2024-12-04T16:29:54Z CommonsDelinker 9184 Removing [[:c:File:20040514_large_hail_5.25".jpg|20040514_large_hail_5.25".jpg]], it has been deleted from Commons by [[:c:User:Krd|Krd]] because: per [[:c:Commons:Deletion requests/File:20040514 large hail 5.25".jpg|]]. 2690282 wikitext text/x-wiki [[Image:Malaspina Glacier in Southeastern Alaska.jpg|thumb|right|300px|Malaspina Glacier in southeastern Alaska is considered the classic example of a piedmont glacier. Credit: NASA STS-97 crew.{{tlx|free media}}]] A cryometeor is a meteor of variable size that has been radiated and is still moving composed of ice, e.g. water or methane ice. A cryometeor that has stopped moving has become a cryometeorite. '''Def.''' a glacier that occurs on a gentle slope leading from the base of mountains to a region of flat land, any "region of foothills of a mountain range", or formed "or lying at the foot of a mountain range"<ref name=PiedmontWikt>{{ cite book |author=[[wikt:User:SemperBlotto|SemperBlotto]] |title=piedmont |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=29 September 2005 |url=https://en.wiktionary.org/wiki/piedmont |accessdate=23 September 2022 }}</ref> is called a '''piedmont glacier'''. {{clear}} ==Ices== {{main|Minerals/Ices}} [[Image:Eiszapfen Schwellenbach.jpg|thumb|right|250px|This is an image of columnar ice crystals. Credit: [[c:User:DrAlzheimer|DrAlzheimer]].{{tlx|free media}}]] '''Def.''' "any frozen volatile chemical, such as water, ammonia, or carbon dioxide"<ref name=IceWikt>{{ cite book |author=[[wikt:User:Długosz|Długosz]] |title=ice |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=23 April 2004 |url=https://en.wiktionary.org/wiki/ice |accessdate=2015-01-05 }}</ref> is called an '''ice'''. The discoveries of water ice on the Moon, Mars and Europa add an extraterrestrial component to the field, as in "astroglaciology".<ref name=RSWilliams>{{ cite journal |title=Annals of Glaciology |volume=9 |page=255 |author=Richard S. Williams, Jr. |url=http://www.igsoc.org/annals/9/igs_annals_vol09_year1987_pg254-255.pdf |year=1987 |publisher=International Glaciological Society |accessdate=7 February 2011}}</ref> "The comet hypothesis of the origin of lunar ice, which was recently discovered in the polar regions of the moon by Lunar Prospector, is examined. It is shown that a comet impact produces a temporary atmosphere whose volatile component accumulates essentially completely in cold traps - the permanently shadowed regions of the Moon."<ref name=Klumov>{{ cite journal |author=B.A. Klumov and A.A. Berezhnoi |title=Possible origin of lunar ice |journal=Advances in Space Research |date=October 2002 |volume=30 |issue=8 |pages=1875-1881 |url=https://www.sciencedirect.com/science/article/abs/pii/S0273117702004891 |arxiv= |bibcode= |doi=10.1016/S0273-1177(02)00489-1 |pmid= |accessdate=19 September 2022 }}</ref> "Due to small oblique angle of the Moon׳s spin axis with respect to ecliptic (1.54°), the plausibility of existence of water ice in cold traps was initially discussed by Watson ''et al''. (1961). Cold traps favorably harbor water ice that originates from occasional comets, water-containing meteorites, and solar-wind-induced iron reduction of regolith; yet ice is lost due to solar wind sputter erosion (Arnold, 1979; Crider and Vondrak, 2002, 2003; Klumov and Berezhnoi, 2002). The processes of deposition and sublimation in these regions have been sustained for nearly 2 Gyr, since the Moon׳s orbital evolution became stable (Arnold, 1979; Bills and Ray, 1999)."<ref name=Wei>{{ cite journal |author=Guangfei Wei, Xiongyao Li and Shijie Wang |title=Thermal behavior of regolith at cold traps on the moon's south pole: Revealed by Chang'E-2 microwave radiometer data |journal=Planetary and Space Science |date=2 March 2016 |volume=122 |issue= |pages=101-10 |url=https://www.sciencedirect.com/science/article/abs/pii/S0032063315300635 |arxiv= |bibcode= |doi=10.1016/j.pss.2016.01.013 |pmid= |accessdate=19 September 2022 }}</ref> "The 31 km diameter and 7.5 km deep de Gerlache crater, located 30 km from the southern pole of the Moon was surveyed. At its bottom a 15 km diameter younger crater can be also found beside many smaller overprinting craters."<ref name=Kereszturi>{{ cite journal |author=A. Kereszturi, R. Tomka, P.A. Gläser, B.D. Pal, V. Steinmann and T. Warren |title=Characteristics of de Gerlache crater, site of girlands and slope exposed ice in a lunar polar depression |journal=Icarus |date=December 2022 |volume=388 |issue= |pages=115231 |url=https://www.sciencedirect.com/science/article/pii/S0019103522003256 |arxiv= |bibcode= |doi=10.1016/j.icarus.2022.115231 |pmid= |accessdate=19 September 2022 }}</ref> "At all locations [these “girland like features” ... which seem to be produced by mass movements on slopes] are superposed by recently formed 10–50 m diameter craters".<ref name=Kereszturi/> "In de Gerlache crater ice occurrences have previously been located on moderately steep slopes, indicating they might be exposed by mass movement processes, where active movements might have happened in the last some 10 Ma using crater statistics based age of the shallow regolith layer."<ref name=Kereszturi/> {{clear}} ==Meteors== '''Def.''' "'''1 :''' a phenomenon or appearance in the atmosphere (as lightning, a rainbow, or a snowfall) '''2 a :''' one of the small particles of matter in the solar system observable directly only when it falls into the earth's atmosphere where friction may cause its temporary incandescence '''b :''' the streak of light produced by the passage of a meteor"<ref name=Gove>{{ cite book |author= |title=Webster's Seventh New Collegiate Dictionary |publisher=G. & C. Merriam Company |location=Springfield, Massachusetts |date=1963 |editor=Philip B. Gove |pages=1221 |bibcode= |doi= |pmid= |isbn= }}</ref> is called a '''meteor'''. '''Def.''' a "fast-moving streak of light in the night sky caused by the entry of extraterrestrial matter into the earth's atmosphere"<ref name=MeteorWikt>{{ cite book |author=[[wikt:User:Xed~enwiktionary|Xed~enwiktionary]] |title=meteor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=3 September 2004 |url=https://en.wiktionary.org/wiki/meteor |accessdate=26 June 2019 }}</ref> is called a '''meteor'''. '''Def.''' "any natural object radiating through a portion or all of the Earth's or another natural, astronomical object's atmosphere"<ref name=Marshallsumter>{{ cite book |author=[[User:Marshallsumter|Marshallsumter]] |title=meteor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California USA |date=September 24, 2011 |url=http://en.wikiversity.org/wiki/Radiation/Meteors |accessdate=2018-01-24 }}</ref> is called a '''meteor'''. ==Cryometeors== "Part two contains contributions focused on the status of near-earth object (NEO) surveys, current knowledge of NEO populations in space, physical properties of NEOs, the quantitative risk of impacts and risk reduction scenarios, the physical terrestrial effects of impacts, the atmospheric and oceanic (tsunami) effects of impacts, case studies including the Kaali meteorite and Tunguska events and cryometeors."<ref name=Bobrowsky>{{ cite book |author=Richard A. F. Grieve and David A. Kring |title=Preface, In: ''Comet/Asteroid Impacts and Human Society: An Interdisciplinary Approach'' |publisher=Springer |location=Berlin |date=November 2006 |editor=Peter T. Bobrowsky and Hans Rickman |pages=546 |url=https://pdfdrive.to/pdfs/cometasteroid-impacts-and-human-society-an-interdisciplinary-approach-pdf |arxiv= |bibcode= |doi= |pmid= |{{isbn|3-540-3270-6}} |accessdate=18 September 2022 }}</ref> "Isotope studies suggest that most of the water did not form on Earth but is the result of the impact of a huge cryometeor that impacted on Earth billions of years ago [Morbidelli ''et al''., 2000]."<ref name=Waitz>{{ cite book |author=Fritz Waitz |title=On the Discrimination and Interaction of Droplets and Ice in Mixed-Phase Clouds |publisher=Karlsruher Institut für Technologie |location= |date=19 November 2021 |editor= |pages=151 |url=https://scholar.archive.org/work/zcneh57gonehzhah7drh6bx33e/access/wayback/https://publikationen.bibliothek.kit.edu/1000140968/138549438 |arxiv= |bibcode= |doi= |pmid= |{{isbn|}} |accessdate=18 September 2022 }}</ref> "Schwerdtfeger (1970, p. 294) notes "With reference to Antarctica, the term ‘cryometeors’ might be more appropriate than "hydrometeors, but it is not used"."<ref name=Turner>{{ cite book |author=John Turner |title=Precipitation/Accumulation, In: ''The International Antarctic Weather Forecasting Handbook'' |publisher=British Antarctic Survey Natural Environment Research Council |location=High Cross, Madingley Road Cambridge, CB3 0ET, UK |date=16 June 2004 |editor=John Turner and Stephen Pendlebury |pages=663 |url=https://nora.nerc.ac.uk/id/eprint/17324/1/handbook_16june04.pdf |arxiv= |bibcode= |doi= |pmid= |{{isbn|1 85531 221 2}} |accessdate=18 September 2022 }}</ref> ==Cryomicrometeoroids== [[Image:PIA17172 Saturn eclipse mosaic bright crop.jpg|thumb|upright=1.5|right|250px|The full set of rings, is imaged as Saturn eclipsed the Sun from the vantage of the Cassini–Huygens orbiter, 1.2 million km distant, on 19 July 2013 (brightness is exaggerated). Earth appears as a pale blue dot at 4 o'clock, between the G and E rings.{{tlx|free media}}]] The rings of Saturn consist of countless small particles, ranging in size from micrometers to meters,<ref name="Questions">{{cite book | last=Porco | first=Carolyn | title=Questions around Saturn's rings|url=http://www.ciclops.org/sci/common_questions.php#ring | accessdate=2012-10-05 }}</ref> that are made almost entirely of water ice, with a trace component of rocky material. The light spectra [of the Upsilon Pegasid fireball], combined with trajectory and light curve measurements, have yielded various compositions and densities, ranging from fragile snowball-like objects with density about a quarter that of ice,<ref name=Povenmire>Povenmire, H. [http://www.lpi.usra.edu/meetings/lpsc2000/pdf/1183.pdf PHYSICAL DYNAMICS OF THE UPSILON PEGASID FIREBLL – EUROPEAN NETWORK 190882A]. Florida Institute of Technology</ref> to nickel-iron rich dense rocks. "It is empirically known that all cooling older stars that possess a global magnetic field have rings. This includes the Earth regardless if they are or not observed with the naked eye."<ref name=Wolynski>{{ cite journal |author=Jeffrey J Wolynski and Stephen Crothers |title=The Earth has Rings |journal=CiteSeer^x |date=27 December 2012 |volume= |issue= |pages=1 |url=https://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=AF51714874998EC06B72646A8C206E96?doi=10.1.1.671.7560&rep=rep1&type=pdf |arxiv= |bibcode= |doi=10.1.1.671.7560 |pmid= |accessdate=19 September 2022 }}</ref> "[W]ater/ice rings will always be oriented in the direction perpendicular to the magnetic field orientation of the cooling star, unless that said star is changing orbits and undergoing a magnetic reversal."<ref name=Wolynski/> Jupiter and Saturn have water ice rings.<ref name=Wolynski/> {{clear}} ==Megacryometeors== [[Image:Megacryometeor1.jpeg|thumb|right|250px|Megacryometeors are something very different, and they are still a mystery to science. Credit: Jesús Martínez-Frias.{{tlx|fairuse}}]] "A '''megacryometeor''' is a very large chunk of ice, weighing at least 10 kg, that are sometimes called huge hailstones, but do not [need] to form in Thunderstorms."<ref name=Megacryometeor>{{ cite book |author=[[w:User:207.61.87.226|207.61.87.226]] |title=Megacryometeor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=2 July 2004 |url=https://en.wikipedia.org/wiki/Megacryometeor |accessdate=16 September 2022 }}</ref> "Over the past decade, over fifty such objects have been recorded worldwide. Some have been as small as about one pound, but one monstrous mass of ice that fell in Brazil weighed about 400 pounds⁠— almost a quarter of a ton⁠— and crashed through the roof of a Mercedes-Benz factory. One recently made headlines in Oakland, California, weighing over 200 pounds and creating a dent in the Earth three feet deep. A similar event occurred in Chicago last February, crashing through the roof of a house."<ref name=Megacryometeor2>{{ cite book |author=Alan Bellows |title=The Peculiar Phenomenon of Megacryometeors |publisher=Damn Interesting |location= |date=April 2006 |url=https://www.damninteresting.com/the-peculiar-phenomenon-of-megacryometeors/ |accessdate=16 September 2022 }}</ref> "The mysterious ice blobs, like hail, have been found to contain air bubbles, onion-like layering, and traces of ammonia and silica. The icy objects also have isotopic distributions of oxygen-18 and deuterium similar to those found in hailstones. Aside from their surprising mass and their tendency to plunge one-at-a-time from clear skies, the ice balls are almost identical to hail."<ref name=Megacryometeor2/> "They are sometimes confused as meteors, because they can leave impact craters. The difference between a megacryometeor and a hailstone is not clearly defined, mostly because the process that creates megacryometeors is not fully understood, but they have been recorded falling out of a clear sky on a hot summer day. They are also not made from airplane toilets or exhaust streams. All analysis of the ice shows it matches normal rain for the region it fell on."<ref name=Megacryometeor/> "A '''megacryometeor''' is a very large chunk of ice which, despite sharing many textural, hydro-chemical and isotopic features detected in large hailstones, is formed under unusual atmospheric conditions which clearly differ from those of the cumulonimbus cloud scenario (i.e. clear-sky conditions). They are sometimes called huge hailstones, but do not need to form in thunderstorms."<ref name=Megacryometeor1>{{ cite book |author=[[w:User:213.0.212.100|213.0.212.100]] |title=Megacryometeor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=24 November 2005 |url=https://en.wikipedia.org/wiki/Megacryometeor |accessdate=16 September 2022 }}</ref> '''Def.''' "a very large water ice object that falls from the sky, similar in composition to hailstones"<ref name=MegacryometeorWikt>{{ cite book |author=[[wikt:User:76.66.203.138|76.66.203.138]] |title=megacryometeor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=14 November 2010 |url=https://en.wiktionary.org/wiki/megacryometeor |accessdate=10 March 2018 }}</ref> is called a '''megacryometeor'''. "LARGE icy conglomerates, occasionally falling from a clear sky even when there are no clouds or precipitation, have recently been termed as megacryometeors¹."<ref name=Deshpande>{{ cite journal |author=R. D. Deshpande, A. S. Maurya, R. C. Angasaria, Medha Dave, A. D. Shukla, N. Bhandari and S. K. Gupta |title=Isotopic studies of megacryometeors in western India |journal=Current Science |date=25 March 2013 |volume=104 |issue=6 |pages=728-737 |url=http://professional.thebabyspecialist.com.sg/wp-content/uploads/2015/01/07281.pdf |arxiv= |bibcode= |doi= |pmid= |accessdate=18 September 2022 }}</ref> "That large blocks of ice fall to the ground is evident enough; they are observed to fall and they are collected, but the central question here is did they enter the Earth’s atmosphere from interplanetary space?"<ref name=Beech>{{ cite journal |author=Martin Beech |title=The Problem of Ice Meteorites |journal=Meteorite Quarterly |date=November 2006 |volume=12 |issue=4 |pages=17-19 |url=https://web.archive.org/web/20110927074403/http://hyperion.cc.uregina.ca/~astro/Ice_Mets.pdf |arxiv= |bibcode= |doi= |pmid= |accessdate=19 September 2022 }}</ref> The "solar system contains numerous bodies that have water-ice as a major compositional component."<ref name=Beech/> It "is a certainty that ice-meteoroids exist. The recent outburst of comet 73P/ Schwassmann- Wachmann 3 [...] provides one example of an event that produced icy-nuclei many tens of meters in diameter, and no-doubt smaller icy meteoroids as well."<ref name=Beech/> ==Ice meteorites== '''Def.''' "a meteor that reaches the surface of the earth without being completely vaporized"<ref name=Gove/> is called a '''meteorite'''. '''Def.''' a "metallic or stony object or body that [is the remains of a meteor]<ref name=MeteoriteWikt1>{{ cite book |author=[[wikt:User:SemperBlotto|SemperBlotto]] |title=meteorite |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=28 December 2007 |url=http://en.wiktionary.org/wiki/meteorite |accessdate=2015-03-28 }}</ref>oid]<ref name=MeteoriteWikt2>{{ cite book |author=[[wikt:User:186.74.9.130|186.74.9.130]] |title=meteorite |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=28 May 2019 |url=http://en.wiktionary.org/wiki/meteorite |accessdate=2015-03-28 }}</ref> [or] has fallen to the surface of the Earth from outer space"<ref name=MeteoriteWikt>{{ cite book |author=[[wikt:User:SnoopY|SnoopY]] |title=meteorite |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 January 2006 |url=http://en.wiktionary.org/wiki/meteorite |accessdate=2015-03-28 }}</ref> is called a '''meteorite'''. "In addition, accepting for the moment that ice meteorites might fall to Earth, the question of their origin must also be addressed – literally, where are the ice fragments from."<ref name=Beech/> "One of the key factors in determining the delivery of a meteorite to the Earth’s surface is the meteoroids initial encounter speed: the lower the encounter speed the better. With respect to known cometary meteoroid streams, the smallest known Earth encounter speed is the 15 km/s of the occasionally active τ-Herculid meteor shower. The next lowest encounter speeds being those for the π-Puppid meteoroids (18 km/s) and the Draconid meteoroids (20 km/s)."<ref name=Beech/> "Firstly, “what is the lifetime of a pure water-ice fragment in the inner solar system”, and second “can water-ice meteoroids survive passage through the Earth’s atmosphere”?"<ref name=Beech/> "While ice-meteoroids must exist within our solar system the more important question at this stage is, how long do they exist for?"<ref name=Beech/> "Once any icy nucleus or ice-meteoroid approaches within about 2.5 AU of the Sun then sublimation will become important."<ref name=Beech/> "For a spherical ice-meteoroid moving in an orbit similar to, for example, Comet 73P/ Schwassmann-Wachmann 3 [Aphelion is 5.211 AU, Perihelion is 0.9722 AU] the radius would decrease due to sublimation at a rate of about 1.4 meters per orbit (or 0.25 m/yr). In other words, a 10-m diameter ice-block would disappear within about 4 orbits of the Sun – a timescale of about 20 years. The same sized meteoroid in an orbit similar to that of the Earth would disappear on an even more rapid timescale of about 2 years. Comet’s that move deep into the outer solar system spend much less time close in towards the Sun, and consequently any ice-meteoroids left in their wake will survive longer. A 10-m diameter ice-block with an orbit similar to that of comet C/1861 G1 (Thatcher) [Aphelion is 110 AU, Perihelion is 0.9207 AU], the parent comet to the April Lyrid meteor shower, which has an aphelion distance of about 109 AU, should survive for about 2000 years – but it would encounter the Earth with an initial speed of 48 km/s."<ref name=Beech/> "The problem with respect to the production of ice-meteorites therefore is that they must encounter the Earth within just a few years of being ejected from their parent body, and this dynamically speaking is highly unlikely to happen."<ref name=Beech/> "The lowest speed that any meteoroid can have at the top of the atmosphere is Earth’s escape velocity of 11.2 km/s."<ref name=Beech/> When "the initial velocity at the top of the atmosphere is 11.5 km/s an ice-meteoroid of mass ~50,000-kg (diameter ≈ 4.8-m) is required to produce a 2-kg meteorite on the ground."<ref name=Beech/> "When the initial velocity is 15 km/s, however, even a 1,000,000-kg (diameter ≈ 15-m) ice-meteoroid will only produce an ice meteorite of a few grams mass on the ground."<ref name=Beech/> If "the Earth did encounter a τ-Herculid fragment of several tens of meters in diameter it would probably produce an air-burst explosion similar to that of the 1908 Tunguska impact."<ref name=Beech/> "Catastrophic fragmentation of all large ice-meteoroids in the Earth’s upper atmosphere is almost inevitable, in fact, because the ram pressure due to the on-coming air flow will easily exceed the tensile strength of solid-ice or that of a cometary nucleus. The tensile strength of comet D/1993 F2 (Shoemaker-Levy 9) was estimated to be about 1000 Pa [Scotti and Melosh, 1993]; the tensile strength of water-ice falls between 10⁶ to 10⁷ Pa."<ref name=Beech/> "So, can an ice-meteoroid survive atmospheric passage to hit the ground? Well, the answer is perhaps yes – just maybe! If the encounter velocity is not much greater than the Earth’s escape velocity then a 5 to 10-m diameter ice-meteoroid might just produce a 1 to 10-kg ice-meteorite at the Earth’s surface (provided that the tensile strength of the ice-meteoroid is greater than ~10⁷ Pa)."<ref name=Beech/> "Two main factors argue against ice meteorites. Firstly the velocity restriction requires that the meteoroids must encounter the Earth with very low velocities – certainly less than 12 – 13 km/s. No currently known cometary meteoroid stream, therefore, can produce ice-meteorites."<ref name=Beech/> "The second reason why ice meteorites must, at best, be exceptionally rare relates to their survival lifetime in space. To get close to the Earth means that an ice-meteoroid must become heated, and once this happens lifetimes against mass-loss by sublimation are typically just a few tens of years. In other words an ice-meteoroid is ‘destroyed’ in space long before it might encounter the Earth to produce an ice-meteorite."<ref name=Beech/> It "has been occasionally noted that meteorite falls can precipitate distinct smells; most often described as sulfurous, or ‘metallic’. Berczi and Lukacs (1997) have picked-up on this point and suggested that odors of sulphuric and ammonia compounds might in fact be released by ‘freshly’ fallen ice-meteorites".<ref name=Beech/> Megacryometeors may "form under a rare, clear-sky variant of the nucleation process responsible for the production of ordinary hail (Bosch, 2002). The ‘meteor’ part of megacryometeors, it should be pointed out, relates to the idea that these objects are considered to be meteorological (that is atmospheric) in origin."<ref name=Beech/> ==Selenometeorites== About 371 lunar meteorites have been discovered so far (as of July, 2019),<ref name="MeteorBulletin">{{cite book | url=https://www.lpi.usra.edu/meteor/metbull.php | title=Meteoritical Bulletin Database — Lunar Meteorite search results | publisher=The Meteoritical Society | work=Meteoritical Bulletin Database | date=10 July 2019 | accessdate=20 July 2019}}</ref> perhaps representing more than 30 separate meteorite falls (i.e., many of the stones are "paired" fragments of the same meteoroid).<ref name="wustl.edu">{{cite book|url=http://meteorites.wustl.edu/lunar/moon_meteorites_list_alumina.htm|title=List of Lunar Meteorites - Feldspathic to Basaltic Order|website=meteorites.wustl.edu|accessdate=8 April 2018}}</ref> The total mass is more than {{convert|190|kg}}.<ref name="wustl.edu"/> All lunar meteorites have been found in deserts; most have been found in Antarctica, northern Africa, and the Sultanate of Oman, but none have yet been found in North America, South America, or Europe.<ref>Washington University in St. Louis: [http://meteorites.wustl.edu/lunar/howdoweknow.htm How Do We Know That It's a Rock from the Moon?]</ref> Cosmic ray exposure history established with noble gas measurements has shown that all lunar meteorites were ejected from the Moon in the past 20 million years. Most left the Moon in the past 100,000 years. ==Extraterrestrial megacryometeorites== [[Image:Enceladus fountains.jpg|thumb|right|250px|Enceladus, Saturn's moon, spews out water vapor from its southern pole creating a halo of ice, gas, and dust. Credit: NASA/JPL/Space Science Institute.{{tlx|fairuse}}]] [[Image:Color polar maps of Enceladus PIA18435 Nov. 2014 full size.jpg|thumb|center|400px|These are the north and south polar hemispheres of Enceladus from left to right. Credit: NASA/JPL-Caltech/Space Science Institute/Lunar and Planetary Institute.{{tlx|free media}}]] "The theory of an origin [for megacryometeors] within the Troposphere [...] seems unlikely because there would be significant heating due to an increase in {{chem|CO|2}} concentration (Fu ''et al''. 2011)."<ref name=Snyder>{{ cite journal |author=Duane P Snyder and Rhawn Joseph |title=The Origins of Megacryometeors: Troposphere or Extraterrestrial? |journal=Cosmology |date=2015 |volume=19 |issue= |pages=70-86 |url=https://www.researchgate.net/profile/Rhawn-Joseph/publication/353002917_The_Origins_of_Megacryometeors_Troposphere_or_Extraterrestrial/links/60e37e7ba6fdccb7450ac9f2/The-Origins-of-Megacryometeors-Troposphere-or-Extraterrestrial.pdf |arxiv= |bibcode= |doi= |pmid= |accessdate=19 September 2022 }}</ref> "[M]egacryometeors have been observed and recorded in the mid 1800s, long before the invention of airplanes".<ref name=Snyder/> The "proliferation of reports may be due to increased access to the media, such that, it's not the number of meteors which has increased but the number of people reporting them."<ref name=Snyder/> "In March of 2000 [...] large chunks and a substantial amount of smashed ice [of the Pullman ice meteorite was discovered near a residence] on a clear, cloudless day. The ice was stratified ice, transitioning from clear transparent to translucent to opaque ice. This is indicative of laid down layers of frozen precipitation. The directionally increasing density is suggestive [of] glacial ice."<ref name=Snyder/> "In July of 2000, [...] two vials of melt-water from the suspected ice meteorite [were sent] to Geochronology Labortories, Cambridge, Massachusetts for stable isotope ratio and tritium analysis. Subsequently, high tritium levels were detected, the most likely source being exposure to cosmic radiation."<ref name=Snyder/> An "oval shaped sphere, approximately 300 nm in diameter [was transported] to the Ecloe Polytechnique Surface Analysis Laboratory (LASM), located at the Unversite de Montreal in Montreal, Canada. This sample was bombarded with a pulsed liquid metal ion source at energy of 25 KeV. Both polarities, positive and negative, were registered. The most intense element is the Na (sodium) in positive and Cl (chlorine) in negative [...]. This indicates the presence of sodium chloride salt. Also noticeable is the presence of Ca, K, Si, Al and known and unknown aluminum hydroxides."<ref name=Snyder/> "Melt-water from the suspected ice meteorite, was analyzed by the labs of EAG, [in] Raleigh, North Carolina. The melt-water was sonicated for 10 minutes then transferred to a copper mesh TEM grid. Imaging using STEM (Hitachi HD2700 scanning transmission electron microscope) provided various magnifications in atomic number contrast mode (ZC) and transmitted electron mode (TE). Chemical analysis was preformed with a Bruker Quantax EDS system."<ref name=Snyder/> "Mass spectra of 7 particles [...] indicates high levels of carbon, and Si and O as highly significant particle constituents, as well as Sodium (Na) and chlorine (Cl), being possible salts. When the carbon and the salts are taken into account, the elemental composition of these particles [is] in agreement with the hydrothermal nano-silica ({{chem|SiO|2}}) particles found in the E ring of Saturn. However, carbon was also the most abundant contaminate element found in Saturn’s E ring by the Cassini’s CDA. When the carbon is taken into account, the elemental composition of the particles are in agreement with the hydrothermal nano-silica ({{chem|SiO|2}}) particles found in the E ring of Saturn."<ref name=Snyder/> There "is no evidence megacryometeors are formed in the stratosphere. Moreover, it is a fact that ice chunks, weighing over tens of kilograms (22 pounds), do fall to Earth and it seems highly unlikely such large objects could develop in the stratosphere when there is no evidence that they were formed in the stratosphere in the first place."<ref name=Snyder/> Growth "and layering was [...] observed. Growth, however, requires a place to grow. Micro-Raman spectroscopy of band profiles has indicated that this growth takes place in a range of temperatures (Ruff ''et al''. (2010); and this suggests that the place where these megacryometeors must have been subject to a range of temperatures over a significant duration of time."<ref name=Snyder/> These "ice meteors are formed either in space or they are ejecta from stellar objects consisting of large amounts of water. Be they formed in space or ejecta, these ice meteors would break apart and melt as they enter Earth's atmosphere. Their origin, therefore, could include comets. However, if from a water world, or a planet or moon with ample amounts of water, then the moon Enceladus is one possible candidate."<ref name=Snyder/> "Enceladus, the six largest moon of Saturn has Cryovolcanic ice water vapor plumes that replenish the E ring of Saturn with material. The plumes contain ice particles, salts, organic compounds, water vapor and nano-silica. The gravitational return, to the surface of Enceladus, of some of the frozen precipitation, salts, organic compounds, and dust particles will lay down a glacial like ice surface."<ref name=Snyder/> The "dominant, if not the sole constituent of most E ring stream particles, are {{chem|SiO|2}} (nano-silica) (Hsu ''et al''. 2015)."<ref name=Snyder/> The "nano-silica particles with a radius of ~8 nm (~16 nm dia.), observed by the Cassini mission Cosmic Dust Analyzer (DCA) (Srama ''et al''. 2011) may have been formed over a period of months or years before being ejected into E ring (Hsu ''et al''. 2015)."<ref name=Snyder/> "These nano-silica particles, initially embedded in icy grains, are presumably emitted from Saturn's moon Enceladus’ subsurface waters. They are released by sputter erosion of the icy grains while in Saturns' E ring."<ref name=Snyder/> "Quantitative mass spectra analysis of Saturn’s E ring stream of particles detected by the Cassini mission Cosmic Dust Analyzer (CDA) (Srama ''et al''. 2011), indicates a diameter D_max = 12 to 18 nm for the largest observed stream particles. This is in agreement with the upper particle size limit independently inferred by simulations (R_max= 8 nm) (Hsu ''et al''. 2011)."<ref name=Snyder/> "The plumes of icy particles and water vapor ejected from the south pole of Enceladus have been shown to contain simple organic compounds (McKay ''et al''. 2008). Analysis of the composition of freshly ejected plume particles have found that salt-rich ice particles dominate the total mass flux of ejected particles (Postberg ''et al''. 2011). However, the salt-rich ice particles are depleted in the population escaping into Saturn’s E ring, due to sputter erosion."<ref name=Snyder/> "Salts are found in the mass spectra [..] of the particles found in the melt-water of this suspected ice meteorite. Sodium chloride and known and unknown aluminum hydroxides were found in this ice. The water in this ice is salt-water. Precipitation here on Earth does not contain salt due to the evaporation cycle of water here on earth."<ref name=Snyder/> It "is possible that this suspected ice meteorite [...], is a genuine extraterrestrial ice meteorite because the ice is frozen tritiated salt-water precipitation containing salts and hydrothermal nano-silica, the chemical footprints from the E ring of Saturn."<ref name=Snyder/> "The stratigraphic evolution of the south pole Tiger Stripe surface of Enceladus (Jaumann ''et al''. 2008) is indicative of material being laid down in a glacial like process. The suggested episodically active tectonic events and the proposed localized catastrophic overturn of the rigid ice surface (O’Neill & Nimmo 2010) of Enceladus, allows for the possibility of large bodies of ice to periodically be ejected from Enceladus. The surface of Enceladus and the E ring of Saturn are exposed to cosmic radiation that creates tritium in the exposed water."<ref name=Snyder/> "The data from the analysis of the Pullman ice meteorite is compatible with the possibility that this ice is a genuine ice meteorite. The data is compatible with the possibility that this ice is of extraterrestrial origin. [And] was formed on the surface of Enceladus and constitutes ejecta which eventually fell to Earth."<ref name=Snyder/> {{clear}} ==Theory== Here are theoretical definitions: '''Def.''' a single ice crystal (such as a snowflake) or large ice object that is radiated and still moving is called a '''cryometeor'''. '''Def.''' a cryometeor that has been stopped from moving (such as by impacting the Earth) is called a '''cryometeorite'''. ==Hail== [[Image:Hagelkorn mit Anlagerungsschichten.jpg|thumb|right|250px|A large hailstone (clear and white) with concentric rings is shown. Credit: [[c:User:ERZ|ERZ]].{{tlx|free media}}]] [[Image:Small hail, fractured to show internal structure.jpg|thumb|left|250px|The image shows small hail that has been fractured to show internal structure. Credit: Erbe, Pooley: USDA, ARS, EMU.{{tlx|free media}}]] [[Image:Pital nevada.jpg|thumb|right|250px|On April 13, 2004, a blanket of hail fell during a storm in Cerro El Pital, El Salvador. Credit: [[c:User:Wanakoo|Wanakoo]].{{tlx|free media}}]] [[Image:Bogota hailstorm.jpg|thumb|left|250px|The image captures a hailstorm in progress in Bogotá, D.C., Colombia, on March 3, 2006. Credit: [[w:User:Ju98 5|Ju98 5]].{{tlx|free media}}]] [[Image:Nssl0098 - Flickr - NOAA Photo Library.jpg|thumb|right|250px|This is a very large hailstone from the NOAA Photo Library. Credit: NOAA Legacy Photo; OAR/ERL/Wave Propagation Laboratory.{{tlx|free media}}]] [[Image:Wea02251 - Flickr - NOAA Photo Library.jpg|thumb|left|250px|This hailstone was four inches in diameter and weighed seven ounces. Credit: Archival Photography by Steve Nicklas, NOS, NGS.{{tlx|free media}}]] [[Image:Hailstone.jpg|thumb|right|250px|As of June 22, 2003, this is the largest hailstone ever recovered. Credit: NOAA.{{tlx|free media}}]] [[Image:Record hailstone Vivian, SD.jpg|thumb|right|250px|This is a record-setting hailstone that fell in Vivian, South Dakota on July 23, 2010. Credit: NWS Aberdeen, SD.{{tlx|free media}}]] '''Hail''' is a form of solid [water] precipitation. It consists of balls or irregular lumps of ice, each of which is referred to as a '''hailstone'''.<ref name="webster">{{ cite book |title=Merriam-Webster definition of "hailstone" |publisher=Merriam-Webster |url=http://www.merriam-webster.com/dictionary/hailstone |accessdate=2013-01-23 }}</ref> Unlike graupel, which is made of rime, and ice pellets, which are smaller and translucent, hailstones – on Earth – consist mostly of water ice and measure between {{convert|5|and|200|mm|in}} in diameter. '''Def.''' "balls [or pieces]<ref name=HailWikt1>{{ cite book |author=[[wikt:User:Dcwinds~enwiktionary|Dcwinds~enwiktionary]] |title=hail |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=18 June 2006 |url=http://en.wiktionary.org/wiki/hail |accessdate=2013-02-15 }}</ref> of ice falling as precipitation from the sky [a thunderstorm]<ref name=HailWikt1/>"<ref name=HailWikt>{{ cite book |author=[[wikt:User:Paul G|Paul G]] |title=hail |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 April 2004 |url=http://en.wiktionary.org/wiki/hail |accessdate=2013-02-15 }}</ref> are called '''hail'''. '''Def.''' a "single ball of hail"<ref name=HailstoneWikt>{{ cite book |author=[[wikt:User:Newnoise|Newnoise]] |title=hailstone |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=29 December 2005 |url=http://en.wiktionary.org/wiki/hailstone |accessdate=2013-02-15 }}</ref> is called a '''hailstone'''. Terminal velocity of hail, or the speed at which hail is falling when it strikes the ground, varies by the diameter of the hail stones. A hail stone of 1 cm (0.39 in) in diameter falls at a rate of 9 metres per second (20 mph), while stones the size of 8 centimetres (3.1 in) in diameter fall at a rate of 48 metres per second (110 mph). Hail stone velocity is dependent on the size of the stone, friction with air it is falling through, the motion of wind it is falling through, collisions with raindrops or other hail stones, and melting as the stones fall through a warmer atmosphere.<ref name=NSSL>{{ cite book |url=http://www.nssl.noaa.gov/primer/hail/hail_basics.html |title=Hail Basics |author=National Severe Storms Laboratory |publisher=National Oceanic and Atmospheric Administration |date=2006-11-15 |accessdate=2009-08-28 }}</ref> Unlike ice pellets, hailstones are layered and can be irregular and clumped together. A cross-section through a large hailstone shows an onion-like structure. This means the hailstone is made of thick and translucent layers, alternating with layers that are thin, white and opaque. The image second down on the right shows a blanket of hail precipitated on the ground at Cerro El Pital, El Savador. {{lang|es|"Cerro El Pital se encuentra a 12 kilómetros de La Palma, con una altura de 2730 msnm es el punto más alto del territorio Salvadoreño. Es una montaña en medio de un bosque nebuloso que suele tener una temperatura aproximada de 10 ºC. El 13 de abril de 2004, las temperaturas bajaron tanto que el cerro fue cubierto por una escarcha de hielo que causó conmoción entre los pobladores, atribuyendo el fenómeno a una supuesta "nevada"."}} The third image at left shows a hailstone that fell at Washington, D. C., on May 26, 1953, that was 4 in in diameter and weighed 7 oz. In the fourth image at right is the largest hailstone ever recovered in the United States as of June 22, 2003. This hailstone fell in Aurora, Nebraska. It has a 7-inch (17.8 cm) diameter and an approximate circumference of 18.75 inches.<ref name=Leslie>{{ cite book |author=John Leslie |title=Central Plains Storm Produced Largest Hailstone in U.S. History |publisher=NOAA Satellites and Information |location=Maryland |date=2008 |url=http://www.noaanews.noaa.gov/stories/s2008.htm |accessdate=2012-10-14 }}</ref> The fourth down on the left hailstone image is one, approximately 133 mm (5 1/4 inches) in diameter, that fell in Harper, Kansas on May 14, 2004. After 2003, another record-setting hailstone fell in Vivian, South Dakota, on July 23, 2010. Its diameter is 8 inches with a weight of 1 pound 15 ounces. It's in the fifth image down on the right. {{clear}} ==Graupel== [[Image:Graupel encasing a snow crystal.jpg|thumb|right|300px|Graupel is shown encasing an unseen snow crystal. Credit: Erbe, Pooley: USDA, ARS, EMU.{{tlx|free media}}]] The METAR reporting code for hail {{convert|5|mm|in|abbr=on}} or greater is '''GR''', while smaller hailstones and graupel are coded '''GS'''. ... Hail has a diameter of {{convert|5|mm|in}} or more.<ref name="gloss">{{ cite book |url=http://amsglossary.allenpress.com/glossary/search?id=hail1 |title=Hail |date=2009 |accessdate=2009-07-15 |author=Glossary of Meteorology |publisher=American Meteorological Society }}</ref> Hailstones can grow to {{convert|15|cm|in|0}} and weigh more than {{convert|0.5|kg|lb|1}}.<ref name=NSSL2007>{{ cite book |url=http://www.photolib.noaa.gov/htmls/nssl0001.htm |title=Aggregate hailstone |author=National Severe Storms Laboratory |publisher=National Oceanic and Atmospheric Administration |date=2007-04-23 |accessdate=2009-07-15 }}</ref> '''Graupel''' ... also called '''soft hail''' or '''snow pellets''')<ref name=Webster>{{ cite book |url = http://www.merriam-webster.com/dictionary/graupel |title = Graupel - Definition, In: ''Merriam-Webster Dictionary'' |publisher = Merriam-Webster |accessdate = 15 Jan 2012 }}</ref> refers to precipitation that forms when supercooled droplets of water are collected and freeze on a falling snowflake, forming a {{convert|2|-|5|mm|in|3|abbr=on}} ball of rime. '''Def.''' a "precipitation that forms when supercooled droplets of water condense on a snowflake"<ref name=GraupelWikt>{{ cite book |author=[[wikt:User:Equinox|Equinox]] |title=graupel |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 June 2009 |url=https://en.wiktionary.org/wiki/graupel |accessdate=4 July 2019 }}</ref> is called '''graupel'''. Strictly speaking, graupel is not the same as hail or ice pellets, although it is sometimes referred to as '''small hail'''. However, the World Meteorological Organization defines ''small hail'' as snow pellets encapsulated by ice, a precipitation halfway between graupel and hail.<ref name=Secretariat>{{ cite book |title=International Cloud Atlas |date=1975 |publisher=Secretariat of the World Meteorological Organization |location=Geneva |url=https://books.google.com/books?id=hkTEMgAACAAJ |ISBN=92-63-10407-7 }}</ref> The frozen droplets on the surface of rimed crystals are hard to resolve and the topography of a graupel particle is not easy to record with a light microscope because of the limited resolution and depth of field in the instrument. However, observations of snow crystals with a low-temperature scanning electron microscope (LT-SEM) clearly show cloud droplets measuring up to {{convert|50|µm|in|5|abbr=on}} on the surface of the crystals. The rime has been observed on all four basic forms of snow crystals, including plates, dendrites, columns and needles. As the riming process continues, the mass of frozen, accumulated cloud droplets obscures the identity of the original snow crystal, thereby giving rise to a graupel particle. Graupel commonly forms in high altitude climates and is both denser and more granular than ordinary snow, due to its rimed exterior. Macroscopically, graupel resembles small beads of polystyrene. The combination of density and low viscosity makes fresh layers of graupel unstable on slopes, and layers of {{convert|20|-|30|cm|in|abbr=on}} present a high risk of dangerous slab avalanches. In addition, thinner layers of graupel falling at low temperatures can act as ball bearings below subsequent falls of more naturally stable snow, rendering them also liable to avalanche.<ref>"[http://www.avalanche.org/~moonstone/snowpack/the%20relation%20of%20crystal%20riming%20to%20avalanche%20formation%20in%20new%20snow.htm The Relation of Crystal Riming to Avalanche Formation in New Snow]". ''Department of Atmospheric Sciences, University of Washington.''</ref> Graupel tends to compact and stabilise ("weld") approximately one or two days after falling, depending on the temperature and the properties of the graupel.<ref>[http://www.avalanche.org/~uac/encyclopedia/graupel.htm Graupel], www.avalanche.org.</ref> {{clear}} ==Sleet== [[Image:Sleet (ice pellets).jpg|thumb|right|250px|The image shows ice pellets aka sleet in North America, with a United States penny for scale. Credit: [[c:User:Runningonbrains|Runningonbrains]].{{tlx|free media}}]] Ice pellets (also referred to as sleet by the United States National Weather Service<ref>{{ cite book |url=http://www.weather.gov/glossary/index.php?word=sleet |title= Sleet (glossary entry) |publisher= National Oceanic and Atmospheric Administration's National Weather Service |accessdate=2007-03-20 }}</ref>) are a form of precipitation consisting of small, translucent balls of ice. Ice pellets are usually smaller than hailstones<ref>{{ cite book |url=http://www.weather.gov/glossary/index.php?word=hail |title= Hail (glossary entry) |publisher= National Oceanic and Atmospheric Administration's National Weather Service |accessdate=2007-03-20 }}</ref> and are different from graupel, which is made of rime, or rain and snow mixed, which is soft. Ice pellets often bounce when they hit the ground, and generally do not freeze into a solid mass unless mixed with freezing rain. The METAR code for ice pellets is PL. '''Def.''' rain "which freezes before reaching the ground"<ref name=SleetWikt>{{ cite book |author=[[wikt:User:Dmh|Dmh]] |title=sleet |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 April 2001 |url=https://en.wiktionary.org/wiki/sleet |accessdate=4 July 2019 }}</ref> is called '''sleet'''. '''Def.''' "a single pellet of sleet"<ref name=IcePelletWikt>{{ cite book |author=[[wikt:User:WikiPedant|WikiPedant]] |title=ice pellet |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=17 December 2007 |url=https://en.wiktionary.org/wiki/ice_pellet |accessdate=4 July 2019 }}</ref> is called an '''ice pellet'''. {{clear}} ==Snow== [[Image:Snow Clouds in Korea.jpg|thumb|right|250px|This image is a satellite photo of lake-effect snow bands near the Korean Peninsula. Credit: NASA.{{tlx|free media}}]] '''Snow''' is precipitation in the form of flakes of crystalline water ice that fall from clouds. Since snow is composed of small ice particles, it is a granular material. It has an open and therefore soft structure, unless subjected to external pressure. '''Def.''' a "crystal of snow, having approximate hexagonal symmetry"<ref name="OED">“[http://dictionary.oed.com/cgi/entry/50229479 snowflake]” listed in the '''Oxford English Dictionary''' [2^nd Ed.; 1989]</ref> is called a '''snowflake'''. Snowflakes come in a variety of sizes and shapes. Types that fall in the form of a ball due to melting and refreezing, rather than a flake, are known as hail, ice pellets or snow grains. '''Def.''' water ice crystals falling as light white flakes are called '''snow'''. '''Def.''' the "frozen, crystalline state of water that falls as precipitation"<ref name=SnowWikt>{{ cite book |author=[[wikt:User:Emperorbma|Emperorbma]] |title=snow |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=20 July 2003 |url=https://en.wiktionary.org/wiki/snow |accessdate=23 September 2022 }}</ref> is called '''snow'''. '''Def.''' "[a]ny or all of the forms of water particles, whether liquid or solid, that fall from the atmosphere"<ref name=PrecipitationWikt>{{ cite book |author=[[wikt:User:CORNELIUSSEON|CORNELIUSSEON]] |title=precipitation |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=17 June 2006 |url=http://en.wiktionary.org/wiki/precipitation |accessdate=2013-02-15 }}</ref> is called '''precipitation'''. "Condensation or sublimation of atmospheric water vapor produces a hydrometeor. It forms in the free atmosphere, or at the earth's surface, and includes frozen water lifted by the wind. Hydrometeors which can cause a surface visibility reduction, generally fall into one of the following two categories: # '''Precipitation'''. Precipitation includes all forms of water particles, both liquid and solid, which fall from the atmosphere and reach the ground; these include: liquid precipitation (drizzle and rain), freezing precipitation (freezing drizzle and freezing rain), and solid (frozen) precipitation (ice pellets, hail, snow, snow pellets, snow grains, and ice crystals). # '''Suspended (Liquid or Solid) Water Particles'''. Liquid or solid water particles that form and remain suspended in the air (damp haze, cloud, fog, ice fog, and mist), as well as liquid or solid water particles that are lifted by the wind from the earth’s surface (drifting snow, blowing snow, blowing spray) cause restrictions to visibility. One of the more unusual causes of reduced visibility due to suspended water/ice particles is whiteout, while the most common cause is fog."<ref name=Mireles>{{ cite book |author=Mark R. Mireles |author2=Kirth L. Pederson |author3=Charles H. Elford |title=Meteorologial Techniques |publisher=Air Force Weather Agency/DNT |location=Offutt Air Force Base, Nebraska, USA |date=February 21, 2007 |editor= |pages= |url=http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA466107 |arxiv= |bibcode= |doi= |pmid= |isbn= |accessdate=2013-02-17 }}</ref> '''Def.''' a "storm consisting of thunder and lightning produced by a cumulonimbus, usually accompanied with rain [and sometimes]<ref name=ThunderstormWikt1/> hail,<ref name=ThunderstormWikt>{{ cite book |author=[[wikt:User:Blade Hirato|Blade Hirato]] |title=thunderstorm |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=12 August 2004 |url=http://en.wiktionary.org/wiki/thunderstorm |accessdate=2013-08-03 }}</ref> [sleet, freezing rain, or snow]"<ref name=ThunderstormWikt1>{{ cite book |author=[[wikt:User:2602:304:59b8:1b69:81:5eab:9cfb:718a|2602:304:59b8:1b69:81:5eab:9cfb:718a]] |title=thunderstorm |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=30 June 2013 |url=http://en.wiktionary.org/wiki/thunderstorm |accessdate=2013-08-03 }}</ref> is called a '''thunderstorm'''. {{clear}} ==Rime== [[Image:Snowflake 300um LTSEM, 13368.jpg|thumb|right|300px|Rime occurs on both ends of a columnar snow crystal. Credit: [[w:User:Brian0918|Brian0918]].{{tlx|free media}}]] [[Image:Mwrime.JPG|thumb|left|250px|Rime ice is shown after deposition on a window. Credit: [[c:User:Ws47|Ws47]].{{tlx|free media}}]] '''Def.''' "ice formed by the rapid freezing of cold water droplets of fog onto a cold surface"<ref name=RimeWikt>{{ cite book |author=[[wikt:User:Beobach972|Beobach972]] |title=rime |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=7 March 2008 |url=https://en.wiktionary.org/wiki/rime |accessdate=4 July 2019 }}</ref> is called '''rime'''. Hard rime is a white ice that forms when the water droplets in fog freeze to the outer surfaces of objects. It is often seen on trees atop mountains and ridges in winter, when low-hanging clouds cause freezing fog. This fog freezes to the windward (wind-facing) side of tree branches, buildings, or any other solid objects, usually with high wind velocities and air temperatures between {{convert|−2|and|−8|°C|°F|1}}. {{clear}} ==Firns== [[Image:Taku glacier firn ice sampling.png|thumb|right|250px|In a snow pit, snow layers are composed of progressively denser firn. Credit: USGS.{{tlx|free media}}]] Firn is granular snow, especially on the upper part of a glacier, where it has not yet been compressed into ice. '''Def.''' "a type of old snow which has gone through multiple thaw and refreeze cycles and thus is made of numerous small icy grains, though it is not nearly as saturated with water as snow-cone slush is; can be hard or somewhat soft depending on recent and current weather conditions"<ref name=MartianBachelor>{{ cite book |author=[[wikt:User:MartianBachelor|MartianBachelor]] |title=firn |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=26 August 2006 |url=https://en.wiktionary.org/wiki/firn |accessdate=2014-09-17 }}</ref> is called '''firn'''. '''Def.''' "rounded, well-bonded snow that is older than one year; firn has a density greater than 550 kilograms per cubic-meter (35 pounds per cubic-foot); called névé during the first year"<ref name=Beitler/> is called '''firn'''. At the Dye 3 location in south east Greenland, "it takes roughly 100 years before the surface snow is compressed into solid ice. During this slow process (firnification) a given snow layer sinks to a depth of 80 m below the new surface formed under constant deposition of 1m of snow per year in South Greenland."<ref name=Dansgaard/> In places, "surface melting often occurs in the summer time. The melt water seeps through the porous snow and refreezes somewhere in the cold firn, which disturbs the layer sequence, of course."<ref name=Dansgaard/> "Firn air is the air that is trapped in the porous medium of firn, which is typically the first one hundred meters of an ice core."<ref name=Kaspers>{{ cite book |author= Karsten Adriaan Kaspers |title=Chemical and physical analyses of firn and firn air: from Dronning Maud Land, Antarctica; 2004-10-04 |publisher=University of Utrecht |location=Utrecht, Netherlands |date=4 October 2004 |url=http://dspace.library.uu.nl/handle/1874/1104 |isbn=90-393-3807-8 |accessdate=October 14, 2005 }}</ref> At the South Pole, the firn-ice transition depth is at 122 m, with a CO₂ age of about 100 years. {{clear}} ==Aufeis== [[Image:Aufeis in the Brooks Range of Alaska.JPG|thumb|right|250px|A group of hikers travels over a large sheet of aufeis in the Anaktuvuk River Valley. Credit: [[c:User:Paxson Woelber|Paxson Woelber]].{{tlx|free media}}]] [[Image:Aufeis close.jpg|thumb|left|250px|Laminations of ice occur in a sheet of aufeis. Credit: [[c:User:Nswanson|Nswanson]].{{tlx|free media}}]] [[Image:Aufeis far.jpg|thumb|right|250px|A sheet of aufeis occurs in a glacial valley in Mongolia. Credit: [[w:User:Nswanson|Nswanson]].{{tlx|free media}}]] [[Image:After flood ice.jpg|thumb|right|250px|Ice layers in the trees are formed by an earlier winter flood. Credit: [[c:User:Doronenko|Doronenko]].{{tlx|free media}}]] '''Def.''' a sheet-like layered mass of ice is called '''aufeis'''. '''Def.''' a sheet-like layered mass of ice formed in freezing temperatures from the freezing of successive flows of ground water over previously formed layers of ice is called '''naled'''. {{clear}} ==Ice streams== [[Image:FRicestreams.jpg|thumb|right|250px|Radarsat image is of ice streams flowing into the Filchner-Ronne Ice Shelf. Credit: [[w:User:Polargeo|Polargeo]].{{tlx|free media}}]] [[Image:Wardhunt.jpg|thumb|left|250px|Canadian RADARSAT image shows the shelf in August 2002, when a crack made its way down the length of the shelf. Credit: Alaska Satellite Facility, Geophysical Institute, University of Alaska Fairbanks.{{tlx|fairuse}}]] [[Image:Flow of Ice Across Antarctica.ogv|thumb|right|200px|These animations show the motion of ice in Antarctica. Credit: NASA.{{tlx|free media}}]] [[Image:Bambervelocity.jpg|thumb|right|250px|This is a velocity map of Antarctic ice streams. Credit: Jonathan Bamber, University of Bristol.{{tlx|free media}}]] On the right is a radarsat image of ice streams flowing into the Filchner-Ronne ice shelf. Annotations outline the Rutford ice stream. "One example of an ice shelf composed of compacted, thickened sea ice is the Ward Hunt Ice Shelf off the coast of Ellesmere Island in northern Canada. Canadian RADARSAT image shows the shelf in August 2002, when a crack made its way down the length of the shelf."<ref name=Staff/> The image on the right is a Radarsat portrayal of ice streams flowing into the Filchner-Ronne Ice Shelf. This image uses data from the Radarsat RAMP 125m Mosaic. The dataset is freely available from the National Snow and Ice Data Center. '''Def.''' "a current of ice in an ice sheet or ice cap that flows faster than the surrounding ice"<ref name=Beitler>{{ cite book |author=Jane Beitler |title=Cryosphere Glossary |publisher=National Snow and Ice Data Center |location= |date=19 September 2014 |url=http://nsidc.org/cryosphere/glossary/I |accessdate=2014-09-17 }}</ref> is called an '''ice stream'''. The second image on the right shows animated motions of ice flowing across Antarctica. These animations shows the motion of ice in Antarctica as measured by satellite data from the Canadian Space Agency, the Japanese Space Agency and the European Space Agency, and processed via NASA-funded research from the University of California, Irvine. The background image from the Landsat satellite is progressively replaced by a map of ice velocity, which is colour-coded on a logarithmic scale. The third image on the right shows the ice stream velocities of Antarctic ice from zero (black) up to 250m/yr (cream white). "Although they account for only 10% of the volume of the ice sheet, ice streams are sizeable features, up to 50 km wide, 2000 m thick and hundreds of km long. Some flow at speeds of over 1000 m per year and most of the ice leaving the ice sheet passes through them."<ref name=BritishAntarcticSurvey>{{ cite web |author=British Antarctic Survey |title=Ice Streams in Antarctica |publisher=Natural Environment Research Council (NERC) |location=Cambridge, United Kingdom |date= 2014 |url=http://www.antarctica.ac.uk//about_antarctica/geography/ice/streams.php |accessdate=2014-11-23 }}</ref> "Ice streams generally form where water is present, but other factors also control their velocity, in particular whether the ice stream rests on hard rock or soft, deformable sediments. At the edges of ice streams deformation causes ice to recrystallise making it softer and concentrating the deformation into narrow bands or shear margins. Crevasses, cracks in the ice, result from rapid deformation and are common in shear margins."<ref name=BritishAntarcticSurvey/> {{clear}} ==Glaciers== {{main|Rocks/Glaciers|Glaciers}} '''Def.''' "a mass of ice that originates on land, usually having an area larger than one tenth of a square kilometer"<ref name=Beitler/> is called a '''glacier'''. '''Def.''' "a persistent body of [dense]<ref name=Glacier1>{{ cite book |author=[[wikt:User:141.163.203.132|141.163.203.132]] |title=glacier |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=14 August 2013 |url=https://en.wikipedia.org/wiki/Glacier |accessdate=16 September 2022 }}</ref> ice<ref name=Glacier>{{ cite book |author=[[wikt:User:Wsiegmund|Wsiegmund]] |title=glacier |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=11 September 2010 |url=https://en.wikipedia.org/wiki/Glacier |accessdate=16 September 2022 }}</ref> [that is]<ref name=Glacier2>{{ cite book |author=[[wikt:User:MONGO|MONGO]] |title=glacier |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=4 January 2014 |url=https://en.wikipedia.org/wiki/Glacier |accessdate=16 September 2022 }}</ref> [moving under its own]<ref name=Glacier1/> [weight]"<ref name=Glacier3>{{ cite book |author=[[wikt:User:99.236.178.14|99.236.178.14]] |title=glacier |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=28 February 2014 |url=https://en.wikipedia.org/wiki/Glacier |accessdate=16 September 2022 }}</ref> is called a '''glacier'''. {{clear}} ==Surging glaciers== [[Image:Surging glacier.jpg|thumb|right|200px|In 1941, Hole-in-the-Wall Glacier surged. Credit: W.O. Field, World Data Center for Glaciology, Boulder, CO.{{tlx|fairuse}}]] [[Image:Sermersauq Ice Cap Glacier.jpg|thumb|left|200px|The image shows a glacial surge from the Sermersauq Ice Cap. Credit: Robert Gilbert, Niels Nielsen, Henrik Möller, Joseph R. Desloges, and Morten Rasch.{{tlx|fairuse}}]] '''Def.''' "a glacier that experiences a dramatic increase in flow rate, 10 to 100 times faster than its normal rate; usually surge events last less than one year and occur periodically, between 15 and 100 years"<ref name=Beitler/> is called a '''surging glacier'''. "In 1941, Hole-in-the-Wall Glacier [imaged at the right] surged, also knocking over trees during its advance."<ref name=Beitler/> An "outlet glacier of the Sermersauq Ice Cap [on Disko Island, West Greenland, shown at the left with progressive surges marked] has surged 10.5 km downvalley to within 10 km of the fjord. [...] surging of the glacier, begun in 1995, was undetected until July 1999, when it was discovered during a geomorphic survey of the valley. Mapping from TM, Landsat and SPOT satellite imagery, and subsequent field work have documented the history of the event. On 17 June 1995 the terminus of the glacier was about where it appears in the 1985 air photography [...]. By 24 September 1995 the glacier had advanced 1.25 km and by 12 October another 1.25 km (mean advance during the second period : 70 m day^-1). The advance slowed from 18 m day^-1 in 1996 to 5 m day^-1 in 1997 and <1 m day^-1 between 1997 and 1999. By summer 1999 the advance ceased; the maximum extension of the terminus, about 10.5 km down-valley to about 10 km from the head of the fjord, was mapped from imagery on 9 July 1999 [...]."<ref name=Gilbert>{{ cite journal |author=Robert Gilbert |author2=Niels Nielsen |author3=Henrik Möller |author4=Joseph R. Desloges |author5=Morten Rasch |title=Glacimarine sedimentation in Kangerdluk (Disko Fjord), West Greenland, in response to a surging glacier |journal=Marine Geology |year=2002 |volume=191 |issue= |pages=1-18 |url=http://geog.queensu.ca/gilbert/surge%20paper.PDF |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-09-24 }}</ref> {{clear}} ==Classification of glaciers== [[Image:Glacier mapping.jpg|thumb|center|500px|Glacier mapping is performed with Landsat TM and a GIS. Credit: F. Paul, C. Huggel, A. Kääb, T. Kellenberger, and M. Maisch.{{tlx|fairuse}}]] "The low reflectivity of snow and glacier ice in the middle infrared part of the spectrum allows glacier classification".<ref name=Paul>{{ cite book |author=F. Paul |author2=C. Huggel |author3=A. Kääb |author4=T. Kellenberger |author5=M.Maisch |title=Comparison of TM-derived glacier areas with higher resolution data sets, In: ''Proceedings of EARSeL-LISSIG-Workshop Observing our Cryosphere from Space'' |publisher=EARSeL-LISSIG |location= |date=11 March 2002 |editor= |pages=15 |url=http://eproceedings.org/static/vol02_1/02_1_paul1.pdf |arxiv= |bibcode= |doi= |pmid= |isbn= |accessdate=2014-10-14 }}</ref> In the set of images at the center top of this section, glacier mapping steps are shown from left to right with the Landsat 7 enhanced Thematic Mapper (TM) and a geographic information system (GIS).<ref name=Paul/> The images are part of the "102 glaciers of the Mischabel mountain range."<ref name=Paul/> The first image on the left is a ratio image from TM4 and TM4, specifically (TM4 / TM5).<ref name=Paul/> The second is a "derived glacier map after thresholding (blue) and overlay with digitized basins (red)."<ref name=Paul/> The third image from the left identifies "[i]ndividual glaciers after basin intersection (colour-coded) ready for [digital elevation model] DEM-fusion."<ref name=Paul/> The thermal emission and reflectivity have been measured "using the sensors ASTER (Advanced Spaceborne Thermal Emission and reflection Radiometer) on board [the] Terra [satellite]".<ref name=Paul/> Glaciers may be classified on the basis of areal extent or size. "With [a standard deviation of] σ = 3% the values obtained [...] are (resolution / minimum useful glacier size (in km²)): 5 m / all, 10 m / 0.01, 15 m / 0.03, 20 m / 0.05, 25 m / 0.1, 30 m / 0.2, 60 m / 0.5."<ref name=Paul/> "The comparison with higher-resolution satellite imagery reveals: (a) an overall good corre- spondence of the TM-derived glacier outlines with the manual delineation, (b) mapping of debris-covered glacier ice is not possible with TM data alone, and (c) also manual glacier delineation is problematic in the case of debris cover or snowfields."<ref name=Paul/> {{clear}} ==Alpine glaciers== [[Image:Trips 04 - Mt Wedge - 02 (90961463).jpg|thumb|right|250px|The wedgemount alpine glacier is rapidly receding and used to touch the lake as recently as 1990. Credit: [http://www.flickr.com/people/56796376@N00 McKay Savage from London, UK].{{tlx|free media}}]] '''Def.''' "a glacier that is confined by surrounding mountain terrain; also called a mountain glacier"<ref name=Beitler/> is called an '''alpine glacier'''. For "alpine glaciers the imbalance [the change of mass balance/altitude profiles from years with positive to those with negative mean balance] is nearly independent of altitude, in dry, continental regions the imbalance is largest near the equilibrium line, where albedo changes are most pronounced."<ref name=Kuhn>{{ cite journal |author=Michael Kuhn |title=Mass Budget Imbalances as Criterion for a Climatic Classification of Glaciers |journal=Geografiska Annaler. Series A, Physical Geography |year=1984 |volume=66 |issue=3 |pages=229-38 |url=http://www.jstor.org/discover/10.2307/520696?uid=3739552&uid=2&uid=4&uid=3739256&sid=21104337348461 |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-10-14 }}</ref> {{clear}} ==Maritime glaciers== [[Image:Whaler off of NOAA Ship John N. Cobb-Sawyer Glacier.jpg|thumb|right|250px|Sawyer Glacier is in the background. Credit: Personnel of the NOAA ship John N. Cobb.{{tlx|free media}}]] '''Def.''' a glacier that is # found on the sea, # "bordering on the sea",<ref name=MaritimeWikt/> # in a moist and temperate climate owing to the influence of the sea, # "related to the sea,"<ref name=MaritimeWikt>{{ cite book |author=[[wikt:User:24.175.141.118|24.175.141.118]] |title=maritime |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=13 September 2004 |url=https://en.wiktionary.org/wiki/maritime |accessdate=23 September 2022 }}</ref> # "near or in the sea"<ref name=MaritimeWikt/> is called a '''maritime glacier'''. "Maritime glaciers owe their mass balance variations mainly to changes in the accumulation area".<ref name=Kuhn/> {{clear}} ==Tidewater glaciers== [[Image:2008-05-24 12 Jökulsarlón.jpg|thumb|right|250px|The Jökulsarlón tidewater glacier is in Iceland. Credit: [[c:User:Simisa|Hansueli Krapf]].{{tlx|free media}}]] '''Def.''' a glacier occurring in "water affected by the flow of the tide,<ref name=TidewaterWikt>{{ cite book |author=[[wikt:User:Dmh|Dmh]] |title=tidewater |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=11 January 2005 |url=https://en.wiktionary.org/wiki/tidewater |accessdate=23 September 2022 }}</ref> especially tidal streams"<ref name=TidewaterWikt1>{{ cite book |author=[[wikt:User:71.38.136.94|71.38.136.94]] |title=tidewater |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=25 March 2007 |url=https://en.wiktionary.org/wiki/tidewater |accessdate=23 September 2022 }}</ref> is called a '''tidewater glacier'''. {{clear}} ==Polar glaciers== [[Image:Pensacola Glacier.jpg|thumb|right|250px|The Pensacola glacier in the Pensacola Range of Antarctica is a polar glacier. Credit: NASA / James Yungel.{{tlx|free media}}]] '''Def.''' a high-latitude glacier that is covered by ice is called a '''polar glacier''', or '''napajäätikkö'''. Polar "glaciers [owe their mass balance variations] to the varying duration of ablation in their lowest parts."<ref name=Kuhn/> {{clear}} ==Rock glaciers== [[Image:Rock glacier.jpg|thumb|right|250px|Frying Pan Glacier is almost entirely covered by rocks and debris. Credit: George L. Snyder.{{tlx|fairuse}}]] '''Def.''' "looks like a mountain glacier and has active flow; usually includes a poorly sorted mess of rocks and fine material; may include: (1) interstitial ice a meter or so below the surface ("ice-cemented"), (2) a buried core of ice ("ice-cored"), and/or (3) rock debris from avalanching snow and rock"<ref name=Beitler/> is called a '''rock glacier'''. '''Def.''' "a mass of rock fragments and finer material, on a slope, that contains either an ice core or interstitial ice, and shows evidence of past, but not present, movement"<ref name=Beitler/> is called an '''inactive rock glacier'''. At the right, "Frying Pan Glacier, Colorado, is almost entirely covered by rocks and debris in this photograph from 1966."<ref name=Beitler/> {{clear}} ==Tributary glaciers== [[Image:03 susitna surge moraines.jpg|thumb|left|250px|This shows the many tributary glaciers of the Susitna Glacier, including surge effects. Credit: Brian John.{{tlx|fairuse}}]] The photo on the left shows many tributary glaciers coming into the Susitna Glacier, including surge effects. {{clear}} ==Valley glaciers== [[Image:Branched valley glacier.jpg|thumb|right|250px|In this photograph from 1969, small glaciers flow into the larger Columbia Glacier from mountain valleys on both sides. Credit: United States Geological Survey.{{tlx|fairuse}}]] '''Def.''' a "glacier that has one or more tributary glaciers that flow into it"<ref name=Beitler/> is called a '''branched-valley glacier'''. "In this photograph from 1969 [at the right], small glaciers flow into the larger Columbia Glacier from mountain valleys on both sides. Columbia Glacier flows out of the Chugach Mountains into Columbia Bay, Alaska."<ref name=Beitler/> {{clear}} ==Outlet glaciers== [[Image:Greenland-glacier hg.jpg|thumb|right|250px|An outlet glacier flows down the side of Fønfjord (Scoresby Sund), Greenland. Credit: [[c:User:Hgrobe|Hannes Grobe, AWI]].{{tlx|free media}}]] "Close to the edges [of an ice sheet], much of the ice flows in narrow and fast-moving outlet glaciers along bedrock troughs [...] Roughly half of the mass loss occurs by iceberg calving from the fronts of these outlets; the other half, by surface melt around the periphery of the whole ice sheet."<ref name=Cuffey>{{ cite book |author=Kurt M. Cuffey |author2=W. S. B. Paterson |title=The Physics of Glaciers |publisher=Elsevier |location=Burlington, Massachusetts USA |date=2010 |editor= |pages=708 |url=http://books.google.com/books?hl=en&lr=&id=Jca2v1u1EKEC&oi=fnd&pg=PP2&ots=KLFO4-pikc&sig=nrAWChisiE5anhb1wFr23YlogvI#v=onepage&f=false |arxiv= |bibcode= |doi= |pmid= |isbn=978-0-12-369461-4 |accessdate=2014-10-15 }}</ref> {{clear}} ==Isolated glaciers== [[Image:Kilimanjaro Glaciers.jpg|thumb|right|250px|Annotated NASA image of Mount Kilimanjaro indicates its glaciers. Credit: NASA and [[c:User:MONGO|MONGO]].{{tlx|free media}}]] [[Image:Mount Kilimanjaro.jpg|thumb|left|250px|This is a panorama of Mount Kilimanjaro showing Kibo peak. Credit: [[w:User:Muhammad Mahdi Karim|Muhammad Mahdi Karim]].{{tlx|free media}}]] [[Image:Mount Kilimanjaro Dec 2009 edit1.jpg|thumb|right|250px|Mount Kilimanjaro is imaged from the air. Credit: [[w:User:Muhammad Mahdi Karim|Muhammad Mahdi Karim]].{{tlx|free media}}]] [[Image:Kilimanjaro glacier retreat.jpg|thumb|left|250px|The two images show the glacial retreat on Mount Kilimanjaro between February 17, 1993, upper, and February 21, 2000, lower. Credit: NASA and U.S. Government.{{tlx|free media}}]] [[Image:Kilimanjaro-1938-uwm.png|thumb|right|250px|This aerial view is from 1938 and shows much more snow than the one above from 2009. Credit:Mary Meader, American Geographical Society Library, University of Wisconsin-Milwaukee Libraries.{{tlx|free media}}]] [[Image:Kibo ice fields.jpg|thumb|left|250px|Shown are the outlines of the Kibo (Kilimanjaro) ice fields in 1912, 1953, 1976, 1989, and 2000, using the OSU aerial photographs taken on 16 February 2000. Credit: Lonnie G. Thompson, Ellen Mosley-Thompson, Mary E. Davis, Keith A. Henderson, Henry H. Brecher, Victor S. Zagorodnov, Tracy A. Mashiotta, Ping-Nan Lin, Vladimir N. Mikhalenko, Douglas R. Hardy, Jürg Beer.{{tlx|fairuse}}]] "Mount Kilimanjaro is the highest [...] "stand-alone" [...] mountain in the world. [...] Mount Kilimanjaro started to be formed about 750000 years ago being currently constituted by three major volcanic cones, Kibo, Mawenzi, and Shira. The first reaches approximately 5900m."<ref name=Fernandes>{{ cite journal |author=Rui M. S. Fernandes |author2=John Msemwa |author3=Machiel Bos |author4=Joaquim Luís |author5=Jorge Santos |author6=André Sá |author7=Saburi John |author8=Essau Mligo |author9=Goodchance J. Tetti |author10=Hassan M. Ubwa |author11=John R. Sorwa |author12=Maenda Kwimbere |author13=Elifuraha Saria |author14=Paul Emmanuel |author15=Hussein Farah |author16=Charles M. Kamamia |author17=Elsayed Issawi |author18=Anwar Radwan |author19=Rob Painter |author20=Lívia Moreira |author21=João Ferreira |title=Precise Determination of the Orthometric Height of Mt. Kilimanjaro |journal=Surveyors Key Role in Accelerated Development |month=3-8 May |year=2009 |volume=TS 8C |issue=Instruments and Calibration |pages=1-11 |url=http://www.fig.net/pub/fig2009/papers/ts08c/ts08c_fernandes_teamkili2008_3438.pdf |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-10-03 }}</ref> Its "location [is] close to [the] equator associated with the existence of permanent glaciers and its almost perfect volcano shape"<ref name=Fernandes/> For "the Uhuru Peak with respect to the KILI2008 datum ... a final value of 5890.79m was determined for the orthometric height of the highest point in Africa considering the Tanzanian vertical datum."<ref name=Thompson/> Kilimanjaro is located at 3°04.6'S and 37°21.2'E.<ref name=Thompson>{{ cite journal |author=Lonnie G. Thompson |author2=Ellen Mosley-Thompson |author3=Mary E. Davis |author4=Keith A. Henderson |author5=Henry H. Brecher |author6=Victor S. Zagorodnov |author7=Tracy A. Mashiotta |author8=Ping-Nan Lin |author9=Vladimir N. Mikhalenko |author10=Douglas R. Hardy |author11=Jürg Beer |title=Kilimanjaro Ice Core Records: Evidence of Holocene Climate Change in Tropical Africa |journal=Science |month=18 October |year=2002 |volume=298 |issue= |pages=589-93 |url=ftp://ftp.soest.hawaii.edu/engels/Stanley/Textbook_update/Science_298/Thompson-02.pdf |arxiv= |bibcode= |doi=10.1126/science.1073198 |pmid= |accessdate=2014-10-04 }}</ref> "Aerial photographs taken on 16 February 2000 allowed production of a recent detailed map of ice cover extent on the summit plateau [diagram at the lower left]."<ref name=Thompson/> "Total ice area calculated from successive maps (1912, 1953, 1976, 1989, and 2000) reveals [diagram at the lower left, inset] that the areal extent of Kilimanjaro’s ice cover has decreased approximately 80% from ~12 km² in 1912 to ~2.6 km² in 2000 and that since 1989, a hole has developed near the center of the NIF. A nearly linear relationship (R² = 0.98) suggests that if climatological conditions of the past 88 years continue, the ice on Kilimanjaro will likely disappear between 2015 and 2020."<ref name=Thompson/> {{clear}} ==Crater glaciers== [[Image:Nevados de Sollipulli.jpg|thumb|right|250px|The image shows the crater glacier of the volcano Sollipulli. Credit: [[c:User:Roka1953|Roka1953]].{{tlx|free media}}]] [[Image:Iss038e012569.jpg|thumb|left|250px|The summit of Sollipulli is occupied by a four-kilometer wide caldera, currently filled with a snow-covered glacier. Credit: William L. Stefanov.{{tlx|fairuse}}]] "While active volcanoes are obvious targets of interest because they pose natural hazards, there are some dormant volcanoes that also warrant concern because of their geologic history. One such volcano is Sollipulli, located in central Chile near the border with Argentina. The volcano sits in the southern Andes Mountains within Chile’s Parque Nacional Villarica. This photograph by an astronaut on the International Space Station features the summit (2,282 meters, or 7,487 feet, above sea level) and the bare slopes above the tree line. Lower elevations are covered with green forests indicative of Southern Hemisphere summer."<ref name=Stefanov2013>{{ cite book |author=William L. Stefanov |title=Sollipulli Caldera, Chile and Argentina |publisher=NASA |location=Washington, DC USA |date=23 December 2013 |url=http://earthobservatory.nasa.gov/IOTD/view.php?id=82676 |accessdate=2014-10-15 }}</ref> "The summit of Sollipulli is occupied by a four-kilometer wide caldera, currently filled with a snow-covered glacier. While most calderas form after violent, explosive eruptions, the types of rock and other deposits associated with such events have not been found at Sollipulli. Geologic evidence does indicate explosive activity occurred about 2,900 years ago, and lava flows were produced approximately 700 years ago. Together with the craters and scoria cones along the outer flanks of the caldera, this history suggests Sollipulli could erupt violently again, presenting a potential hazard to towns such as Melipeuco and the wider region."<ref name=Stefanov2013/> {{clear}} ==Cirque glaciers== [[Image:Backed up Against the Wall.jpg|thumb|right|250px|A quarter mile of glacial ice is all that remains from the retreat of the glacier of Southwind Fiord, Baffin Island, Nunavut, Canada. Credit: [http://www.flickr.com/people/31856336@N03 Mike Beauregard from Nunavut, Canada].{{tlx|free media}}]] [[Image:Glacial Cirque Formation EN.svg|thumb|left|250px|Schematic profile of a cirque and cirque glacier, shows Bergschrund, randkluft and the headwall gap. Credit: [[c:User:ClemRutter|Clem Rutter]].{{tlx|free media}}]] Cirques, as diagrammed at the left, are formed by a glacier (the cirque glacier) and usually exhibit a Bergschrund, randkluft and the headwall gap. The image at the right shows a glacier on Baffin Island that has retreated back to a cirque glacier. {{clear}} ==Temperate glaciers== [[Image:HafrahvammagljúfurIV 02092006.jpg|thumb|right|250px|The canyons of Hafrahvammar are shown. Credit: [[c:User:Fbd|Friðrik Bragi Dýrfjörð]].{{tlx|free media}}]] At the right is an image of a temperate glacier; i.e., one flowing through a temperate region, as evidenced by the green plants. {{clear}} ==Ice shelves== [[Image:Alfred Ernest Ice Shelf.jpg|thumb|right|250px|This is a radar image of Alfred Ernest Ice Shelf on Ellesmere Island, taken by the ERS-1 satellite. Credit: NASA.{{tlx|free media}}]] [[Image:Moa_iceshelves.jpg|thumb|right|250px|Antarctica's major ice shelf areas are indicated. Credit: National Snow & Ice Data Center.{{tlx|free media}}]] [[Image:Antarctic shelf ice hg.png|thumb|right|250px|This is a schematic of glaciological and oceanographic processes along the Antarctic coast. Credit: [[c:User:Hgrobe|Hannes Grobe]], Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany.{{tlx|free media}}]] [[Image:2008 Wilkins 1.png|thumb|right|250px|A 430-square-kilometer section of the 13,680-square-kilometer Wilkins Ice Shelf on the Antarctic Peninsula rapidly disintegrated. Credit: National Snow & Ice Data Center.{{tlx|free media}}]] [[Image:Iceshelf 03.jpg|thumb|right|250px|This satellite image shows floating chunks of ice from the 2008 Wilkins Ice Shelf collapse. Credit: Cheng-Chien Liu and An-Ming Wu, National Space Organization, Taiwan.{{tlx|free media}}]] [[Image:A-62 iceberg connected to Fimbul Ice Shelf at Queen Maud Land.jpg|thumb|right|250px|Iceberg A 62 was connected to the Fimbul Ice Shelf by a mere 800-metre-wide bridge. Credit: DLR - German Space Agency.{{tlx|free media}}]] [[Image:Wardhunt.jpg|thumb|right|250px|Canadian RADARSAT image shows the shelf in August 2002, when a crack made its way down the length of the shelf. Credit: Alaska Satellite Facility, Geophysical Institute, University of Alaska Fairbanks.{{tlx|fairuse}}]] [[Image:Iceberg A-38.jpg|thumb|left|250px|This is an image of iceberg A-38 after it detached from the Ronne Ice Shelf. Credit: National Ice Center/National Oceanic and Atmospheric Administration.{{tlx|free media}}]] On the right is a radar image of Alfred Ernest Ice Shelf on Ellesmere Island, taken by the ERS-1 satellite. '''Def.''' a thick, floating platform of ice that forms where a glacier or ice sheet flows down to a coastline and onto the ocean surface is called an '''ice shelf'''. "Ice shelves are permanent floating sheets of ice that connect to a landmass."<ref name=Staff>{{ cite book |author=Staff |title=Quick Facts on Ice Shelves |publisher=National Snow & Ice Data Center |location= |date= 2014 |url=https://nsidc.org/cryosphere/quickfacts/iceshelves.html |accessdate=2014-10-31 }}</ref> "Most of the world's ice shelves hug the coast of Antarctica [as shown in the image on the right]. However, ice shelves can also form wherever ice flows from land into cold ocean waters, including some glaciers in the Northern Hemisphere. The northern coast of Canada's Ellesmere Island is home to several well-known ice shelves, among them the Markham and the Ward Hunt ice shelves."<ref name=Staff/> "Ice from enormous ice sheets slowly oozes into the sea through glaciers and ice streams. If the ocean is cold enough, [...] newly arrived ice doesn't melt right away. Instead it may float on the surface and grow larger as glacial ice behind it continues to flow into the sea. Along protected coastlines, the resulting ice shelves can survive for thousands of years, bolstered by the rock of peninsulas and islands. Ice shelves grow when they gain ice from land, and occasionally shrink when icebergs calve off their edges."<ref name=Staff/> The schematic on the right presents glaciological and oceanographic processes along the Antarctic coast. Snow falling in the accumulation zone creates an upstream stress. The ice shelf has built up to a thickness of about 4 km. The ice flows along the glacier and in ice streams within. On the coast the ice loses contact with its bedrock at the grounding line and becomes significantly thinner by some 100 m. It forms an ice shelf over the continental shelf. At the edge of the continental shelf, tabular icebergs calve. "Satellite imagery [third image on the right] revealed that the western front of the 13,680 square kilometer (5,282 square mile) Wilkins Ice Shelf began to collapse because of rapid climate change in a fast-warming region of Antarctica."<ref name=Staff/> "This satellite image [fourth on the right] shows floating chunks of ice from the 2008 Wilkins Ice Shelf collapse."<ref name=Staff/> "Most ice shelves are fed by inland glaciers. Together, an ice shelf and the glaciers feeding it can form a stable system, with the forces of outflow and back pressure balanced. Warmer temperatures can destabilize this system by increasing glacier flow speed and—more dramatically—by disintegrating the ice shelf. Without a shelf to slow its speed, the glacier accelerates. After the 2002 Larsen B Ice Shelf disintegration, nearby glaciers in the Antarctic Peninsula accelerated up to eight times their original speed over the next 18 months. Similar losses of ice tongues in Greenland have caused speed-ups of two to three times the flow rate in just one year."<ref name=Staff/> "Ice shelves fall into three categories: (1) ice shelves fed by glaciers, (2) ice shelves created by sea ice, and (3) composite ice shelves (Jeffries 2002). Most of the world's ice shelves, including the largest, are fed by glaciers and are located in Greenland and Antarctica."<ref name=Staff/> "A small island obstructs the constant flow of the ice shelf on Queen Maud Land – it is the lighter area at the bottom left of the image [on the right]. From September 2010 until it broke off, Iceberg A 62 was connected to the Fimbul Ice Shelf by a mere 800-metre-wide bridge. Two fissures in the ice from different sides of the bridge approached one another until the break occurred. The images transmitted by the radar satellite TerraSAR-X over a long period of time should enable researchers to achieve a better understanding of how icebergs calve."<ref name=DLR>{{ cite book |author=DLR |title=The iceberg breaks free |publisher=Deutsches Zentrum für Luft- und Raumfahrt |location=Cologne |date=20 January 2011 |url=http://www.dlr.de/media/en/desktopdefault.aspx/tabid-4986/8423_read-18675/8423_page-2 |accessdate=2016-09-20 }}</ref> '''Def.''' a "thick, floating platform of ice that forms where a glacier or ice sheet flows down to a coastline and onto the ocean surface"<ref name=IceShelfWikt>{{ cite book |title=ice shelf |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=24 May 2014 |url=https://en.wiktionary.org/wiki/ice_shelf |accessdate=2014-10-30 }}</ref> is called an '''ice shelf'''. "Ice shelves are permanent floating sheets of ice that connect to a landmass."<ref name=Staff>{{ cite book |author=Staff |title=Quick Facts on Ice Shelves |publisher=National Snow & Ice Data Center |location= |date= 2014 |url=https://nsidc.org/cryosphere/quickfacts/iceshelves.html |accessdate=2014-10-31 }}</ref> "Ice shelves fall into three categories: (1) ice shelves fed by glaciers, (2) ice shelves created by sea ice, and (3) composite ice shelves (Jeffries 2002). Most of the world's ice shelves, including the largest, are fed by glaciers and are located in Greenland and Antarctica."<ref name=Staff/> "One example of an ice shelf composed of compacted, thickened sea ice is the Ward Hunt Ice Shelf off the coast of Ellesmere Island in northern Canada. Canadian RADARSAT image shows the shelf in August 2002, when a crack made its way down the length of the shelf."<ref name=Staff/> The Ronne Ice Shelf has a nominal location of 78°30'S 61°W. {{clear}} ==Ice fields== [[Image:Southern Patagonian Ice Field.jpg|thumb|right|250px|Southern Patagonian Ice Field, Argentina-Chile, with the volcano Lautaro visible in the upper portion of this image, and Mount Fitz Roy is in the lower left corner. Credit: NASA.{{tlx|free media}}]] [[Image:St Elias Ice-field.jpg|thumb|left|250px|Photo from a plane shows the St Elias Ice-field. Credit: [[c:user:Kitrabbit|Kitrabbit]].{{tlx|free media}}]] [[Image:Ice field near the shore at sunset.jpg|thumb|right|250px|Ice field is near the shore at sunset. Credit: [[c:user:Александр Байдуков|Александр Байдуков]].{{tlx|free media}}]] '''Def.''' a network of interconnected glaciers or ice streams having a common source or a large expanse of floating ice (several miles long) is called an '''ice field'''. Komarovsky Beach: on both sides of the Primorsky Highway between Morskaya, Sportivnaya and Kurortnaya Streets and the Gulf of Finland, including coastal shallow water, Komarovo, Kurortny District, Saint Petersburg. Several ice fields can become an ice cap. {{clear}} ==Ice caps== [[Image:Drill sites on Greenland.jpg|thumb|right|250px|The most important drill sites on the inland ice and on two small separate ice caps: Hans Tavsen in Peary Land in the north and Renland in the east are indicated. Credit: Willi Dansgaard.{{tlx|fairuse}}]] [[Image:Aaj-13201212214-1377205687.jpeg|thumb|left|250px|Looking south on Renland is across the Edward Bailey Glacier into the Alpine Bowl. Credit: Silvan Schüpbach.{{tlx|fairuse}}]] [[Image:Ice cap.jpg|thumb|left|250px|This is an aerial image of the ice cap on Ellesmere Island, Canada. Credit: National Snow and Ice Data Center.{{tlx|fairuse}}]] [[Image:Vatnajökull.jpeg|thumb|left|250px|Vatnajökull, Iceland has an ice cap. Credit: NASA.{{tlx|free media}}]] '''Def.''' "a dome-shaped mass of glacier ice that spreads out in all directions"<ref name=Beitler/> is called an '''ice cap'''. In addition to many of the ice core drilling sites on Greenland in the image at the right, there are the separate ice caps on Hans Tavsen in Peary Land way to the north and Renland in the east. In "1985, when [the final version of “the Rolls Royce drill”] penetrated the separate, high-lying Renland ice cap in the Scoresbysund Fiord [...] down to 325 m, world record for this type of drill".<ref name=Dansgaard/> The Renland ice core from East Greenland apparently covers a full glacial cycle from the Holocene into the previous Eemian interglacial. The Renland ice core is 325 m long.<ref name=Hansson1>{{ cite journal |author=Hansson M, Holmén K |title= |journal=Geophy Res Lett. |month=November |year=2001 |volume=28 |issue=22 |pages=4239-42 |doi=10.1029/2000GL012317 }}</ref> "The δ-profile [...] proved that the Renland ice cap has always been separated from the inland ice. Since all of the δ-leaps revealed by the Camp Century core recurred in the small Renland ice cap, the Renland peninsula cannot have been overrun by ice streams from the inland ice, not even during the glaciation.<ref name=Dansgaard/> The Penny Ice Cap is on Baffin Island, Canada, at 67° 15'N, 65° 45'W, 1900 masl. In April 1998 on the Devon Ice Cap filtered lamp oil was used as a drilling fluid. In the Devon core it was observed that below about 150 m the [[stratigraphy]] was obscured by microfractures.<ref name=P1386j>{{ cite book |author=C. SIMON L. OMMANNEY | title=HISTORY OF GLACIER INVESTIGATIONS IN CANADA | url=http://pubs.usgs.gov/prof/p1386j/history/history-lores.pdf | accessdate=October 14, 2005 }}</ref> "Beginning in 1995, a large outlet glacier of the Sermersauq Ice Cap on Disko Island [Greenland] surged 10.5 km downvalley to within 10 km of the head of the fjord, Kuannersuit Sulluat, reaching its maximum extent in summer 1999 before beginning to retreat. Sediment discharge to the fjord increased from 13 x 10³ t day^-1 in 1997 to 38 x 10³ t day^-1 in 1999. CTD results, sediment traps and cores from the 2000 melt season document the impact of the surge on the glacimarine environment of the fjord."<ref name=Gilbert>{{ cite journal |author=Robert Gilbert |author2=Niels Nielsen |author3=Henrik Möller |author4=Joseph R. Desloges |author5=Morten Rasch |title=Glacimarine sedimentation in Kangerdluk (Disko Fjord), West Greenland, in response to a surging glacier |journal=Marine Geology |year=2002 |volume=191 |issue= |pages=1-18 |url=http://geog.queensu.ca/gilbert/surge%20paper.PDF |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-09-24 }}</ref> "Short gravity cores were taken and CTD profiles were recorded at stations throughout Kuannersuit Sulluat [...]. Positions located by GPS are accurate to ±10 m or less. The stream flowing over the sandur to the head of the fjord was gauged and integrated suspended sediment samples were recovered from primary channels."<ref name=Gilbert/> "The cores were photographed, X-rayed and logged. X-radiographs provided measures of the number and size of gravel particles interpreted as ice-rafted debris (IRD) and the grey-scale (GS) of the scanned images was plotted as a measure of the properties of the sand and silt."<ref name=Gilbert/> "The twelve layers in core D4 [imaged at the right] suggest a mean period of about 20 days for these events based on the accumulation rates in the traps [...]. In general, these layers have both higher MS and X-radiographs have lighter toned GS, the former related to lower water content and the latter also related to greater absorption of X-rays by the larger rock and mineral fragments."<ref name=Gilbert/> There "are notable differences in the surge-generated sediments. The proximal sediments [such as in core D4 at the right] are more clearly laminated and layered in visual examination of the cores and as seen in the X-radiographs [compared to distal sediments as imaged on the left for core D20]. These consist both of the subtle differences in the fine-grained sediments on a millimetre scale, and of the sand layers up to 8 cm thick representing more energetic processes (Ó Cofaigh and Dowdeswell, 2001). Both are a response to greater sediment input to the fjord."<ref name=Gilbert/> The ice core drilled in Guliya ice cap in western China in the 1990s reaches back to 760,000 b2k; farther back than any other core at the time, though the EPICA core in Antarctica equalled that extreme in 2003.<ref name=Bowen>{{ cite book |author=Mark Bowen |year=2005 |title=Thin Ice |publisher=Henry Holt Company |isbn=0-8050-6443-5 }}</ref> Ice cores from Sajama in Bolivia span ~25 ka and help present a high resolution temporal picture of the Late Glacial Stage and the Holocene climatic optimum.<ref name=Thompson1>{{ cite journal |author=Thompson LG |author2=Mosley-Thompson EM |author3=Henderson KA |title=Ice-core palaeoclimate records in tropical South America since the Last Glacial Maximum |journal=J Quaternary Sci. |volume=15 |issue=4 |year=2000 |pages=377-94 |doi=10.1002/1099-1417(200005)15:4<377::AID-JQS542>3.0.CO;2-L }}</ref> Although the ice cores from Quelccaya ice cap only go back ~2 ka,<ref name=Thompson1/> others may go back ~5.2 ka. The Quelccaya ice cores correlate with those from the Upper Fremont Glacier. {{clear}} ==Greenland ice sheets== [[Image:Greenland 42.74746W 71.57394N.jpg|thumb|right|250px|Satellite composite image shows the ice sheet of Greenland. Credit: NASA.{{tlx|free media}}]] [[Image:Une partie de l'hémisphère nord de la Terre avec la banquise, nuage, étoile et localisation de la station météo en Alert.jpg|thumb|left|250px|Earth's northern hemisphere polar ice sheet includes sea ice. Credit: NASA/Goddard Space Flight Center.{{tlx|free media}}]] [[Image:Grl18577-fig-0001.png|thumb|right|250px|(a) The probability is for of a pixel melting at least as many times as observed during the 1995, 1998 and 2002 melt seasons given the last 25 years of melt observations. (b) Melt extent is for 2002: Pixels are color coded for number of melt days during the season. (c) Slopes of the trend lines are fit to the areas observed to melt between April and November from 1979 to 2003. Credit: K. Steffen, S. V. Nghiem, R. Huff, and G. Neumann.{{tlx|fairuse}}]] [[Image:Grl18577-fig-0002.png|thumb|left|250px|Half-decade records for ETH/CU Camp station: (a) Top panel is for QSCAT backscatter, (b) middle panel for QSCAT diurnal signature, and (c) bottom panel for air temperature measured at the AWS site. Credit: K. Steffen, S. V. Nghiem, R. Huff, and G. Neumann.{{tlx|fairuse}}]] [[Image:Grl18577-fig-0003.png|thumb|right|250px|QSCAT melt maps are shown on the climatological peak-melt day (1 August). Red color represents current active melt areas, light blue is for areas that have melted but currently refreeze, white is for areas that will melt later, and magenta is for areas that do not experience any melt throughout the melt season. The dark blue color surrounding Greenland is the ocean mask. Credit: K. Steffen, S. V. Nghiem, R. Huff, and G. Neumann.{{tlx|fairuse}}]] [[Image:Grl18577-fig-0004.png|thumb|left|250px|QSCAT maps of number of melt days (violet to red for 1 to 31 days) in 2000–2003 with the overlaid black contours representing melt extent derived from PM data are shown. Credit: K. Steffen, S. V. Nghiem, R. Huff, and G. Neumann.{{tlx|fairuse}}]] '''Def.''' "a dome-shaped mass of glacier ice that covers surrounding terrain and is greater than 50,000 square kilometers (12 million acres)"<ref name=Beitler/> is called an '''ice sheet'''. At the right is a satellite composite image of the ice sheet over Greenland. "Active and passive microwave satellite data are used to map snowmelt extent and duration on the Greenland ice sheet. The passive microwave (PM) data reveal the extreme melt extent of 690,000 km² in 2002 as compared with an average extent of 455,000 km² from 1979–2003."<ref name=Steffen>{{ cite journal |author=K. Steffen |author2=S. V. Nghiem |author3=R. Huff |author4=G. Neumann |title=The melt anomaly of 2002 on the Greenland Ice Sheet from active and passive microwave satellite observations |journal=Geophysical Research Letters |month=21 October |year=2004 |volume=21 |issue=20 |pages= |url= |arxiv= |bibcode= |doi=10.1029/2004GL020444 |pmid= |accessdate=2014-09-28 }}</ref> "Several PM-based melt assessment algorithms [Mote and Anderson, 1995; Abdalati and Steffen, 1995] are applicable to Scanning Multi-channel, Microwave Radiometer (SMMR) and Special Sensor Microwave/Imager (SSM/I) instruments providing near-continuous coverage since 1979. The PM data as gridded brightness temperatures on polar stereographic grids (25 km resolution) [used] are from the National Snow and Ice Data Center [Maslanik and Stroeve, 2003], containing daily data spanning 25 melt seasons from 1979 to 2003."<ref name=Steffen/> In the second image on the right, (a) "shows the probabilities of the observed melt behavior on the Greenland ice sheet for several large melt years and indicates the extreme melt anomaly observed in northeastern Greenland in 2002."<ref name=Steffen/> "Prior to 2002, both 1995 and 1998 were extreme melt years in terms of maximum areal extent and total melt. During 1995 melt was dominated by a high frequency of melt along the western margin of the ice sheet. During 1998 melt was spatially diverse with slightly more melt than usual in the northeast and southwest. However, the high frequency melt in 2002 in the northeast and along the western margin is unprecedented in the PM record with a log likelihood of occurrence that is 35% lower than the previous record melt anomaly in 1991."<ref name=Steffen/> (c) "depicts the magnitude of the increasing trends in melt extent on a daily basis over the last 25 years. Although there is a large amount of inter-annual variability in melt extent on a given day, 56 days show statistically significant (alpha = 0.1) increasing trends in melt area."<ref name=Steffen/> "Melt along the west coast was extensive during 2002 but not atypical for large melt years. However melt in the north and northeast was highly irregular both in terms of extent and frequency. Nearly 3,000 km²[(b)] were classified as melting during 2002 that had not previously melted during any other year between 1979 and 2003."<ref name=Steffen/> The figure at the left "presents QSCAT backscatter and diurnal signatures, and ETH/CU AWS air temperature."<ref name=Steffen/> Half-decade records for ETH/CU Camp station: (a) Top panel is for QSCAT backscatter, (b) middle panel for QSCAT diurnal signature, and (c) bottom panel for air temperature measured at the AWS site.<ref name=Steffen/> At the lower right QSCAT melt maps are shown on the climatological peak-melt day (1 August). Red color represents current active melt areas, light blue is for areas that have melted but currently refreeze, white is for areas that will melt later, and magenta is for areas that do not experience any melt throughout the melt season. The dark blue color surrounding Greenland is the ocean mask. "QSCAT mapping can reveal details of the spatial pattern of surface melt evolution in time. There are large variabilities in melt extent and melt timing over different regions. [The figure at tje lower right] confirms that 2002 has the most extensive areal melt. In 2002, the northeast quadrant of the Greenland ice sheet, extending well into the dry snow zone, experienced at least some melt where melt never happened before (from satellite data records to date). Since the beginning of the QSCAT data record (July 1999), the smallest spatial extent of melt occurred in 2001, and melt extent was similar for years 2000 and 2003."<ref name=Steffen/> "To provide a direct comparison of PM and QSCAT results, we overlay results for PM melt extent and QSCAT number of melt days in [the figure at the lower left] for years 2000–2003. PM XPGR melt extent is approximately confined to QSCAT melt areas experiencing 2 weeks or more of melting time [the figure at the lower left]. QSCAT melt areas outside of the PM melt extent represent the surface that has less melt corresponding to about 15 melt days or less. This is consistent with the relationship of relative melt strength measured by active and passive data as discussed above. Note that such areas can total up to a large region in year 2002. Surface albedo can reduce considerably once the snow melts for a period of 2 weeks. The albedo reduction may significantly impact the surface heat balance and thus change the mass balance. The large number of melt days around the northern perimeter of the ice sheet, which is shown as the narrow dark-red band in north Greenland in the 2003 map was an anomalous feature [the figure at the lower left]. This band was wider as defined by the PM melt extent in 2002 than in 2003. However, there were more QSCAT melt days in the 2003 northern melt band."<ref name=Steffen/> "The comparison reveals that the PM cross-polarized gradient algorithm classifies melt more conservatively than the scatterometer algorithm. The active microwave identifies melt approximately up to two weeks more than the PM at higher elevation in the percolation zone toward the dry snow zone [the figure at the lower left]. Both methods (active and passive microwave) consistently identify melt areas that have a melt duration of at least 10–14 days. The longer snowmelt duration can be sufficient to decrease surface albedo and affect surface heat and mass balance."<ref name=Steffen/> {{clear}} ==Antarctic ice sheets== [[Image:Antarctica 6400px from Blue Marble.jpg|thumb|right|250px|A satellite composite image shows the ice sheet of Antarctica Credit: [[c:User:Dave Pape|Dave Pape]].{{tlx|free media}}]] [[Image:Antarktyda i Antarktyka.jpg|thumb|left|250px|A satellite composite image shows a global view of the sea ice and ice sheet of Antarctica. Credit: NASA Scientific Visualization Studio Collection.{{tlx|free media}}]] [[Image:Antarctic-ice-flow inline.png|right|thumb|300px|Velocity of ice flowing across Antarctica varies by location. Credit: Jeremie Mouginot, University of California Irvine.{{tlx|fairuse}}]] "The only current ice sheets are in Antarctica and [[Greenland]]; during the last glacial period at Last Glacial Maximum (LGM) the Laurentide ice sheet covered much of North America, the Weichselian ice sheet covered northern Europe and the Patagonian Ice Sheet covered southern South America."<ref name=IceSheet>{{ cite book |title=Ice sheet |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=June 6, 2013 |url=http://en.wikipedia.org/wiki/Ice_sheet |accessdate=2013-06-23 }}</ref> At the south pole, Antactica, there is also an extensive ice sheet shown in the second image on the right. Seasonally, when the North polar sea ice and ice sheet has been contracting, the South polar sea ice and ice sheet has been expanding. Apparent global warming that was progressively melting more and more of the north polar ice sheet each year has been countered by progressive expansion of the south polar ice sheet. "Decades of satellite observations have now provided the most detailed view yet [second image down on the right] of how Antarctica continually sheds ice accumulated from snowfall into the ocean."<ref name=Temming>{{ cite book |author=Maria Temming |title=A new map is the best view yet of how fast Antarctica is shedding ice |publisher=Science News |location= |date=9 August 2019 |url=https://www.sciencenews.org/article/new-map-best-view-yet-how-fast-antarctica-shedding-ice?utm_source=Editors_Picks&utm_medium=email&utm_campaign=editorspicks081119 |accessdate=12 August 2019 }}</ref> The "first comprehensive view of how ice moves across all of Antarctica, [includes] slow-moving ice in the middle of the continent rather than just rapidly melting ice at the coasts."<ref name=Temming/> Subtle "movements of Antarctic ice [were detected] with a kind of measurement called synthetic-aperture radar interferometric phase data."<ref name=Temming/> "By using a satellite to bounce radar signals off a patch of ice, [...] how quickly that ice is moving toward or away from the satellite [can be determined]. Combining observations of the same spot from different angles reveals the speed and direction of the ice’s motion along the ground."<ref name=Temming/> "Inland ice moves incredibly slowly — much of it plods along at fewer than 10 meters per year. Closer to the ocean, ice can travel hundreds to thousands of meters per year."<ref name=Temming/> "To get multiple vantage points of the same swathes of ice, [...] data from about half a dozen satellites launched by Canada, Europe and Japan since the early 1990s [was put together]."<ref name=Temming/> "Each brought a little piece of the puzzle."<ref name=Rignot>{{ cite book |author=Eric Rignot |title=A new map is the best view yet of how fast Antarctica is shedding ice |publisher=Science News |location= |date=9 August 2019 |url=https://www.sciencenews.org/article/new-map-best-view-yet-how-fast-antarctica-shedding-ice?utm_source=Editors_Picks&utm_medium=email&utm_campaign=editorspicks081119 |accessdate=12 August 2019 }}</ref> "Surface ice velocity is a fundamental characteristic of glaciers and ice sheets that quantifies the transport of ice. Changes in ice dynamics have a major impact on ice sheet mass balance and its contribution to sea level rise. Prior comprehensive mappings employed speckle and feature tracking techniques, optimized for fast‐flow areas, with precision of 2‐5 m/yr, hence limiting our ability to describe ice flow in the slow interior. We present a vector map of ice velocity using the interferometric phase from multiple satellite synthetic aperture radars resulting in ten‐times higher precision in speed (20 cm/yr) and direction (5o) over 80% of Antarctica. Precision mapping over areas of slow motion (< 1 m/yr) improves from 20% to 93%, which helps better constrain drainage boundaries, improve mass balance assessment, evaluate regional atmospheric climate models, reconstruct ice thickness, and inform ice sheet numerical models."<ref name=Mouginot>{{ cite journal |author=Jeremie Mouginot |author2=E. Rignot |author3=B. Scheuchl |title=Continent‐wide, interferometric SAR phase, mapping of Antarctic ice velocity |journal=Geophysical Research Letters |date=29 July 2019 |volume= |issue= |pages= |url=https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2019GL083826 |arxiv= |bibcode= |doi=10.1029/2019GL083826 |pmid= |accessdate=12 August 2019 }}</ref> {{clear}} ==Himalayas ice sheets== [[Image:Himalayas landsat 7.png|thumb|right|250px|This is a Landsat 7 image of the Himalayas. NASA.{{tlx|fairuse}}]] [[Image:HIRES for web g 453284734.jpg|thumb|center|400px|The Tibetan plateau, often called the third pole, will be monitored by balloons, drones and ground sensors. Credit: Wolfgang Kaehler/LightRocket via Getty.{{tlx|fairuse}}]] Often called the third pole, the image on the right shows the rocky ice sheet over the top of the Himalayas. "Sitting at an average height of around 4,000 metres above sea level, the plateau protrudes into the middle of the troposphere, where most weather events originate. As the biggest and highest plateau in the world, it disturbs this part of the atmosphere like no other structure on Earth."<ref name=Qiu>{{ cite journal |author=Jane Qiu |title=Tibetan plateau gets wired up for monsoon prediction |journal=Nature |volume=514 |issue= |pages=16-7 |location= |month=01 October |year=2014 |url=http://www.nature.com/news/tibetan-plateau-gets-wired-up-for-monsoon-prediction-1.16030 |doi=10.1038/514016a |accessdate=2014-10-02 }}</ref> "The plateau’s remoteness, altitude and harsh conditions — it is often called the third pole because it hosts the world’s third-largest stock of ice — mean that even basic weather stations are few. Satellite data are also plagued by large errors owing to lack of calibration from ground observations."<ref name=Qiu/> {{clear}} ==Glaciations== [[Image:GlaciationsinEarthExistancelicenced annotated.jpg|thumb|center|450px|Geologic time is annotated with glacial or ice age periods. Credit: [[c:User:William M. Connolley|William M. Connolley]].{{tlx|free media}}]] [[Image:IceAgeEarth.jpg|thumb|right|200px|Earth at the last glacial maximum of the current ice age. Credit: [[c:User:Ittiz|Ittiz]], based on: "Ice age terrestrial carbon changes revisited" by Thomas J. Crowley (Global Biogeochemical Cycles, Vol. 9, 1995, pp. 377-389.{{tlx|free media}}]] [[Image:Iceage north-intergl glac hg.png|thumb|left|200px|Recent (black) and maximum (grey) glaciation of the northern hemisphere are during the Quaternary climatic cycles. Credit: [[c:User:Hgrobe|Hannes Grobe]]/AWI.{{tlx|free media}}]] [[Image:Iceage south-intergl glac hg.png|thumb|right|200px|Recent (black) and maximum (grey) glaciation of the southern hemisphere are during the Quaternary climatic cycles. Credit: [[c:User:Hgrobe|Hannes Grobe]]/AWI.{{tlx|free media}}]] '''Def.''' the "process of covering with a glacier,<ref name=GlaciationWikt1>{{ cite book |author=[[wikt:User:66.32.178.103|66.32.178.103]] |title=glaciation |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=22 April 2010 |url=https://en.wiktionary.org/wiki/glaciation |accessdate=4 July 2019 }}</ref> or the state of being glaciated;<ref name=GlaciationWikt/> the production of glacial phenomena;<ref name=GlaciationWikt>{{ cite book |author=[[wikt:User:Poccil|Poccil]] |title=glaciation |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=20 October 2004 |url=https://en.wiktionary.org/wiki/glaciation |accessdate=4 July 2019 }}</ref> an ice age<ref name=GlaciationWikt2>{{ cite book |author=[[wikt:User:SemperBlotto|SemperBlotto]] |title=glaciation |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=1 November 2018 |url=https://en.wiktionary.org/wiki/glaciation |accessdate=4 July 2019 }}</ref>" is called a '''glaciation'''. The ice ages or glaciations on Earth occurred from the early Proterozoic (Huronian), late proterozoic (Cryogenian), early Paleozoic (Andean-Saharan) during the Ordovician and Silurian periods, late Paleozoic (Karoo Ice Age) during the Carboniferous and early Permian periods, and lately the Quaternary glaciation. Although these ice ages are widely separated in geological time, "in most parts of the Earth major climatic and palaeoenvironmental units typically have a duration of the order of half a precession cycle (around 10 ka) rather than half an eccentricity cycle (around 50 ka) so that the level of stratigraphic resolution provided by the Middle Pleistocene [Marine Isotope Stage] MIS (typical duration 50 ka) is not sufficiently fine to constitute a universal stratigraphic template."<ref name=Shackleton>{{ cite journal |author=Nicholas J. Shackleton |author2=Maria Fernanda Sánchez-Goñi |author3=Delphine Pailler |author4=Yves Lancelot |title=Marine Isotope Substage 5e and the Eemian Interglacial |journal=Global and Planetary Change |year=2003 |volume=36 |issue= |pages=151-5 |url=http://www.colorado.edu/geography/class_homepages/geog_5241_f09/media/Readings/shackletonetal.pdf |arxiv= |bibcode= |doi=10.1016/S0921-8181(02)00181-9 |pmid= |accessdate=2014-10-11 }}</ref> {{clear}} ==Icequakes== [[Image:Stations-1.jpg|thumb|right|250px|This map of Antarctica shows the icequakes triggered by Chile's 2010 earthquake. Credit: Zhigang Peng, Georgia Tech.{{tlx|fairuse}}]] "Only 12 of Antarctica's 42 seismometers picked up icequakes after the Maule earthquake, but the signals seemed to fit a pattern. The pattern suggests that opening or closing of shallow crevasses generated the tiny tremors. For example, seismic stations near Antarctica's mountain ranges and fast-flowing ice rivers known as ice streams were more likely to see icequakes. These are areas with a lot of crevasses. The high-frequency shaking also fits with cracking of '''brittle ice'''."<ref name=Oskin>{{ cite book |author=Becky Oskin |title=Faraway Earthquake Triggered Antarctica Icequakes |publisher=LiveScience.com |location= |date=10 August 2014 |url=http://www.livescience.com/47282-chile-earthquake-caused-antarctica-icequakes.html |accessdate=2014-08-16 }}</ref> Bold added. "Antarctica's ice snapped and popped because of a major earthquake in Maule, Chile, halfway around the world [...] Antarctica has been touched by great earthquakes before. In March 2011, Japan's Tohoku tsunami tore off two Manhattan-size icebergs from the Sulzberger Ice Shelf, more than 8,000 miles (13,000 kilometers) south. Sailors also reported a massive Antarctica iceberg-calving event after Chile's 1868 great earthquake."<ref name=Oskin/> "Icequakes are seismic tremblings caused by sudden movement within a glacier or ice sheet, such as from a fracturing crevasse. (Anyone who has dropped an ice cube into a glass of water knows ice snaps under stress.)"<ref name=Oskin/> "Chile's magnitude-8.8 earthquake on Feb. 27, 2010, set off a flurry of Antarctic icequakes, each lasting from one to 10 seconds, researchers report today (Aug. 10) in the journal Nature Geoscience. The epicenter was 2,900 miles (4,700 km) north of Antarctica."<ref name=Oskin/> "We think the crevasses are being activated by the surface waves from this big earthquake coming through, and that's making the icequake."<ref name=Walter>{{ cite book |author=Jacob Walter |title=Faraway Earthquake Triggered Antarctica Icequakes |publisher=LiveScience.com |location= |date=10 August 2014 |url=http://www.livescience.com/47282-chile-earthquake-caused-antarctica-icequakes.html |accessdate=2014-08-16 }}</ref> "Regular icequakes probably occur all the time in Antarctica and other polar regions."<ref name=Peng>{{ cite book |author=Zhigang Peng |title=Faraway Earthquake Triggered Antarctica Icequakes |publisher=LiveScience.com |location= |date=10 August 2014 |url=http://www.livescience.com/47282-chile-earthquake-caused-antarctica-icequakes.html |accessdate=2014-08-16 }}</ref> "What we found is that they occurred more during the seismic waves of the Maule event."<ref name=Peng/> "Many different kinds of icequakes rumble across Antarctica and Greenland. Known icequake triggers include opening and closing of the fractures called crevasses; glaciers tearing away from sticky bedrock; water runoff; and calving, the breaking off of an iceberg. Spooky underwater sounds from melting, cracking icebergs were once called The Bloop."<ref name=Oskin/> Just "one kind of seismic wave, a surface wave, gets the blame for most of Antarctica's icequakes. [...] a Rayleigh wave [...] travels close to the Earth's surface, rolling along like a wave in a lake or the ocean. [...] At some stations, there was also a short icequake burst from a seismic "P wave," which travel through the Earth's interior."<ref name=Oskin/> {{clear}} ==Sea ices== [[Image:Greenland East Coast 7.jpg|thumb|right|250px|This is an aerial view of the pack ice off the eastcoast of Greenland. Credit: [[c:user:Michael Haferkamp|Michael Haferkamp]].{{tlx|free media}}]] [[Image:Vaxholm ice.jpg|thumb|left|250px|This is pack ice off the coast of Vaxholm, Sweden. Credit: [[c:User:Cyberjunkie|Cyberjunkie]].{{tlx|free media}}]] [[Image:Line3892 - Flickr - NOAA Photo Library.jpg|thumb|right|250px|Pack-ice-covered Auke Bay Harbor, Alaska, in winter. Credit: David Csepp, NOAA/NMFS/AKFSC/ABL.{{tlx|free media}}]] [[Image:Seaice 04.jpg|thumb|left|250px|When waves buffet the freezing ocean surface, characteristic "pancake" sea ice forms. Credit: Ted Scambos, NSIDC.{{tlx|fairuse}}]] A "climate interpretation was supported by very low δ’s in the 1690’es, a period described as extremely cold in the Icelandic annals. In 1695 Iceland was completely surrounded by sea ice, and according to other sources the sea ice reached half way to the Faeroe Islands."<ref name=Dansgaard>{{ cite book |author=Willi Dansgaard |title=Frozen Annals Greenland Ice Cap Research |publisher=Niels Bohr Institute |location=Copenhagen, Denmark |year=2005 |editor=The Department of Geophysics of The Niels Bohr Institute for Astronomy Physics and Geophysics at The University of Copenhagen Denmark |pages=123 |url=http://www.iceandclimate.nbi.ku.dk/publications/FrozenAnnals.pdf/ |arxiv= |bibcode= |doi= |pmid= |isbn=87-990078-0-0 |accessdate=2014-10-05 }}</ref> "The correlation is astonishing, because it implies that the dramatic climate changes during the first more than 50 kyrs of the glaciation elapsed nearly in parallel on both sides of the North Atlantic Ocean, presumably controlled by varying sea ice cover. Thus, the Gulf Stream was not just deflected toward North Africa in cold periods, it was rather turned off."<ref name=Dansgaard/> '''Def.''' a "large consolidated mass of floating sea ice"<ref name=PackIceWikt>{{ cite book |title=pack ice, In: ''Wiktionary'' |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=29 May 2014 |url=https://en.wiktionary.org/wiki/pack_ice |accessdate=2014-11-01 }}</ref> is called '''pack ice'''. Pack ice in the image on the right is drifting southward in the East Greenland current during July 1996. In the second image on the left, when "waves buffet the freezing ocean surface, characteristic "pancake" sea ice forms."<ref name=Scambos>{{ cite book |author=Ted Scambos |title=Quick Facts on Arctic Sea Ice |publisher=National Snow & Ice Data Center |location= |date= 2004 |url=http://nsidc.org/cryosphere/quickfacts/seaice.html |accessdate=2014-11-03 }}</ref> "Sheets of sea ice form when frazil crystals float to the surface, accummulate and bond together. Depending upon the climatic conditions, sheets can develop from grease and congelation ice, or from pancake ice."<ref name=Scambos1>{{ cite book |author=Ted Scambos |title=Ice formation |publisher=National Snow & Ice Data Center |location= |date= 2004 |url=http://nsidc.org/cryosphere/seaice/characteristics/formation.html |accessdate=2014-11-03 }}</ref> "If the ocean is rough, the frazil crystals accummulate into slushy circular disks, called pancakes or pancake ice, because of their shape. A signature feature of pancake ice is raised edges or ridges on the perimeter, caused by the pancakes bumping into each other from the ocean waves. If the motion is strong enough, rafting occurs. If the ice is thick enough, ridging occurs, where the sea ice bends or fractures and piles on top of itself, forming lines of ridges on the surface. Each ridge has a corresponding structure, called a keel, that forms on the underside of the ice. Particularly in the Arctic, ridges up to 20 meters (60 feet) thick can form when thick ice deforms. Eventually, the pancakes cement together and consolidate into a coherent ice sheet. Unlike the congelation process, sheet ice formed from consolidated pancakes has a rough bottom surface."<ref name=Scambos1/> {{clear}} ==Icebergs== [[Image:Iceberg in the Arctic with its underside exposed.jpg|right|thumb|250px|When the polar sea is calm, the underside of icebergs can easily be observed in the clear waters of the Arctic Ocean. Credit: [[c:User:AWeith|AWeith]].{{tlx|free media}}]] [[Image:Black ice growler upernavik 2007-07-07a.jpg|right|thumb|250px|Black ice growler from a recently calved iceberg is closing in on the shore at the old heliport in Upernavik, Greenland. Credit: [[c:User:Slaunger|Kim Hansen]].{{tlx|free media}}]] [[Image:Black ice growler texture upernavik 2007-07-07.jpg|left|thumb|250px|Surface texture on a growler of black ice. Credit: [[c:User:Slaunger|Kim Hansen]].{{tlx|free media}}]] The first image on the right shows that when the polar sea is calm, the underside of icebergs can easily be observed in the clear waters of the Arctic Ocean. Centered in the image second down on the right is a black ice growler from a recently calved iceberg closing in on the shore at the old heliport in Upernavik, Greenland. Such black ice growlers originate from glacial rifts, or crevasses, filled with melting water, which freezes into transparent ice without air bubbles. On the left is an image of the surface texture on a black ice growler. There are bowl-like depressions in the surface created by the melting process of sea water. {{clear}} ==Lahars== [[Image:MSH82 lahar from march 82 eruption 03-21-82.jpg|thumb|right|250px|An explosive eruption of Mount St. Helens on March 19, 1982, sent pumice and ash 9 miles (14 kilometers) into the air, and resulted in a lahar (the dark deposit on the snow) flowing from the crater into the North Fork Toutle River valley. Credit: Tom Casadevall.{{tlx|free media}}]] "Because the volcano itself is covered by 15 square miles of glaciers, the lava that flows down the side and mixes with ice and snow to form lahars — a mudflow slurry that can move extremely quickly and destroy towns in their path. According to the Smithsonian, "lahars have damaged towns on Villarica's flanks." The BBC reports that more than 100 people are believed to have been killed by the volcano's mudflows in the past century."<ref name=Plumer>{{ cite book |author=Brad Plumer |title=Chile's recent volcanic eruption looked absolutely stunning — and terrifying |publisher=Vox |location=Villarrica, Chile |date=4 March 2015 |url=http://www.vox.com/2015/3/4/8147751/chile-volcano-villarrica |accessdate=2015-03-27 }}</ref> '''Def.''' a "volcanic mudflow"<ref name=LaharWikt>{{ cite book |author=[[wikt:User:Emperorbma|Emperorbma]] |title=lahar |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=1 December 2004 |url=https://en.wiktionary.org/wiki/lahar |accessdate=4 May 2019 }}</ref> is called a '''lahar'''. Part of the Mount St. Helens lahar entered Spirit Lake (lower left corner of the image on the right) but most of the flow went west down the Toutle River, eventually reaching the Cowlitz River, 50 miles (80 kilometers) downstream. {{clear}} ==Lightning== {{main|Lightning}} [[Image:Rinjani 1994.jpg|thumb|right|250px|The 1995 eruption of Mount Rinjani in Indonesia exhibits volcanic lightning. Credit: [[commons:User:Spolloman|Oliver Spalt]].{{tlx|free media}}]] [[Image:Galunggung.jpg|thumb|left|250px|The slide depicts a spectacular view of lightning strikes during a third eruption on December 3, 1982. Credit: R. Hadian, U.S. Geological Survey.{{tlx|free media}}]] Many volcanic eruptions put on impressive lightning displays such as during the 1995 eruption of Mount Rinjani in Indonesia shown in the image on the right which exhibits many leaders. The image on the left shows spectacular lightning strikes around Galunggung, including multiple leaders apparently involved in cloud to cloud lightning. "This stratovolcano with a lava dome is located in western Java. Its first eruption in 1822 produced a 22-km-long mudflow that killed 4,000 people. The second eruption in 1894 caused extensive property loss. The photo depicts a spectacular view of lightning strikes during a third eruption on December 3, 1982, which resulted in 68 deaths. A fourth eruption occurred in 1984."<ref name=Hadian>{{ cite book |author=R. Hadian |title=Galunggung, Indonesia |publisher=NOAA National Geophysical Data Center |location= |date=3 December 1982 |url=http://www.ngdc.noaa.gov/nndc/servlet/ShowDatasets?EQ_0=603&bt_0=&st_0=&EQ_1=&bt_1=&st_1=&query=&dataset=101634&search_look=2&group_id=null&display_look=4,44&submit_all=Select+Data |accessdate=2015-03-24 }}</ref> Volcanic lightning arises from colliding, fragmenting particles of volcanic ash (and sometimes ice),<ref>{{Cite news |url=https://www.washingtonpost.com/news/capital-weather-gang/wp/2016/04/13/scientists-think-theyve-solved-the-mystery-of-how-volcanic-lightning-forms/?utm_term=.80462ad8051d |title=Scientists think they've solved the mystery of how volcanic lightning forms |last=Fritz |first=Angela |date=2016 |work=The Washington Post |accessdate= }}</ref><ref>{{Cite news |url=https://www.seeker.com/mystery-of-volcano-lightning-explained-1771209774.html |title=Mystery of Volcano Lightning Explained |last=Mulvaney |first=Kieran |date=2016 |work=Seeker |accessdate= }}</ref> which generate [[static electricity]] within the volcanic plume.<ref>{{Cite news |url=https://blogs.agu.org/geospace/2016/04/12/new-studies-uncover-mysterious-processes-generate-volcanic-lightning/ |title=New studies uncover mysterious processes that generate volcanic lightning |last=Lipuma |first=Lauren |date=2016 |work=American Geophysical Union GeoSpace Blog |accessdate= }}</ref> Volcanic eruptions have been referred to as '''dirty thunderstorms'''<ref>{{Cite journal |last=Hoblitt |first=Richard P. |date=2000 |title=Was the 18 May 1980 lateral blast at Mt St Helens the product of two explosions? |journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences |volume=358 |issue=1770 |pages=1639–1661 |doi=10.1098/rsta.2000.0608 |doi-access=free }}</ref><ref name=":6">{{Cite journal |last=Bennett |first=A J |last2=Odams |first2=P |last3=Edwards |first3=D |last4=Arason |first4=Þ |date=2010-10-01 |title=Monitoring of lightning from the April–May 2010 Eyjafjallajökull volcanic eruption using a very low frequency lightning location network |journal=Environmental Research Letters |volume=5 |issue=4 |pages=044013 |doi=10.1088/1748-9326/5/4/044013 |doi-access=free |issn=1748-9326 }}</ref> due to [http://glossary.ametsoc.org/wiki/Moist_convection moist convection] and ice formation that drive the eruption plume dynamics<ref name=":7">{{Cite journal |last=Woods |first=Andrew W. |date=1993 |title=Moist convection and the injection of volcanic ash into the atmosphere |journal=Journal of Geophysical Research: Solid Earth |volume=98 |pages=17627–17636 |doi=10.1029/93JB00718 }}</ref><ref name=":8">{{Cite journal |last=Van Eaton |first=Alexa R. |last2=Mastin |first2=Larry G. |last3=Herzog |first3=Michael |last4=Schwaiger |first4=Hans F. |last5=Schneider |first5=David J. |last6=Wallace |first6=Kristi L. |last7=Clarke |first7=Amanda B. |date=2015-08-03 |title=Hail formation triggers rapid ash aggregation in volcanic plumes |journal=Nature Communications |volume=6 |issue=1 |doi=10.1038/ncomms8860 |doi-access=free |issn=2041-1723 }}</ref> and can trigger volcanic lightning.<ref>{{Cite journal |last=Williams |first=Earl R. |last2=McNutt |first2=Stephen R. |date=2005 |title=Total water contents in volcanic eruption clouds and implications for electrification and lightning |url=http://www.giseis.alaska.edu/Input/steve/PUBS/williams-mcn-signpost.PDF |journal=Proceedings of the 2nd International Conference on Volcanic Ash and Aviation Safety |volume= |pages=67–71 }}</ref><ref name=":9">{{Cite journal |last=Van Eaton |first=Alexa R. |last2=Amigo |first2=Álvaro |last3=Bertin |first3=Daniel |last4=Mastin |first4=Larry G. |last5=Giacosa |first5=Raúl E. |last6=González |first6=Jerónimo |last7=Valderrama |first7=Oscar |last8=Fontijn |first8=Karen |last9=Behnke |first9=Sonja A. |date=2016-04-12 |title=Volcanic lightning and plume behavior reveal evolving hazards during the April 2015 eruption of Calbuco volcano, Chile |journal=Geophysical Research Letters |volume=43 |issue=7 |pages=3563–3571 |doi=10.1002/2016gl068076 |doi-access=free |issn=0094-8276 |via= }}</ref> But unlike ordinary thunderstorms, volcanic lightning can also occur before any ice crystals have formed in the ash cloud.<ref>{{Cite journal |last=Cimarelli |first=C. |last2=Alatorre-Ibargüengoitia |first2=M.A. |last3=Kueppers |first3=U. |last4=Scheu |first4=B. |last5=Dingwell |first5=D.B. |date=2014 |title=Experimental generation of volcanic lightning |journal=Geology |volume=42 |issue=1 |pages=79–82 |doi=10.1130/g34802.1 |doi-access=free |issn=1943-2682 }}</ref><ref>{{Cite journal |last=Cimarelli |first=C. |last2=Alatorre-Ibargüengoitia |first2=M. A. |last3=Aizawa |first3=K. |last4=Yokoo |first4=A. |last5=Díaz-Marina |first5=A. |last6=Iguchi |first6=M. |last7=Dingwell |first7=D. B. |date=2016-05-06 |title=Multiparametric observation of volcanic lightning: Sakurajima Volcano, Japan |journal=Geophysical Research Letters |volume=43 |issue=9 |pages=4221–4228 |doi=10.1002/2015gl067445 |doi-access=free |issn=0094-8276 }}</ref> {{clear}} ==Blues== {{main|Radiation astronomy/Blues}} [[Image:Argentina - Bariloche trekking 013 - Glacier Castaño Overo spilling water and ice over the cliff on Cerro Tronador (6797419529).jpg|thumb|right|250px|This image shows the Glacier Castaño Overo spilling blue water ice, or blue ice. Credit: [https://www.flickr.com/people/56796376@N00 McKay Savage from London, UK].{{tlx|free media}}]] '''Blue ice''' occurs when snow falls on a glacier, is compressed, and becomes part of a [[w:glacier|glacier]] ... blue ice was observed in [[w:Tasman Glacier|Tasman Glacier]], New Zealand in January 2011.<ref name="NZ_Herald_10699700">{{ cite book |url=http://www.nzherald.co.nz/travel/news/article.cfm?c_id=7&objectid=10699700 |title=NZ blue ice sighting an unexpected treat for tourists, In: ''The New Zealand Herald'' |author=Harvey, Eveline |date=14 January 2011 |accessdate=21 September 2011 }}</ref> Ice is blue for the same reason water is blue: it is a result of an [[w:overtone|overtone]] of an oxygen-hydrogen (O-H) bond stretch in water which absorbs light at the red end of the visible spectrum.<ref name=Dartmouth>[http://www.dartmouth.edu/~etrnsfer/water.htm Why Is Water Blue]</ref> {{clear}} ==Venus== [[Image:Maxwell_Montes_of_planet_Venus.jpg|thumb|right|250px|Brightening of the radar reflection from the surface of Venus at high elevations such as Maxwell Montes. Credit: NASA/JPL.{{tlx|free media}}]] While there is little or no water on Venus, there is a phenomenon which is quite similar to snow. The Magellan probe imaged a highly reflective substance at the tops of Venus's highest mountain peaks which bore a strong resemblance to terrestrial snow. This substance arguably formed from a similar process to snow, albeit at a far higher temperature. Too volatile to condense on the surface, it rose in gas form to cooler higher elevations, where it then fell as precipitation. The identity of this substance is not known with certainty, but speculation has ranged from elemental tellurium to lead sulfide (galena).<ref name=Otten>{{ cite book |title='Heavy metal' snow on Venus is lead sulfide |author=Carolyn Jones Otten |publisher=Washington University in St Louis |url=http://news-info.wustl.edu/news/page/normal/633.html |date=2004 |accessdate=2007-08-21}}</ref> {{clear}} ==Mars== {{main|Liquids/Liquid objects/Mars}} [[Image:2005-1103mars-full.jpg|thumb|right|250px|This Hubble Space Telescope image shows a dust storm, just above center and lighter in contrast than the surface of Mars. Credit: NASA, ESA, The Hubble Heritage Team (STScI/AURA), J. Bell (Cornell University) and M. Wolff (Space Science Institute).{{tlx|free media}}]] [[Image:Icy Crater on Mars ESP 016954 2245.jpg|thumb|left|250px|A newly formed impact crater is observed by HiRISE on Mars Reconnaissance Orbiter. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]] [[Image:Icy Crater on Mars ESP 016954 2245 subimage 2.jpg|thumb|right|250px|Another newly formed impact crater is observed by HiRISE on Mars Reconnaissance Orbiter. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]] [[Image:Small Crater on Planum Boreum PSP 009942 2645 subimage 1.jpg|thumb|left|250px|An impact crater on Planum Boreum, or the North Polar Cap, of Mars, is observed by HiRISE on the Mars Reconnaissance Orbiter. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]] [[Image:ESP 011425 1775.jpg|thumb|right|250px|This freshly formed impact crater occurred on Mars between February 2005 and July 2005. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]] Martian meteors are thought to be from Mars because they have elemental and isotopic compositions that are similar to rocks and atmosphere gases analyzed by spacecraft on Mars.<ref name=Treiman>{{ cite journal |author=A. H. Treiman, coauthors=''et al.'' |title=The SNC meteorites are from Mars |journal=Planetary and Space Science |volume=48 |issue=12–14 |month=October |year=2000 |pages=1213–30 |bibcode=2000P&SS...48.1213T |doi=10.1016/S0032-0633(00)00105-7 }}</ref> At right is a Hubble Space Telescope image of a dust storm on Mars. The picture was snapped on October 28, 2005. The regional dust storm on Mars had "been growing and evolving over the past few weeks. The dust storm, which is nearly in the middle of the planet in this Hubble view is about 930 miles (1500 km) long measured diagonally, which is about the size of the states of Texas, Oklahoma, and New Mexico combined. No wonder amateur astronomers with even modest-sized telescopes have been able to keep an eye on this storm. The smallest resolvable features in the image (small craters and wind streaks) are the size of a large city, about 12 miles (20 km) across. The occurrence of the dust storm is in close proximity to the NASA Mars Exploration Rover Opportunity's landing site in Sinus Meridiani. Dust in the atmosphere could block some of the sunlight needed to keep the rover operating at full power. ... The large regional dust storm appears as the brighter, redder cloudy region in the middle of the planet's disk. This storm has been churning in the planet's equatorial regions for several weeks now, and it is likely responsible for the reddish, dusty haze and other dust clouds seen across this hemisphere of the planet in views from Hubble, ground based telescopes, and the NASA and ESA spacecraft studying Mars from orbit. Bluish water-ice clouds can also be seen along the limbs and in the north (winter) polar region at the top of the image."<ref name=Bell>{{ cite book |author=Jim Bell |author2=Mike Wolff |author3=Keith Noll |title=Mars Kicks Up the Dust as it Makes Closest Approach to Earth |publisher=HubbleSite NewsCenter |location= |date=November 3, 2005 |url=http://hubblesite.org/newscenter/archive/releases/2005/34/image/a/ |accessdate=2013-02-24 }}</ref> At left is an image of a "newly formed impact crater, observed by HiRISE on Mars Reconnaissance Orbiter. The impact that formed the crater exposed the water ice beneath the surface. Some of the ice can be seen scattered at the adjascent area in the subimages. The blast zone (excavated dark material) is almost 800 meters (half a mile) across. The crater itself is just over 20 meters (66 feet) across".<ref name=Byrne>{{ cite book |author=Shane Byrne |title=Icy Craters on Mars |publisher=NASA/JPL/University of Arizona |location=Tucson, Arizona USA |date=April 21, 2010 |url=http://www.uahirise.org/ESP_016954_2245 |accessdate=2013-05-25 }}</ref> "This crater is one of a special group that have excavated down to buried ice. This ice gets thrown out of the crater onto the surrounding terrain. Although buried ice is common over about half the Martian surface, we can only easily discover craters in dusty regions. The overlap between areas that both have buried ice and surface dust is unfortunately small. So even though we have discovered over 100 new impact craters we have only discovered 7 new craters that expose buried ice."<ref name=Byrne/> "When craters excavate this buried ice it tells us something about the extent and depth of buried ice on Mars (controlled by climate); this information is used by planetary scientists to figure out what the recent climate of Mars was like. It has also been a surprise that this ice is so clean. Scientists expected this buried ice to be a mixture of ice and dirt; instead this ice seems to have formed in pure lenses. Yet another surprise that Mars had in store for us!"<ref name=Byrne/> The ice (presumably water ice) is white in the image, but take note of the blue dust or regolith also exposed. The second image at right is a subimage of the one at left. It is natural color and shows in better detail both the ice (white) and the blue material. At second left is an image showing an impact crater on Planum Boreum, or the North Polar Cap, of Mars, as observed by HiRISE on Mars Reconnaissance Orbiter in natural color. "Impact craters on the surface of Planum Boreum, popularly known as the north polar cap, are rare. This dearth of craters has lead scientists to suggest that these deposits may be geologically young (a few million years old), not having had much time to accumulate impact craters throughout their lifetime."<ref name=Fishbaugh>{{ cite book |author=Kate Fishbaugh |title=Small Crater on Planum Boreum |publisher=NASA/JPL/University of Arizona |location=Tucson, Arizona USA |date=October 15, 2008 |url=http://www.uahirise.org/ESP_016954_2245 |accessdate=2013-05-25 }}</ref> "It is also possible that impacts into ice do not retain their shape indefinitely, but instead that the ice relaxes (similar to glass in an old window), and the crater begins to disappear. This subimage shows an example of a rare, small crater ( approximately 115 meters, or 125 yards, in diameter). Scientists can count these shallow craters to attain an estimate of the age of the upper few meters of the Planum Boreum surface."<ref name=Fishbaugh/> "The color in the enhanced-color example comes from the presence of dust and of ice of differing grain sizes. The blueish ice has a larger grain size than the ice that has collected in the crater. The reddish material is dust. The smooth area stretching to the upper right, away from the crater may be due to winds being channeled around the crater or to fine-grained ice and frost blowing out of the crater."<ref name=Fishbaugh/> The third image at right shows a freshly formed impact crater that occurred on Mars between February 2005 and July 2005.<ref name=Team1>{{ cite book |author=HiRISE Team1 |title=Fresh Impact Crater Formed between February 2005 and July 2005 |publisher=NASA/JPL/University of Arizona |location=Tucson, Arizona USA |date=January 2, 2009 |url=http://hirise.lpl.arizona.edu/ESP_011425_1775 |accessdate=2013-05-25 }}</ref> Note the blue material expelled from the crater rock onto the nearby Martian landscape. Very light snow is known to occur at high latitudes on Mars.<ref name=Minard>{{ cite book |url=http://news.nationalgeographic.com/news/2009/07/090702-snow-mars-phoenix.html |title="Diamond Dust" Snow Falls Nightly on Mars |author=Anne Minard |date=2009-07-02 |publisher=National Geographic News }}</ref> {{clear}} ==Europa== {{main|Rocks/Ice sheets/Europa}} [[Image:PIA02500.jpg|thumb|right|250px|Frozen sulfuric acid on Jupiter's moon Europa is depicted in this image produced from data gathered by NASA's Galileo spacecraft. Credit: NASA/JPL.{{tlx|free media}}]] [[Image:Europa densely packed plates.jpg|thumb|right|250px|This chaotic terrain on Europa has areas consisting of densely packed blocks with fractures and narrow lanes of matrix between them. Credit: G. C. Collins, J. W. Head III, R. T. Pappalardo, and N. A. Spaun.{{tlx|fairuse}}]] [[Image:Europa mostly matrix.jpg|thumb|left|250px|The image shows areas on Europa consisting of almost all matrix and no blocks. Credit: G. C. Collins, J. W. Head III, R. T. Pappalardo, and N. A. Spaun.{{tlx|fairuse}}]] [[Image:Conamara Chaos.jpg|thumb|right|250px|Conamara Chaos, the most intensely studied chaos area, lies near the middle of this continuum. Credit: G. C. Collins, J. W. Head III, R. T. Pappalardo, and N. A. Spaun.{{tlx|fairuse}}]] [[Image:High resolution Conamara Chaos.jpg|thumb|left|250px|High-resolution (10 m/pixel) image shows a plate surrounded by matrix material within Conamara Chaos. Credit: G. C. Collins, J. W. Head III, R. T. Pappalardo, and N. A. Spaun.{{tlx|fairuse}}]] [[Image:Europa Chaos.jpg|thumb|right|250px|This view from the Galileo spacecraft of a small region of the thin, disrupted, ice crust in the Conamara region of Jupiter's moon Europa shows the interplay of surface color with ice structures. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]] [[Image:PIA01296 Conomara Chaos regional view.jpg|thumb|left|250px|This Galileo spacecraft image of Jupiter's icy satellite Europa shows surface features such as domes and ridges. Credit: NASA/Jet Propulsion Laboratory/University of Arizona.{{tlx|free media}}]] [[Image:Europa chaotic terrain.jpg|thumb|right|260px|Craggy, 250&nbsp;m high peaks and smooth plates are jumbled together in a close-up of Conamara Chaos. Credit: NASA/JPL.{{tlx|free media}}]] [[Image:PIA01125 Europa chaos and gray band.jpg|thumb|left|250px|Chaotic terrain is typified by the area in the upper right-hand part of the image. Credit: NASA / JPL.{{tlx|free media}}]] "Frozen sulfuric acid on Jupiter's moon Europa is depicted in this image produced from data gathered by NASA's Galileo spacecraft. The brightest areas, where the yellow is most intense, represent regions of high frozen sulfuric acid concentration. Sulfuric acid is found in battery acid and in Earth's acid rain."<ref name=Lavoie09301999>{{ cite book |author=Sue Lavoie |title=PIA02500: Sulfuric Acid on Europa |publisher=NASA's Office of Space Science |location=Washington DC USA |date=September 30, 1999 |url=http://photojournal.jpl.nasa.gov/catalog/PIA02500 |accessdate=2013-06-24 }}</ref> "The morphology of chaotic terrain forms a continuum from areas consisting of densely packed blocks with fractures and narrow lanes of matrix between them ([second image at the right]), to areas consisting of almost all matrix and no blocks ([first image at the left]). Conamara Chaos, the most intensely studied chaos area ([third image at the right]), lies near the middle of this continuum, with -60% of its area consisting of matrix and the remainder consisting of blocks [Spaunet al., 1998]. In addition to these large chaos areas, chaotic terrain also occurs in the interiors of some small (-10 km diameter) features [Spaun et al., 1999] known as "lenticulae.""<ref name=Collins/> "In Conamara Chaos, where data with spatial resolution of up to ten meters per pixel were obtained, the hummocky matrix appears to be a jumbled collection of ice chunks of all sizes, from a kilometer to tens of meters across ([second image on the left])."<ref name=Collins/> "Galileo spacecraft observations of Europa suggest the existence of a brittle ice crust (or lithosphere) at most -2 km thick, and maybe thinner locally, overlying a liquid water or ductile ice layer [''Carr et al''., 1998; ''Pappalardo et al''., 1998, 1999]. Elastic and viscous models of buckling based on the spacing between possible folds in the Astypalaea Linea region give a thickness for the buckling layer of -2 km [''Prockter and Pappalardo'', 2000]. Evidence derived from the width troughs (interpreted as possible grabens) in the surroundings of Callanish, a possible impact structure, might denote a brittle-ductile transition locally as shallow as 0.5 km [''Moore et al''., 1998]. Besides this, study of ice flexion induced by a dome-type structure located close to Conamara Chaos suggests an elastic lithosphere thickness of only -0.1-0.5 km [''Williams and Greeley'', 1998]."<ref name=Ruiz>{{ cite journal |author=Javier Ruiz |author2=Rosa Tejero |title=Heat flows through the ice lithosphere of Europa |journal=Journal of Geophysical Research |date=25 December 2000 |volume=105 |issue=E12 |pages=29,283-9 |url=http://onlinelibrary.wiley.com/store/10.1029/1999JE001228/asset/jgre1197.pdf;jsessionid=8B4256297C7AAD49444947999112809F.f02t01?v=1&t=hzbw8b6c&s=8cf7d27f3f3cf8ddd191c3dcfb213c138e6b08ea |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-08-26 }}</ref> The "odd surface terrain patterns [of Europa] likely come about due to convection. [...] The ice shell of Jupiter’s moon Europa is marked by regions of disrupted ice known as chaos terrains that cover up to 40% of the satellite’s surface, most commonly occurring within 40° of the equator. Concurrence with salt deposits implies a coupling between the geologically active ice shell and the underlying liquid water ocean at lower latitudes. Europa’s ocean dynamics have been assumed to adopt a two-dimensional pattern, which channels the moon’s internal heat to higher latitudes. [...] heterogeneous heating promotes the formation of chaos features through increased melting of the ice shell and subsequent deposition of marine ice at low latitudes."<ref name=Goodman>{{ cite book |author=Jason Goodman |title=Scientists Detect Hidden Ocean on Jupiter’s Moon |publisher=Astro Watch |location= |date=December 2, 2013 |url=http://www.astrowatch.net/2013/12/scientists-detect-hidden-ocean-on.html |accessdate=2014-06-11 }}</ref> The fifth image at the right is a "view of the Conamara Chaos region on Jupiter's moon Europa taken by NASA's Galileo spacecraft shows an area where the icy surface has been broken into many separate plates that have moved laterally and rotated. These plates are surrounded by a topographically lower matrix. This matrix material may have been emplaced as water, slush, or warm flowing ice, which rose up from below the surface. One of the plates is seen as a flat, lineated area in the upper portion of the image. Below this plate, a tall twin-peaked mountain of ice rises from the matrix to a height of more than 250 meters (800 feet). The matrix in this area appears to consist of a jumble of many different sized chunks of ice. Though the matrix may have consisted of a loose jumble of ice blocks while it was forming, the large fracture running vertically along the left side of the image shows that the matrix later became a hardened crust, and is frozen today. The Brooklyn Bridge in New York City would be just large enough to span this fracture."<ref name=Lavoie1998>{{ cite book |author=Sue Lavoie |title=PIA01177: Chaotic Terrain on Europa in Very High Resolution |publisher=NASA's Office of Space Science |location=Washington, DC USA |date=March 2, 1998 |url=http://photojournal.jpl.nasa.gov/catalog/PIA01177 |accessdate=2013-06-24 }}</ref> "North is to the top right of the picture, and the sun illuminates the surface from the east. This image, centered at approximately 8 degrees north latitude and 274 degrees west longitude, covers an area approximately 4 kilometers by 7 kilometers (2.5 miles by 4 miles). The resolution is 9 meters (30 feet) per picture element. This image was taken on December 16, 1997 at a range of 900 kilometers (540 miles) by Galileo's solid state imaging system."<ref name=Lavoie1998/> "Chaotic terrain on Europa is interpreted to be the result of the breakup of brittle surface materials over a mobile substrate."<ref name=Collins>{{ cite journal |author=G. C. Collins |author2=J. W. Head III |author3=R. T. Pappalardo |author4=N. A. Spaun |title=Evaluation of models for the formation of chaotic terrain on Europa |journal=Journal of Geophysical Research |date=25 January 2000 |volume=105 |issue=E1 |pages=1709-16 |url=http://onlinelibrary.wiley.com/store/10.1029/1999JE001143/asset/jgre1144.pdf?v=1&t=hzbx3jkf&s=502393cfea3bb6d9420615af0ca826e8ea8a6a57 |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-08-26 }}</ref> At the third left, "the mottled appearance results from areas of the bright, icy crust that have been broken apart (known as "chaos" terrain), exposing a darker underlying material. This terrain is typified by the area in the upper right-hand part of the image. The mottled terrain represents some of the most recent geologic activity on Europa. Also shown in this image is a smooth, gray band (lower part of image) representing a zone where the Europan crust has been fractured, separated, and filled in with material derived from the interior. The chaos terrain and the gray band show that this satellite has been subjected to intense geological deformation."<ref name=Ciclops>{{ cite book |author=Ciclops |title=Regional Mosaic of Chaos and Gray Band on Europa |publisher=NASA/JPL |location=Pasadena, California USA |date=6 November 1997 |url=http://ciclops.org/view.php?id=4352&js=1 |accessdate=2014-08-26 }}</ref> {{clear}} ==Technology== [[Image:250mm Rain Gauge.jpg|thumb|upright|right|125px|The image shows a standard rain gauge. Credit: [[c:User:Bidgee|Bidgee]].{{tlx|free media}}]] The standard way of measuring rainfall or snowfall is the standard rain gauge, which can be found in 100-mm (4-in) plastic and 200-mm (8-in) metal varieties.<ref name=NationalWeatherService>{{ cite book |author=National Weather Service Office, Northern Indiana 2009 |url=http://www.crh.noaa.gov/iwx/program_areas/coop/8inch.php |title=8 Inch Non-Recording Standard Rain Gauge |accessdate=2009-01-02 }}</ref> The inner cylinder is filled by {{convert|25|mm|in|abbr=on}} of rain, with overflow flowing into the outer cylinder. Plastic gauges have markings on the inner cylinder down to {{convert|0.25|mm|in|abbr=on}} resolution, while metal gauges require use of a stick designed with the appropriate {{convert|0.25|mm|in|abbr=on}} markings. After the inner cylinder is filled, the amount inside it is discarded, then filled with the remaining rainfall in the outer cylinder until all the fluid in the outer cylinder is gone, adding to the overall total until the outer cylinder is empty.<ref name=Lehmann>{{ cite book |author=Chris Lehmann 2009 |url=http://nadp.sws.uiuc.edu/CAL/2000_reminders-4thQ.htm |title=10/00 |publisher=Central Analytical Laboratory |accessdate=2009-01-02 }}</ref> {{clear}} ==Global Precipitation Measurement== [[Image:Visualization of the GPM Core Observatory and Partner Satellites.jpg|thumb|right|250px|This image depicts the GPM Core Observatory satellite orbiting Earth, with several other satellites from the GPM Constellation in the background. Credit: NASA.{{tlx|free media}}]] "The Global Precipitation Measurement (GPM) mission is an international network of satellites [shown in the image at right] that provide the next-generation global observations of rain and snow. Building upon the success of the Tropical Rainfall Measuring Mission (TRMM), the GPM concept centers on the deployment of a “Core” satellite carrying an advanced radar / radiometer system to measure precipitation from space and serve as a reference standard to unify precipitation measurements from a constellation of research and operational satellites. Through improved measurements of precipitation globally, the GPM mission will help to advance our understanding of Earth's water and energy cycle, improve forecasting of extreme events that cause natural hazards and disasters, and extend current capabilities in using accurate and timely information of precipitation to directly benefit society. GPM, initiated by NASA and the Japan Aerospace Exploration Agency (JAXA) as a global successor to TRMM, comprises a consortium of international space agencies, including the Centre National d’Études Spatiales (CNES), the Indian Space Research Organization (ISRO), the National Oceanic and Atmospheric Administration (NOAA), the European Organization for the Exploitation of Meteorological Satellites (EUMETSAT), and others."<ref name=Hou>{{ cite book |author=Arthur Hou |title=Precipitation Measurement Missions |publisher=Goddard Space Flight Center |location=Greenbelt, Maryland USA |date=July 26, 2013 |url=http://pmm.nasa.gov/ |accessdate=2013-08-03 }}</ref> The launch occurred on February 28, 2014 at 3:37am JST on the first attempt.<ref>{{cite book|title=GPM Launch Information|date=22 January 2014|url=http://www.nasa.gov/mission_pages/GPM/launch/index.html|publisher=NASA|accessdate=2014-02-19}}</ref> {{clear}} ==See also== {{div col|colwidth=20em}} * [[Rocks/Rocky objects/Callisto|Callisto]] * [[Rocks/Rocky objects/Ceres|Ceres]] * [[Radiation astronomy/Comets|Comets]] * [[Rocks/Rocky objects/Dione|Dione]] * [[Rocks/Rocky objects/Earth|Earth]] * [[Rocks/Ice sheets/Enceladus|Enceladus]] * [[Rocks/Meteorites|Meteorites]] * [[Minerals/Mineralogy|Mineralogy]] * [[Liquids/Liquid objects/Rains|Rains]] {{Div col end}} ==References== {{reflist|2}} ==External links== <!-- footer templates --> {{Radiation astronomy resources}}{{Sisterlinks|Cryometeor radiation astronomy}} <!-- footer categories --> [[Category:Radiation astronomy/Lectures]] ge52az5rfjhvudz19p916t03zm0m4ha 2 240847 2690251 2689978 2024-12-04T12:55:28Z Jason M. C., Han 2696291 delete not relative picture 2690251 wikitext text/x-wiki '''No Orders, Beautiful flowers were blooming from Childhood! Let us pick up your most beautiful moments...''' [[File:Salzburg Cathedral (Inside) - Mozart Baptized place 1. - Mozart-Complexes World under Water (Modernist View).jpg|thumb|Piano Kids, on that day, modern colours, lights and technologies made Mozart's birthplace (Washing) -Salzburg Cathedral like a world (palace) under the biggest water - Blue Sea]] [[File:Beautiful landscapes surrounding Mozart's birthplace - Heavenly Spring flowing from Alps Moutain.jpg|thumb|Heavenly Ice-spring from Alps Mountain]] [[File:Mozart from Green Nature 1.jpg|thumb|Mozart comes from Green Mountain and Nature]] [[File:Beautiful Sunset View of Oceanic City from top building during COVID-19 period.jpeg|thumb|Beautiful Sunset View of Oceanic City during COVID-19 period]] [[File:Small items in Commons Life of 2019 Spring 2.jpg|thumb|Flaming Katy and piano-peers as Micro-landscape in classroom ]] [[File:An Oil Painting watched since born.jpeg|thumb|Oil Painting: Forest Cabin in front of Fuji(or Helan) Mountain 'hanging on the wall']] Oh, dear piano pupils, we have seen so many beautiful landscapes, across the times and spaces. Now, firstly, getting into East- your nature and self-meditation, let us try to take the Pentatonic Scales Exercises for relaxing you 'busy heart' and focusing your attention back to piano... *[[Portal:Pentatonic Impressionism (China Wu Sheng) in the view of Neo-classical Piano Techniques-training]] '''Afterwards, we can take further great life-tasks in piano...''' [[File:Piano Lecture of Dalian Library (Music) delivered by Jason (Jixun) on October 13th, 2024.jpg|thumb|Piano Lecture of Dalian Library (Music) Delivered by Jason (Jixun), Han and His Piano Pupils Team Associated by Family members and Library workers]] '''New Time and Fresh Air from 2024''' *[[Portal:Fresh Air of Piano after COVID-19 and beginning up a Recovering New Time of Openness from 2024]] ===='''Firstly, passing by a Skyline of Citylife Night-Skyline'''==== [[File:Flowing City (Lin Hai), Player and Teacher JMC,Han (Jason).ogg|thumb| Flowing City from 'Forest Sea', JMC,Han [[File:Dalian Urban Night Skyline from Xiao Ping Island Mountain-top No.2.jpg|thumb|Dalian Urban Night Skyline from Xiao Ping Island Mountain-top No.2]] ]] ===='''New Era is coming...:(New Teaching List)'''==== [[File:Skiing Statue made during my rollers-skating.jpg|thumb|Skiing Statue shows Spirit (like piano)]] [[File:The new situation of bubbles-fish statue.jpg|thumb|Ice & Snow Bubbles-fish are welcoming]] [[File:Mmexport1614470203508.jpg|thumb|Picture of piano lessons still under Covid-19 situation]] Etude in the Advanced level - Sir Chopin's famous Black-keys (Flatten G Major) Etude: [[File:Chopin Black-keys Flat G Etude (Exercise), player JMC, Han.ogg|thumb| Black-keys: An Etude your teacher has no chance in childhood, but still hurries up to be better]] [[File:Chopin Impromptus Op.29, JMC, Han.ogg|thumb|Chopin Impromptus Op.29, JMC, Han - Deep Fast Waltz-thinking in Quiet Gentleman's Self-Expression (Improvising)]] [[File:Sogdian whirl with large pipa.jpg|thumb| (Public Domain Work: See original page: https://commons.m.wikimedia.org/wiki/File:Sogdian_whirl_with_large_pipa.jpg#mw-jump-to-license): Could you find some musicality similarly with Chopin's Black-keys' Pentatonic?]] True Fairland - a piano melody of The Nutcracker (Dance of Sugar Plum Fairy): the advanture from childhood realizes your piano dream-"Fingering Ballet" [[File:The Nutcracker (Dance of Sugar Plum Fairy), Piano performer JMC, Han.ogg|thumb|The piano performance of <Dance of the Sugar Plum Fairy> in 'The Nutcracker' (Ballet Re-edited Piano Melody)]] Czerny 299 etudes were designed for the smart&strong fingering&modelling of hands and a pair of Vienna school's ears for the harmony...such as No.23... [[File:Czerny 299 No. 23 JMC,Han.ogg|thumb|Czerny 299 No. 23, player JMC,Han]] A peacefully praying Sinfonia of Sir Bach is freshly added in the "New exam book's list' during this COVID-19 period, to which you can have a relaxing hear and try (Don't worry, listening, it's enough time, you knew, 'Andatino'-Peacefully walking, and to sing by hands, in a small Baroque place : [[File:Sinfonia No.11 - 3Ps Invention, Bach, JMC,Han.ogg|thumb|Sinfonia No.11 Andatino- 3Ps Invention, Bach, Teacher JMC,Han (COVID-19 Protection Time)]] Encouragement in Italy Smart Fashion, but needs the very carefully fingering-techniques training (Long time, advanced), feet-edges' staccato, sentence-Pizzicato, flowing streams...Italian artist techniques always attract our eyeballs... Let us attempt to... make out your own Italy style! Cheer up! [[File:Domenico Scarlatti G Major Sonata, JMC,Han.ogg|thumb| Domenico Scarlatti G Major Sonata (In COVID-19 Pandemic Period), JMC,Han.ogg]] '''A good teaching video result played by Yixuan, Wang:''' http://m.kugou.com/mv/?hash=f00b36624f27b091b79e3f30e158aa03&sruserid=640650901 Baroque staccato techniques were always in a reasonable, confident, relaxing(wrists), fluent and vivid - 'Enough manners' of the Era, which needs us apply very careful fingering trainings. In a view of the whole structure, according to ears' musical suitable habits(psychological), I gave 1st section a twice repetitions, and then a throughout 2nd section to the Code. Hopefully, French Suite would make us brave, confident and relaxed. (but it also need years' accumulation of hand working to let those out and better) Have a try? Good luck! [[File:BWV.816 Gigue-French Suites No.5 Bach, Player JMC, Han.ogg|thumb|BWV.816 Gigue-French Suites No.5 Bach, Player JMC, Han]] (By turning to Wikimedia Commons, you will find two versions under its 'historical tree', currently. They are showing different stages we can reach. The first version was kept because it's relatively slower and more stable that in the basic stage we can make notes heard staccato and clear. After feeling suitable in this stage, we need to improve its tempo and get Gigue Dances' happy, vivid,,dialoguing with moods, jumping and wrists' breathing naturally. It needs time to train your hands frequently, untill relaxing but accurately. Main Difficulties: Stiffen wrists, Cramp and Tiredness... Now, it's the time of yourself...) Hi! We, piano kids: Imaging a scene, let us hands-dances with the good manners and a earnest mind in a beautiful Baroque palace. It's easy and natural... [[File:Primary Bach No. 16 March, Player JMC, Han.ogg|thumb|'Primary Bach No. 16 March' -A peaceful Bach-melody for all 'Piano Kids'...]] Sir Debussy's Arabesque Suites (The second suite) is also in the list. The musicality in my world is: It's the legend of Butterfly in birds' chorus... it took us to a Life-mountain behind our living garden facing a quiet sea... Watching, in some time of one section, you can also hear Monet's 'Quiet Morning' upon the sea... Alongside Butterfly's dance-suite, imagination is beginning. Oh, listening... (Main meaning referenced from my main page) What about it in your world and imagination? '''Sound teaching demonstration:''' [[File:Claude Debussy - 2nd Arabic Suite (Arabesque) - Spring Butterfly, Performer JMC, Han.wav|thumb|2nd Arabic Suite- Spring Butterfly (Impressionism Singing -Main Natural Lines&Breathes from Sir Debussy)]] '''Good video teaching result from a 12 years' old little girl piano-pupil Mo Zhou:''' http://m.kugou.com/mv/?hash=b50e133a360fa8d30cdcd9fca4163e73&sruserid=640650901 (Photographer: Ms. Yang, Gao) Listening! boys and girls, Dvorak's Humor-jumping and Homesick-expressing: A true Czech-homeland heart, but Dancing... somewhere in American Countryside [[File:Flatten G Humoresque Dvorak, Player JMC,Han.ogg|thumb|Flatten G Humoresque, A. Dvorak, Player JMC,Han]] [[File:One Town-view from Cesky Krumlov Castle.jpeg|thumb| Krumlov Castle-town's view]] ====Xinran, Yu - a lovely Chinese little piano girl's 'Ink-Mountain & Green Rivers' view of <The Cowherd's Flute> ==== '''Comment:''' "Before taking the national examination and the exhibition competition, we together listened and learned to the net-editions of young master Lang Lang and Yujia Wang...(regarding with this famous little melody of Chinese tradition) I think in this melody, she tried out her best for the techniques-training and the musicality in her age... from a little performer's view. Therefore,I gave the comment-Excellent. Close your eyes, thinking of a little lovely girl happily playing among ink-mountains and the green rivers, with a water buffalo, some birds followed, and her smart flute... let us relax in the Chinese Ink-Landscape and listen to this little melody...(referenced partly from the writing in Wikimedia Commons page) [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 02:28, 27 May 2021 (UTC) ==== Zichen Tao - A little Chinese snowgirl's hardworking of D Major Sonatina==== '''Comment:''' D Major Sonatina is still piano Children's favorites to perform and show... taken to the national grade examination, daily performed with each other, and also to city's piano Competition &Exhibition. That's an educational case lasting for many years in Dalian. This edition is played by a lovely and white Chinese girl - Zichen, Tao. She and her mother took the very responsibility to check the wrong notes, improve the learning progress, and make the performing manners and designs for the stage-show... Therefore, in my view as her piano teacher, this edition is already great in her age...(though hand-running details need to improve for her age). Hopefully, her family can enjoy this piano experience, companying with this melody in her childhood. (Partly referenced from her Wikimedia Commons' page) [[File:D major Sonatina , Piano student Zichen Tao.ogg|thumb| D major Sonatina (Kuhlau's) played by piano pupil Zichen Tao]] ==== Meng's Performance and Comments after learning in the reality from Jason M. C.,Han in Children's Corner: ==== [[File:Children's corner of Meng.ogg|thumb| Meng Meng (nick name)'s edition of Doctorial in Children's Corner: Currently, the Fourth Version was her most beautiful one self-made in classroom before Piano Grade Test Exanimation. Regarding with all editions' comments and reasons, please reference to the original file in Wikimedia Commons]] [[File:MM Good classroom F major 1838 Grande valse brillante.ogg|thumb|MM Good classroom F major 1838 Grande valse brillante]] '''[[Portal: Part of Comments - 'for students' Examination Performance, Piano tutor's teaching self-reflexivity and possible some requirements of Pedalling Sound-effects with Artist Fashion of Post-impressionism' | Part of Comments - 'for students' Examination Performance, Piano tutor's teaching self-reflexivity and possible some requirements of Pedalling Sound-effects with Artist Fashion of Post-impressionism']]''' [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 04:15, 12 October 2018 (UTC) ==== Kim Hui's 'Doctoral Training','Under the sunshine' in Children's Corner: ==== '''Comments for Kim Hui's first draft:''' 1. She did the second theme (associated) well 'much deeper like vocally singing out a better life in New Era under the sunshine, on a piece of small area in a rainforest'... 2. Her Korean dance (Wikipedia introduction: https://en.wikipedia.org/wiki/Korean_dance) has been done well, in which I can hear traditional drum-points in bass-part and crossing hand to tremble part. 3. I can hear Time-travelling and space-shaking to the past through a 'Dark-cave', from..., minutes 1.30-1.40... But, I think: if 'dramatically' and 'significantly' in dynamics (loudness), it would be better to show... 4. I can hear Forest's Evensong in Coda part - 'dim.' to the silence of night and a 'rit.' slowing down to the sleeping dream, and even several night-birds' dreaming voices...But, please make a much gentler taste (not so hurry up and not so strong) of those pictures. Meanwhile, I hope you can get a better & coherent control of the rhythm among different sections. 5. I knew, regarding with 'peak-parts', she had made many attempts 'drumming beats rights and keeping those connections clear'. However, still, in minutes 1.04 and 1.59, I felt it's a little bit 'rough', and needed to be handled in of the solidification... Oh, maybe, I am so severe... sorry, I should give you the encouragement. Main comment: 'Under the sunshine' is suitable to Kim Hui's fashion and can be kept in her performance list. Her first draft and its preparation has given me an enjoyable teaching experience and many beautiful memories of life. It's fluent and vivid, expressive and dedicated. Thanks, Kim Hui! More colours and lights would be added from technique details, from her independent fingering and some traditional piano manners, meanwhile, the rhythm should be balanced well in the future. There are many developing zones of 'this painting' she can better and draw out for her future.[[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 21:22, 17 October 2018 (UTC) ===== Kim Hui's second 'Show-Time' in classroom - Poem of Music (Piano Etude) ===== '''Teacher's Demonstration:''' [[File:Poem of music.ogg|thumb|Poem of music JMC, Han]] '''Student's Second Performance - Poem of Music: ''' [[File:Poem of Music (Piano Etude) - Student Kim Hui.ogg|thumb|Poem of Music (Piano Etude) - Student Kim Hui]] '''Comments for Kim Hui's first draft:''' 1. She has mainly got the technique-points, but a little bit of rough in some details, such as the minutes 0.30-0.31 - 'Tail-closing part' of a sentence - in the progression of 'Diminished Seventh Chord-Arpeggio'... However, as her first draft and the random collection from a normal classroom, I thought it's well-done. we can wish its further 'Developing Zone', in the view of piano education. 2. In 'Coda Part' of Poem of Music (Piano Etude), she was able to show a great controllability of the 'Legato' between two hands, as the pieces of falling leaves slowly flying-upon the surface of water. Sometimes, it was evenly better than mine. I hoped she could manage it in a better way. 3. She showed some thoughts of musicality... However, 'Techniques-points' still wasted much of her energy. I think the total Dynamics in physics will be improved soon. (waiting more) [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 14:53, 23 May 2019 (UTC) ===== Kim Hui can reach the advanced level of piano performance in this 'Little Children's Corner' ===== Though It's still Covid-19 Health-protection Time, after Home-self-Training Time, some classroom face-mask covering & health-good-protection lessons and an examination of China Culture&Tourism Ministry, She can play this 'Doctoral Etude' which is dreamed many piano pupils, with impressionism style. As her teacher, I didn't think It's a simple Etude which was expressed in many scenes, but with the big universe imagination and impression. Therefore, we have trained it as Meng's approach and further developed it. Indeed, I think she performed far greater than this edition, right in that online national examination. She got it, Congratulation! Let us listen to her ... [[File:Debussy-Children's Corner-Doctorial Etude, Piano pupil Jinghui Jin (Kim).ogg|thumb|Debussy-Children's Corner-Doctorial Etude, Piano pupil Jinghui Jin (Kim)]] ==== 'Colourful Clouds Running After Moon' impressed into the Heart of Xinyi, Hua (Hua family's Heart-sweet girl from 'Painting Imagery')==== '''Comments for the first draft of Xinyi, Hua:''' 1. I like her treatment of the prelude part in 'Colourful Clouds Running After Moon'. It's light and soft like silky clouds up-bridging alongside moonlight towards a round moon above the dark-blue sea. However, please try to link each silky pentatonic-arpeggio weave as a smooth whole from the bottom to the top, and from the left hand to the right hand. If so, her progress will be enlarged; 2. I can hear the situation 'Colourful Clouds Running After Moon' appeared in many linking parts before and later. She was attempting to give an acceleration imitating this procedure from a slower speed to a fast one, and between two hands' echo-following from a loose density to a tight one... However, if obviously, it will be better; 3. Like 0.58 to 1.03 minutes, I can hear that in some parts, she would like to make a returning sound-boomerang (Wikipedia introduction: https://en.wikipedia.org/wiki/Boomerang) up-rising 'to the moon' and down-landing upon the sea-surface. If a small time of middle reaction was canceled out by her proficiency, we will appreciate the musical beauty in a much more advanced situation; 4. In 1.22 to 1.30 minutes, I know she would like to make a silky veil, with the colourful clouds as material, upon moon's beautiful face by her right hand. It's a little bit of pity that the controlling ability of relative loudness made she carried this willing but harder to realize. Meanwhile, this veil needed to be smooth. Oh, sorry, I am so critical... indeed, she did not bad; 5. There is a hard hurt in 1.36 minute - it's still a repetition of bass-chord though she has already attempted her best to grasp the bass large chord through left hand's opening degrees (Little girl, I knew you had tried your best. Though the momentum was great, I still need to point it out.); 6. 2.12 - 2.40 minutes is the part - 'Bright Moon up-rising above the harvest sea'. This is a grand scene which needs great forces from students' forearms and a fast reactions for some flexible connections to arpeggio-parts... Congratulations, little girl, she have got it, though it's a little bit slower. She have given out a great momentum; 7. In the Moon-tail part, she has expressed her great musicality to make moon disappear in the dawn of sea; 8. Many ornamentations she has done well, though still some need to be gently breezing in the impressionism of Chinese landscape painting. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 12:22, 24 October 2018 (UTC) ===== 'Pure White Dove' in Young Teenager Xinyi's Eyes ===== [[File:Dove in the eyes of Young Teenager girl Xinyi, Hua.ogg|thumb|'Pure White Dove' ('La Paloma' - 'No More' in English) in the eyes of Young Teenager girl Xinyi, Hua]] '''Congratulation to your beautifully singing of the 'Melodic lines' behind the right hand's octaves-grasping!''' '''Comments:''' 1. Four biggest designs appeared: around B50 (Minutes 1.36), B55 (Minutes 1.48), B58 (Minutes 1.54) and B62 (Minutes 2.01) - four Peak-currents, we'd like to throw (rit.) the 'missing notes' into the air and rotate them a little bit more slowly - like to send, wait and feel Dove's messages across the ocean in a self-holding & self-releasing intoxication. She tried her best to make them out, but not quite clearly and still need much time to grow up... 2. I liked her coda part (from B65 Minutes 2.08 to the end): She was so sure about two hands' March-doubling, as a confirmation of future and belief; or to say, she transformed her 'missing' in the melody to be a true hope of tomorrow, or someday... Evenly, I thought it's better than mine... 3. For more than half years, we have worked hard to help her link all octave-grasping pearls out of melodic lines in singing breaths. She almost got it successfully, through small breaks... 4. Some 'Spanish Dotted notes' and 'Triples-wandering', with the rhythm of Spanish Dance Habanera-Andante (Wikipedia introduction: https://en.wikipedia.org/wiki/Contradanza) can be fulfilled, but some not really... I am happy she recognized them and paid more attentions to... It's waiting time that she could perform much better. 5. Yes, I had to say: still some small faults there... The good usage of pedalling almost hide some, but... also a little bit rough... Oh, I didn't want to be a so severe teacher. Rather than, much more good wishes of her growth should be given. Okay, hopefully, she enjoyed it. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 12:57, 4 April 2019 (UTC) '''Teacher's Demonstration for standards above and Further Progress (Advanced Level):''' [[File:Dove With Spanish Sense in Piano JMC.Han.ogg|thumb|Dove With Spanish Sense in Piano JMC.Han (It was once used by the photography gallery of music friend User:PigeonIP - https://commons.wikimedia.org/wiki/User:PigeonIP/Tauben/2019_April_1-10 and the main page La Paloma in wikipedia )]] '''More information and reading - articles (list) about 'Dove', please watch:''' https://en.wikipedia.org/wiki/La_Paloma ==== Malagueña Dream from a little Chinese girl - Yinuo's heart (A promise ... to piano) in a small beach-side classroom ==== [[File:Student edition - Malagueña Suite (modified for piano) played by Yinuo, Liu.ogg|thumb|Student edition - Malagueña Suite (modified for piano) played by Yinuo, Liu]] '''Comments for the first draft of Yinuo, Liu:''' 1. We can hear the impression of Flamenco rhythmic pattern (Compás) (Wikipedia introduction: https://en.wikipedia.org/wiki/Flamenco) diffusing from some simple rhythm-components of a parts in a total ABA structure of Malagueña Suite. This is what I - the tutor and the little girl -learner would like to express through three more different accompaniment bass-forms, including pizzicatos, small slurring breathes and some opposite weights..., which imitated some of Classical Guitar's handling ways. Thanks to the little girl Yinuo, you have realized most of our designs! Congratulations! 2. I really like her beautiful Cante jondo - associated 'vocal' lines by right hand which was balanced & flying above the flamenco accompaniment of the left hand when the second thematic melody began. It's a deep, profound and emotions-rich singing, almost from a beautiful Spanish girl's natural expression for the missing, the reasoning of life & Universe when facing a 'deep and far' sea. Though if the dynamics would be dramatically and the singing would be much deeper, the emotional atmosphere would be better: I thought to only a girl of her 11 years' old age, she has already attempted her best to understand those across cultures; 3. I like the middle B's fantastical view of holiday beach under the sunshine, which was almost formed by white waves from blue sea. It's relaxable, dreamful and graceful, like a girl poet's walk alongside a small sand bay... (Yinuo, you knew, if you can make the 'rit. - A Tempo' much more nature like the real tides of sea and the speed tiny faster, the progress zone will be enlarged); 4. I know in two middle long 'vocal' ornaments, she would like to show us ' the blackbirds or the nightingales of its gardens...' However, if making it much more smoothly, expressively, and flexibly, even a little bit down-slowed, her Spanish 'tasteful' fashion will be more beautiful; 5. Repeated A part is better to be different in small details which can show the ability of hands and the variation of music. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 13:35, 24 October 2018 (UTC) ==== Für Elise in piano boy Zhe's eyes ==== '''Teacher's demonstration in classroom for better:'''[[File:For Elise (Für Elise) Beethoven JMC Han.ogg|thumb|For Elise (Für Elise) Beethoven JMC Han]] '''Student's performance in classroom:''' [[File:Für Elise -Student Performance Zhe,Zhang.ogg|thumb|Für Elise in piano boy Zhe's eyes]] ==== D major Sonatina (Kuhlau) - Piano-pupil girl Mengshuang's 'Strong Willpower and Persistence' ==== '''Teacher's demonstration in classroom for better:''' [[File:D major sonatina 2nd movement Kuhlau (played by Jason).ogg|thumb|D major sonatina 2nd movement Kuhlau - Teacher JMC. Han]] '''Student's performance in classroom:''' [[File:D major Sonatina (Kuhlau) - the version from piano-student Meng Shuang, Wang.ogg|thumb|D major Sonatina (Kuhlau) - the version from piano-student Meng Shuang, Wang. This classroom version has been selected by https://commons.wikimedia.org/wiki/User:Rsteen/Artists_from_Denmark/2019_August_1-10]] '''Comments and Statements:''' 1. Totally to say, the main melody fast-run by the right hand has kept its fluency, transparency and clearance. It's very hard in piano training for herself, owe to that her hands-shape was a little bit of 'frozen'. Thanks for your hard-working in the training. Congratulation! 2. Her musicality in this melody has also been motivated out - unrestrained and natural in the expression. 3. Left hand's accompaniment was in good triplet-treatment, but please light and dedicate a little bit... It's to say: the controllability still needs to improve. 4. Some heads of sentences and smaller phrases need to be match together between two hands in a better way - some parts, because of small ornamentations and dotted notes, weren't quite well... 5. I am very happy that you (in your 12th year of life) were willing to play out the middle 'rit. - A Tempo' in a comparison ('rit.' was slowing down the waiting, then, 'A Tempo' for the Peak expression in return). However, it was still a little bit rough (before its right time). You can try to modify it in a better view. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 03:08, 2 August 2019 (UTC) ==== How to play Mozart's Classics? ==== ===== Work from Mozart's earlier time - Turkish March ===== '''Teacher's Demonstration in classroom:''' '''Video:''' [[File:Turkey March Video-Mozart-Jason Han.webm|thumb|Turkish March (Video) for Mozart's; JMC, Han]] '''Sound:''' [[File:Turkey March for Mozart and Memory JMC, Han.ogg|thumb|Turkish March (Sound) for Mozart's and Memory (many times used in peer Tokfo/Vienna Gallery - such as https://commons.wikimedia.org/wiki/User:Tokfo/Vienna/2019_January_26-28 ); JMC, Han]] ==== Sure, Ma's boy-view of Mozart's 'Turkish March' ==== [[File:Sure, Ma's Version of 'Turkish March' in piano classroom for piano education.ogg|thumb|'Turkish March' - an old mysterious Turkey story in piano student (Jason's piano pupil) boy Sure, Ma's classroom edition. Thanks, this good teaching & Learning result was selected by Tokfo/Vienna Gallery: https://commons.wikimedia.org/wiki/User:Tokfo/Vienna/2019_April_25-27 ]] '''Further Comments for his first draft:''' 1. It's very difficult for a young boy to manage Turkish March's speed in a smooth way... He tried his best to keep it stable and unified, and almost did achieve it. (Turkish March is easily to make people play faster and faster until crazily broken. He tried to solve it by giving a slower beginning ) But, it's a little bit of too serious,afraid to touch wrong. Indeed, I heard his another time, in which he totally open himself and relax from nature... We could give him more hopes. 2. His melodic flow of scale-phrases (legato) are quite fluent and natural, which shows his scale-playing and fingering were quite great. But, a small break occurred around 1.14 to 1.17 minutes could be caused by the stiff right wrist (too tried) and no-good fingering design. He should frequently move second and third fingers in a much smarter way. To a young boy in his age, it should already be 'okay'. 3. When the theme occurred in the second time, it's better to give a dynamics-difference in contrast. My mother-Ms Song said: it's like an old story (sound) heard from a far distance to near somewhere - mysteriously. However, he gave a very tight connection, as if it was linked with the previous section. 4. He tried his best to take the Worldwide difficult challenge - 'Broken-chordal Arpeggiated-octaves' (Around 2.00 to 2.14 minutes). I gave him a 'LIKE' that he had taken this challenge which even many pianists or teachers made some 'faults' as their heart pities - You can hear the edition of Romuald Greiss' in Wikipedia and several my previous times... However, this boy achieved it after many trainings time after time... Though later half one, compared with the beginning, might be in lower distinguishing degree, he didn't make any 'breaks', which comforted my teaching way so much. Thanks, boy Sure! 5. The final problem would occur in 'Alberti Bass' (left hand) of Coda part. Coming to it, you will feel easy to give up, which required more endeavors to control your hands in narrow and elaborate dealing way. He did it good, but lost in the counting of number (B111), and further, the connection with the final 'Square-opening Dance' (a small break). other things, such as the strength, are fine in his age. 6. In addition, I am planning to add a 'Turkish Stop' by a final pedaling. I didn't know whether he could, someday. Overall, I gave him an Excellent Comment. Hopefully, he will play better after better in his growth. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 14:13, 8 March 2019 (UTC) '''Further Comments for his second draft:''' 1. As his teacher, I am very happy to hear several great designs we have made in classrooms can be achieved in the second draft, according to the background knowledge of 'Turkey March' I taught, such as the final 'Turkey Stop' (not really in modern piano, but a little bit similar) and the Bass-points-layer (simulating the military drum) beneath the long fast running scale-phrases of right hand (middle section)... Cool boy, thanks that you can remember your teacher's words! Congratulation! 2. Yes, right after the chance of Music library Report-performance in local we have made and getting back, you improved the edition's speed and fluency. You can evenly save 15 seconds, contrasted with before, which showed that your fingering & running ability of hands had been greatly improved. However, the disadvantage is that it's easier to make some small motives uneven and rough (touching wrong notes) without purpose, which needs more your careful attention and exactness about details. 3. I knew you tried your best to face the peak challenge Mozart made to all people - making broken-progression of octaves message (middle part) and hearing out its hidden melodic lines. Great! However, it's still a little bit beyond your ability that its distinction with chords-effect weren't so clear. No matter, Sure boy, more exercises, it will be better. In all, progressing soon which shows the potential, thanks to your performances1 There is still the developing zone waiting for you. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 15:08, 25 April 2019 (UTC) ==== Clever Girl Jia Xin's Clever view of Sir Bach's 3-parts Invention ==== '''Teacher's Demonstration for Future Development:''' [[File:3 Parts Invention 8th- from Sir Bach.ogg|thumb|3 Parts Invention 8th- from Sir Bach's BWV 794 – Sinfonia No. 8 in F major ]] ''' Jia Xin's Performance:''' [[File:3-parts Invention No. 8 (F major) in piano-girl Jia Xin's view.ogg|thumb|3-parts Invention No. 8 (F major) in piano-girl Jia Xin's view]] '''Comments of her first draft:''' 1. Totally, 3 parts are ranged in Bach's harmony, to a girl in her age - 12 years old. It's not easier to make so clear layers out. I was satisfied with this point, after heard every time; 2. I can hear piano techniques for polyphonic & counterpoint music like Bach's, such as cannon, intimation between two hands, up-climbing shoulder by shoulder, dialogues, long-notes down-pressed for different parts' SHE (sentence-head-enhancement), long-notes kept for parts' division and maintaining, fingering grouping in one hand for 2 parts, parts' continuously melting into one for the summarization, and..., Baroque ornamentation... mentioned for long. For those trainings, and further, the internalization into her own mind-analysis, we had searched information & knowledge through Wikipedia, two more manuscripts, books and other webs online, further, spent classroom time to reason, analyse, train and fix note by note for long time... In this case, I gave her hard-working a 'Like' again; 3. There would also be some problems regarding with recording pressure and her memory...: some big ones - left hand's relatively weak ability in managing two parts, small mistakes (like B18's f note played as sharp f - around 1.13 minute, B21's final g isn't raised there - around 1.23-1.24 minutes, and others...), small ornaments in a little bit of rough view and a much more graceful manner in Coda part. (sorry, to such a 12 years old girl, my suggestions could be so severe. But when listening, they are directly in my ears...) 4. She did really pay her attention to Dynamics, but please better - lighter, smarter and more obviously... 5. The speed of later part is better than the slower earlier part. Overall, I also gave her an excellent comment for her performance (Live) in classroom. For further development, she can listen to my edition and the one in 'Inventions and Sinfonias (Bach)' - Wikipedia article (I thought it's great, but I didn't like too many speed-variations in Bach's works. It's better more reasonable: https://en.wikipedia.org/wiki/Inventions_and_Sinfonias_(Bach) ) [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 07:33, 9 March 2019 (UTC) ==== Yixuan's piano-view and traveling landscapes ==== '''First view (only 9 years old) of Sir Bach's 2-parts Inventions''' [[File:Bach 2-parts Invention, played by Yixuan Wang (Only 9 years old).ogg|thumb|Bach 2-parts Invention, played by Yixuan Wang (Only 9 years old)]] '''Comments from teacher:''' 1. Musical parts and space-dimensions were much clearer than before, which reflected the little girl Yixuan's hard-working continuously after her piano examination... 2. I can hear the heads of musical sentences which were highlighted by each hand when required. I can also hear cannon-following, doubling and countermelody which were clearly shown into her performance. It's quite necessary for students in this age - 9 years old. In certain degree, she is already a good and careful student in piano. 3. Still, the controllability and the stability of hands, especially the turn of her 3rd, 4th and 5th fingers, need to be improved, which caused some small faults, such as 0.49-0.50 minutes (Bar 22) - a recovered B in right 3rd finger, 0.54-0.55 minutes (Bar 25) - 3rd, 4th & 5th fingers of left hand, and a small disharmonic note - flatten B in the right hand - 0.32 (Bar 14)... 4. It's great that I can hear Baroque staccatos were in their graceful manners - like imperfect pearls required by its era. She almost did it... 5. Totally saying: it must be a very hard-job for a student in 9 years old to play Bach's 2-parts Invention. She bravely took up this life-task and successfully completed - this point should be affirmed. Congratulation! You can do more further... [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 02:28, 6 September 2019 (UTC) '''MARIAGE D'AMOUR (Dreaming Wedding Ceremony) and its Educational Story:''' It's only no more than 2 weeks she did get the main techniques, after taking a Covid-19 Protection piano lesson and further test(Face-mask covering and breathe-prevention…).Then, she went back home and made a hard self-working exercise. Afterwards, around 10 days, this edition can come out. Why was she so keen on making it? She told me... One of her family's friends would like a piece of background music for his wedding ceremony, and they knew she was a good piano pupil.They invited her to take this task. She online self found out her long dreaming piece, and felt very happy for them. She thought only hard-working at home can realize this dream in this 'Hard Recovery Time'. She has beautifully taken this life-task for a very short time, and finally I could find a beautiful smile on her face... Though there were still some small faults in teacher's view, such as the biggest 1.02-1.03... ('''Problems:''' Mentioned one is because of the distance of the Tenth-grasping is out of her hand-shape and ability in this age - rolling but touching a wrong note; In addition, the breathing of each sentence's tail somehow is with a longer responded break... Further, the Pedaling for the coherence from natural breathing need to improve; The final departing dropping notes were too noisy... which needs to be quiet,rit. and peaceful...), her hands' ability (especially the big chords-grasping, whole-viewing, locating, and sight-reading) was improved by her own endeavors (Maybe... subjects-divided examination-taking online through self-video-recording,in this special time, motivated her self-management...). This point made me feel happy... Hopefully, the friend of her family enjoyed their wedding ceremony with this own and LIVE background music, luckily as in a fantastic, peaceful and forever-lasting life-dream of happiness. In future, Hoping: Yixuan, you can play this fantastic wedding song of piano (fluently and heart-touchingly) for more families and share their friendship, love and happiness... Little girl pupil, thank you!^_^ [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 01:55, 24 September 2020 (UTC) [[File:CCzerny 299 Etude No. 27, Piano student Yixuan Wang, Tutor Jason Han.oggzerny 299 Etude No. 27, Piano student Yixuan Wang, Tutor Jason Han.ogg|thumb| Czerny 299 Etude No. 27, Piano student Yixuan Wang, Tutor Jason Han.ogg]] I have taught two children this Etude-a girl and also a boy (with outcomes). They played all well in very different musicalities. One is like a fast gym meeting 299's standards. The other-hers is with a good sound effect -light and peaceful after her grade examination. Both I all like. Regarding with how to train this sound effect with pedal, Please see my etudes'platform: https://en.m.wikiversity.org/wiki/Portal:Piano_Etudes_as_Poems '''G major Sonata L.349 - Yixuan Wang's New Attempt of Italian Baroque Style of A. Scarlatti''' A. Scarlatti's Sonatas are quite hard for young students and young teenagers to train and perform.However, Yixuan is fine, I thought. It needs a very fast & light fingering of Scale & Arpeggios and different STACCATOing keyboard-touching way, meanwhile, the exaggerating fluency of simple patterns... I thought she somehow had touched at her own little age. Just, more from nature, more details-care and the flexibility of hands&body could make things better. At her age, it's already fine.Thanks to the recent striving in this still hard time of COVID-19 Recovery. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 02:31, 20 August 2021 (UTC) [[File:G major Sonata L.349, Piano Pupil Yixuan, Wang.ogg|thumb|A, Scarlatti L.349 Sonata, Piano-pupil Yixuan, Wang]] It's very hard to train Scarlatti's Italian Style of technique skills... However, Yixuan, Wang has never given up... during this COVID-19 Time...-Protecting herself with masks, meanwhile, playing times after times... Finally, we can get some senses of fast, airily,lightly and breezily... Yes, there may also be some problems, like- it's very easy to be stressful and breathless on the stage...But to her age, do you think it's already good... Therefore, I recorded it in a video and published it on a musical platform -Kugou, and a educational platform - Youku, as to remember her growth: http://m.kugou.com/mv/?hash=f00b36624f27b091b79e3f30e158aa03&sruserid=640650901 ====Piano Pupil Mo Zhou's Smart Growth and Hard-working learning of Techniques ==== The video of Mo Zhou's most beautiful performance of Debussy's work- 2re Arabesque (I call it 'Butterfly's Dream') : Kugou musical platform - http://m.kugou.com/mv/?hash=547cb2c1e2f57a9e8ec66e8ecf36c269&sruserid=640650901 Youku educational part - http://v.youku.com/v_show/id_XNTgxNzA2Njk2MA==.html?x&sharefrom=android&sharekey=9631de9a76de1af3d601221019590cd26 (Published on the musical platform of Kugou and the educational part of Youku; the classroom volunteering photographer is Ms. Yang Gao) '''Piano Pupil Mo, Zhou's Violin simulation of Cremer's Etude's Art''' Catching the hands' positions (somehow borrowed from voilin's) is almost the hardest point to train.One focuses on left hand's Notes-Slipping; the other regards with the interval Position-switching (2 notes) check of right hand frequently. Though this little girl has a pair of smart&slim hands, she attempted her best. You can hear the most part's effect LIVE in classroom... In this point, I gave a 'LIKE'. [[File:Cramer Etude, Performed by Piano Pupil Mo, Zou.ogg|thumb|Cramer Etude, Performed by Piano Pupil Mo, Zhou]] '''A, Scarlatti L.349 Sonata - A Italian Style Taste of Baroque Music''' ''Comments:'' Mo,Zhou's hands are very smart, regarding with which some very tiny actions she can take, though they aren't quite big. Yes, she has been always willing to enlarge her hands. This point, but somehow, associate her to take this Italian Baroque Style (Rocca) quite easy. Yes, I thought she was fine regarding with much more details.(though it's LIVE that very few unexpected faults could be caused by the stress of the recording). I thought: to her performance, my teaching is working well. She did many requirements... Let us listen to her.[[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 03:25, 20 August 2021 (UTC) [[File:G major Sonata L.349, Piano pupil Mo, Zhou.ogg|thumb|File:G major Sonata L.349, Piano pupil Mo, Zhou.ogg]] ==== Brilliant Snow-ball boy (Yu) of Zhang family is praying for his father working in New Zealand ==== [[File:Pupil Yu,Zhang's edition - e minor sonata of Sir Haydn.ogg|thumb|E minor sonata of Sir Haydn was played by Piano student Yu, Zhang in classroom]] '''(Waiting better)''' Could you understand how hard Sir Haydn's & Mozart's mature sonata-structuralism and Classical Countermelody (from String Quartet: https://en.wikipedia.org/wiki/Wolfgang_Amadeus_Mozart) were for the training of such a young boy or some students around this aget? Oh, looking back, I, myself, also did feel hard... However, this boy and another older piano sister did really insist on doing so. Today, they can give their own editions - very different with own personalities and natures. Another point I would like to say: It's only one week's time that this boy was fighting for 'a good hearing' of his father. Afterwards, a modified recording edition soon got out, which showed his proficiency and quality...Good boy! To be honest, reviewing the past year, in order to train Sir Haydn's melody, we researched many ways together, including mathematics... Sometimes, evenly felt hopeless... Playing from childhood, Haydn's style is quite simple to me - models, switching, sonata structure..., but to students, they didn't quite like the sense of thinking being structured... And at the very beginning, I even didn't understand why they felt difficult... Recognizing something, We began to make many games, and evenly counting out some scores for the achievement of his 'fried chicken legs'... Here, from rhythm to notation, and from melodic interaction to parts-division, I felt it's much clearer, more fluent and stable, than before... His ability of coordination has also been improved, though still some problems. I dared and felt confidential to say: it's a great edition of himself. Hopefully, he can progress further. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 16:08, 4 April 2019 (UTC) Now, he made it much more fluent and accurate, and also played out his own fashion, though some details still need to be modified. Honestly to say, I thought somehow he got his progress in this period which we can hear...[[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 03:28, 2 August 2019 (UTC) '''Comments''' It's the second time of piano boy Yu Zhang's teaching result show. In this time, he chose the melody <Under the Sunshine> - a Chinese fork teaching melody as one subject of national examination and also a performance of one piano competition in Dalian. In my view as his teacher, he gave a very different view of this melody, compared with girls'. He paid more attentions to the whole view of melody's energy, strength, fluency and the joyfulness in Under the Sunshine, but didn't too much care about the details of some parts. However, on the stage, it showed a very great expression as from boy's situation... Good luck and happy experiment. After some practices, in the classroom we together recorded it and submit it up... (Referenced Partly from his page in Wikimedia Commons) ==== How to play Chinese Folk tune - 'Kids' Dance' with Chinese kids' fashion? Listen to little girl Kunlu's performance ==== '''Teacher's Demonstration in classroom:''' [[File:Kid's Dance Chinese Folk Piano Player Jason M. C.,Han.ogg|thumb|Kid's Dance - 'Kid's Dance', from a folk piano-tune in China National Grading Book, was personally performed here, as a gift for all piano-kids' 'Happy 2019 Lucky Pig Year']] '''Student Kunlu's Performance in classroom:''' [[File:Kid's Dance (Chinese) - Student Kunlu, Han.ogg|thumb|Student Kunlu, Han's (Han family's girl born as bright as dewdrop in Kun - Saturn of Wuxing) good performance of Kid's Dance (Chinese)]] '''Teacher's Comments:''' Totally to say: Though She can play better editions (many better ones, last winter), in this sound file, she showed the coherence, fluency, flexibility and stability ( as Chinese fork-tune required). Hearing such a smart Chinese girl playing such a fugue-cannoning song, you will feel: it's a right song designing for a right girl... I think that's one meaning of piano-performance. Though spending much time, We did also research special 'Chinese supplemental positions & Dialogues' in polyphony together, which gave us many beautiful memories... Further more, in this age, her staccatos, slurs and Tenuto have been performed quite well, which helped her to keep a unified speed to the end. Taking back a step, there must still be some small faults in classroom (without purposes) that I have to point out: such as B18's #C blowing to D a little bit, the attention didn't get back in B41 head A which made a small break, and a small mistake of 'Recovered C' rather than #C... In order to dream of its accuracy and pentatonic harmony, it's a hard-working that we have already come over many problems and mistakes... Therefore, I think she fulfilled herself and achieved many things from 'Kid's Dance'. Hopefully, she enjoys the procedure of music-carving. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 15:07, 4 April 2019 (UTC) ===== Kunlu's Crystal Heart on International Children's Day - Kleine Kinder Kleine Sorgen (Little Child) ===== '''Teacher's Comments:''' 1. The degrees of proficiency, fluency (and internal speed) have been improved, right on International Children's Day. 2. I preferred her treble part very much - so cool, pure, clean and refreshing, which reflected her crystal heart in childhood. 3. The grasping of big chords - stronger, that's great - but needs to be more accurately and deep (The word 'deep' wasn't always 'loud' and 'heavy'). Please try to understand this point. Yes, it needs to show the hardness of growth (to young teenager), but also the achievement 'to be stronger and more confident of yourself...' [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 03:32, 28 June 2019 (UTC) [[File:Internationl Children&#039;s Day Gifts - 2. Kleine Kinder Kleine Sorgen (Little Child, Piano-modification of Germany Song).ogg|thumb| Kleine Kinder Kleine Sorgen (Little Child, Piano-modification of Germany Song), played by Kunlu, Han]] ==== Teacher's Shares of his own home-works from childhood (Open) - Jason, Han==== ====='''M. Moszkowsky Etude (Op.72 No.5) - "C major's Fluency, Clarification, Sunshine and Love'''===== New Beginning with...: M. Moszkowsky Etude (Op.72 No.5) played by Jixun Han (Jason) for piano teaching. It's long time that my piano classroom on the cloud in wikiversity hasn't update its situation. After so many things, now I can partly return to English writing world. The first Etude I would like to upload is still MM Edute which gave me so fluent and clean mind in my childhood. Oh, 38 years old, and after a wedding ceremony with my real lover, my fingers would not be so great as around 15s'... However, I would like to update its situation and new editions untill great someday. Now, let's began with this new melody. It's taught to my good Chinese boy pupil named Guoguo (fruit zeyu, Cui) when I grasped up and recorded. Yes, this little boy will also play well. Let's listen to my version, firstly. Thanks Jason M. C., Han (talk) 13:26, 20 November 2024 (UTC) More information, please see https://en.wikipedia.org/wiki/List_of_compositions_by_Moritz_Moszkowski Homework Requirements (challenges): 1. B23-B24(B stands for Musical Bar): By right hand, heads of every 4-notes group make a down-going semi-notes scale, which needs a very careful&exact arpegio-fingering with a whole—palm holded and also thumb-measuring ability. Meanwhile, the left hand is making a whole-tenth measure, but arranged upon every two chords' link. The semi-notes scale is also its fixed channel accordingly. This point is very different to follow and be made accurately and perfectly, which needs long-time training. 2. B49-50 It's almost a two-hands doubling for playing arpeggio-phrase.But not really! You can watch the second phrase- fingering! Your left hand need a smallish shape. Meanwhile, the little finger's head of last phrase need to jump out a minor third distance down. It's very hard to control and also not a doubling. (Hard for playing, but good for sounding, if out. Therefore, dears, have a try like mine...) Yours little uncle Han [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 14:15, 25 November 2024 (UTC) [[File:M. Moszkowsky Etude (Op.72 No.5) played by Jixun Han (Jason) for piano teaching.wav|thumb|M. Moszkowsky Etude (Op.72 No.5) played by Jixun Han (Jason) for piano teaching]] ===== '''Beethoven's Moonlight Sonata''' ===== Indeed, Beethoven left a historical challenge (difficulty) but the continuously creative inspiration to understand the techniques & musicality of all his movements, equally to all people. We can attempt different approaches and own personal life-experiences to understand them, and discuss out some possible results. [[File:Moonlight Sonata - 3rd Movement of Sharp c minor Sonata Beethoven.ogg|thumb|Moonlight Sonata - 3rd Movement of Sharp c minor Sonata after a library presentation of Beethoven (Further, thanks to the April-collections of Tokfo Gallery (Vienna: https://commons.wikimedia.org/wiki/User:Tokfo/Vienna/2019_April_28-30) and Sir James Gallery (Bonn:https://commons.wikimedia.org/wiki/User:Sir_James/Bonn/2019_April_29) - great encouragements!)]] '''3rd Movement- 'Moonlights Storming' - Techniques Analysis from Notation-reading ('Presto agitato' of Breitkopf & Hartel Company and Berlin Arts Collage also compared with Old New York Edition - as the remembrance of one monitor):''' '''Musicality:''' In a grandly general view, it's like...in a crazily running (very fast) race, viewed from the window, moonlights have been dismembered upon deep Lake Lucerne (many fragmental sections composed together). [[File:Vienna Beethoven Monument (with angels and children surrounding).jpg|thumb| Beethoven's Monument in Vienna]] [[File:Beside Beethoven's Musicality.jpg|thumb| A third-person's Watching of Beethoven's Musicality]] For its musicality cultivation, I could give a similar sense of its situation, like in Picasso's works- such as Picasso's Guernica (Ceridwen's Creative Commons Attribution-Share Alike 2.0 Generic license) For achieving it, a little bit of dark-moods anger and sadness faced from the unfairness and out of control could be inputted, after all technique points were trained in the dexterity. Therefore, from emotion to say, I thought the video right after getting back from UK and the lost one in Newcastle central station were better than this time. [[File:3rd movement of Sonata &#039;Moonlight&#039; Rocking Video JMC, Han (Jason).webm|thumb|3rd movement of moonlight sonata; Rocking Video JMC, Han (Jason)]] However, I satisfied with it, right like in life and after the presentation. From this point, we can see: Beethoven, as a piano master, has super-reached too much before the time - even abstractionism and postmodernism (deconstructionism). '''2nd Movement - 'a little Fantasy Moonflower blooming between two rocky layers' - Techniques Analysis from Notation-reading (Allegretto of Breitkopf & Hartel Company and Berlin Arts Collage):''' 1. Parts-distinguishing way can be applied to pick up the main melodic points from its background and legato them into lines. '''( Notice: Here, from the historical observation, a thing needs to be clarified: Baroque-regression (back-reasoning) was usually made by classical composers (in Vienna school: Mozart, Haydn and Beethoven etc.), especially in their later years of life for calming down the dramatical emotions, and keeping Life's Reasonability. Meanwhile, from Haydn, they discussed and created classical counterpoints from symphony and string quartet together, to modify creative inspirations. Beethoven also inherited it. Therefore, when we play some in piano, we need to analyse and apply some special techniques, commonly used in classical polyphony, to pick up the main from the background, sentence by sentence, as an era-responding.)[[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 03:38, 15 May 2019 (UTC)''' 2. Octaves-bridging and chords-connections for big hands, into their hidden melodies, are the most difficulties, which need your frequent exercises, sentence by sentence. (Painful but worthful! Finally, flexible and skillful... ) 3. Long keeping-notes, in certain parts, are important for the continuity of the tune and the texture, without broken. 4. It's better in light and tender keyboard-touching way to make melodic lines clear and 'the little flower' smile lightly. 5. 'Rondo' (ABA) formation can be applied to understand its repetitions, responding and structure. [[File:Moonlight Sonata (Sharp c minor Sonata) 2nd Movement Beethoven JMC,Han.ogg|thumb|Moonlight Sonata - a little fantasy flower between two rocky layers]][[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 14:51, 14 May 2019 (UTC) ===== Blue Danube is always flowing from heart and life, with the vitality as spring: ===== Children and young teenagers, let us swim in this life-long river, to see some beautiful landscapes! [[File:Blue Denube in my heart.jpg|thumb|Blue Denube in my heart]] [[File:Blue Danube (Exercise and variations-collections in piano) JMC, Han.ogg|thumb|Blue Danube (Exercise and variations-collections in piano) JMC, Han]] ===== Pachelbel's Canon is always the canon (polyphonic technique) since Baroque Era, but in '''Modern Piano's Pop Variations''' ===== [[File:Pachelbel&#039;s Canon in Pop Variations (Geoge Winston Notation) Player Jason, Han.ogg|thumb| Pachelbel's Canon in Pop Variations (George Winston Notation) Player Jason, Han]] '''Story of teaching & learning (from Wikimedia Commons):''' Regarding with this piano melody, there is a long story in my heart... Oh, did you hear Mag-pie's singing (I like 'pie' in the tail of this word) in the first draft? Yes, it was attracted and landing on the tree outside my balcony... You can clearly hear it at the beginning and in the tail in my first draft... Almost, it would like to share my memory... Long long ago, my old brother on my mother's side used to be one hero of my life and fashion... On each holiday, he was always able to find great music pieces, MTVs, transcripts , and scientific fictions, from foreign countries, such as American and Japan (Summer)... and brought & shared with me... Then, I attempted my best to exercise them into the reality, which included this song - Canon Variations from pianist George Winston... Those memories have never faded out, but in my deep sea. To now, evenly did I think Canon was from US and a POP song... After seeking the exact information in Wikipedia, I found it's Pachelbel's Canon in D and Baroque Era and German, rather than C and Modern and Pop in American... and with a 'Gigue for Violins and Basso Continuo', it's not only for piano in many parts than our 3 parts in original piano edition. However either, I still like it very much and would like call it American POP in my music world... Then after, a male colleague in my working college said to me: Jason, on my wedding ceremony, I would like to play it for a girl... Could you give me a simple one? Then, searching online, I found a simple (middle level) notation and an original (advanced level) notation, I downloaded both, and chose the simple one for him Three months, he was able to play it from 0 level (he wasn't able to read the notation)... I thought piano would have give him a good memory of wedding... Following, I found a girl felt bored about her piano examination... Then, by choosing the simple transcript and inserting into her lessons... it made my tutoring classrooms really beautiful, relaxable, magical and peaceful... Now, I have time to play the original edition out... One long dream of my heart is going to be fullfilled... Though my hands in several points didn't make my perfectionism satisfied contrasted with before, especially the tenth-cross design between the left hand and the right hand, I knew it's my life, and fate?... I prefered to update its situations for bettering continuously... if having time... Compared with the firstly draft, I thought the second was much down-calmed and peaceful...Somehow, I preferred the first draft, but a little bit of 'fast'... I cannot make the decision...then, kept two. However either, I still felt very happy the little natural friend - mag-pie can join... For this reason, I kept it. Hopefully, you will enjoy... [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 04:50, 16 September 2019 (UTC) ===== KV.265 12 Variations on Ah vous dirai-je, Maman - 'Twinkle Twinkle Little Star' ===== Analysis (Waiting) [[File:KV.265 12 Variations on Ah vous dirai-je, Maman Mozart JMC, Han.ogg|thumb|KV.265 12 Variations on Ah vous dirai-je, Maman Mozart JMC, Han]] [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 08:43, 7 October 2019 (UTC) =====''' 'The beautiful views of Scotti-highlands' from Thompson's Book (Advanced Level) - Teaching and demonstration '''===== [[File:The beautiful views of Scotti-highlands (Teaching demonstration - video; Jason, Han).webm|thumb| After the performance in local Crossing-year 2019-2020 Library Concert and many further exercises, a good edition in classroom got out - Piano kids, always, you knew: No pains, No gains]] For its musicality and academic thoughts, please visit Wikiverty's Portal of Piano education(The Section: The beautiful views of Scotti-highlands' in a Far-land Home (Academic thoughts, musicality, literature-writing and case-realization)): *[[Portal:Green Sleeves (Impressionist Visualization)]] *[[Portal: Sir J.S.BACH and His contributions to Piano Kids' Reasonable Life]] [[File:C minor Prelude Bach (BWV847), Performer JMC, Han.wav|thumb|C minor Prelude Bach (BWV847)]] ==== 'Swan's Dream Upon the lake' - Little Girl 'Wenxin's' (Brilliant & Sweet literatures in arts ) Performance==== '''Teacher's Demonstration:''' [[File:The Dying Swan - black angel JMC Han.ogg|thumb| Musicality from watching 'The Dying Swan - the black angel', performed by JMC Han]] '''Wenxin's Performance:''' 1. Techniques-recovery: The Arpeggio-training of left hand in the accompaniment was the biggest challenge to not only a piano-child at her age - no more than 12 (In Chinese culture, Kid's first year was in mother's womb. Thereby, I asked her - how old are you, and she gave the number '13'...), to me and evenly some expertise pianists. (Camille Saint-Saëns's 'The Swan' on wikipedia or other social editions). The arpeggio-accompaniment is travelling in rich variations of tunes, which caused left hand much harder to expand, shrink and positions-change. Therefore, it spent us more than half a year to train and recover her hand's dexterity from a small failure of her piano life in the Grade Test, just like 'Princess Swan's' experience. Now, totally to say, she got an excellent situation in which children at her age can perform. Thanks to your hard-working! 2. Musicality-cultivation: Usually, she showed a very great musicality in the first page - to the minute (Approximately 1.05) - tender, expending, lyrical and expressive... However, it's really a hardness to keep it throughout the second section - a shading & wandering heart-road in the growth. The attention has to be paid too much on the exactness of left hand's arpeggio-travelling. With a pity, still, some notes were beaten wrongly. But oppositely again, we can see: Princess Swan, in her period of Darkness growth - facing Satan, turning into a dark angel and only appearing in night... She really faced a hardness and the difficulty of life, right as beating wrong notes, getting out some noises and travelling a little bit slowly and roughly in a channel. In this view, perhaps that the difficulties can be transformed- in the musical needs and with a small fashion. Congratulation, more exercises, haha! 3. Together, we gave two great designs: one is the 'Big Brightness' began from the minute (Approximately 2.03) when the main theme happens again; and the other is 'Swan's Departure like Sound of Fall-Leaves rotating upon Lake's Surface' (from minute 2.49 to the end)... She almost achieved some - the mood calmed down very much and stably progressed to further with a confidence. However, a little bit of disfluency made the impression fade, somehow. Meanwhile, a 'rit. to a tempo' turned inversely - what a pity. Totally to say, musicality, at her age, was preciously showing in this time's performance. The hard-working of recovery and exercises, during many classes, touched my heart very much. (I knew that...) More trainings of Arpeggio-running (dominate sevenths) and its fluency can help her achieve more in the future. Wenxin, thanks to you for letting us appreciate this world-famous melody in piano. [[File:Growth of Swan in eyes of the little girl - Wenxi, Zhang.ogg|thumb|Growth of Swan in the eyes of the little girl - Wenxi, Zhang]] [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 16:06, 5 May 2019 (UTC) ==== Listen to Mother's Old Story! - A beautiful and quiet little girl's Good Wish ==== '''Teacher's Demonstration:''' [[File:Mother&#039;s Old story - China Impression.ogg|thumb| Listen to Mother's Old story - China Impression (JMC, Han - teacher's domenstration)]] '''Student's Performance:''' [[File:Listen to Mother&#039;s Old Story - Piano Pupil Yiwen, Cui.ogg|thumb| Listen to Mother's Old Story- Piano Pupil Yiwen, Cui]] '''Teacher's Comments:''' Yiwen, Cui (Direct translation of her Chinese name - A beautiful girl who is good at the translation of art and literature, from Cui family), at the age of 10, is a quite and beautiful girl. She got a good life effect from this Chinese piano-kid's song - 'Listen to Mother's Old Story': making her family and parents happy, getting some confidences through this piano song from the examination, showing her fashion in my library concert held for piano kids... After those more above, frequent exercises, and getting her permission, I can submit this classroom-recording edition. Though in the tail I found a note lost... and some parts of her left hand might run much more fluently... , I think her emotional background of this music reached to a good level, and those polyphonic parts can be clearly heard two layers, their cannoning, and so on... Congratulation! '''(Words from the description in Wikimedia Commons page)''' [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 01:04, 8 August 2019 (UTC) ==== Moonlight upon Lotus Pool (Summer) - Letters-Accompaniment Improvising Chinese Pop-song with both Classical Tradition and Pentatonic Scale ==== '''Teacher's Comment:''' 1. I am very excited that you (only 10-years-old) understood Letter-To-Accompaniment Improvising sheet and its approach in a very fast way. 2. It's great you can use both Pentatonic Arpeggios and Tenth-Rolling-Bass-dropping in your accompaniment (You can make Tenth-rolling Bass in a more fluent view, I thought) 3. We can feel the musical scene from your musicality - In a beautiful summer night, Walking along a lotus pool, you and your family members were enjoying the moonlight and a breeze of cool wind... 4. In future, hopefully, you can improve your 'new learns' to a higher level. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 01:04, 8 August 2019 (UTC) ''' Xuan, Lee's Second Attempt - Pearl of the Orient''' ==== 'Mother in the candlelight' - A little girl (Siqi)'s heart-words for her mother's birthday - in the growth and in the dreams... ==== '''Heart-story:''' Regarding with this piano-song, there is a little story about this little commons' girl: Usually, her parents were very busy in the family's restaurant... I and my mother saw she had independently managed herself well and grew up alone for many years... In this year - 2019, time was near her mother's birthday. In the KTV (a place like karaoke bar, but for small single groups of people in rooms, with TV in the middle for singing ), she heard this song - 'Mother in the candlelight' and found her parents enjoyed singing it very much. Then, she decided to play it as a gift to her mother, right on mother's birthday. It's my biggest honour to be together sight-reading the notation, making the re-designs and re-editions of this song into piano - like, Prelude, Introduction-theme 1st, theme 2nd, Development and Peak, a small Repetition and Coda... She learnt in a very fast and hard-working way that merely around one month she played it in this level. And finally, she got her heart-sweet - playing it for her mother, as a birthday gift.(Wikimedia Commons' original page, 2019) '''Comments from teacher:''' 1. The musical emotions were very rich and expressive, especially the 'Peak-Calling for mother' (2.53 minutes - 3.53 minutes). I almost can hear 'Mum...' (or Mumu...) for many times in a kid's tear-drops and in the candlelight... by your right hand's touchable singing... 2. I liked our 'Flanger tr. Ornaments' very much (I thought it's from Mozartian). I am very happy you can put it in for soon time... 3. I am very happy in the Coda-tail, you can get my suggestion - ending by a Major Seventh Progression-Arpeggio. This point should give the thanks to my mentor - Ray. I quite enjoy its special colour... 4. Your strong and mixed left hand accompaniment must have been trained for many times. I knew it's a hard-working job, but tender and flexible a little bit... better? 5. The singing of right hand and its 'breathing' were quite natural and fine, sentence by sentence..., but the total speed is too slower than normal, which reflect the running ability of the left hand needed to improve. I knew: to your 9-years-old hands, it's a very hard requirement... However, waiting the up-grown, I have the confidence you can hands-sing it in a much more fluent way... [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 14:10, 5 September 2019 (UTC) '''Night's Piano-Song - The depth of China Pop-Piano (and Siqi's Heart-Try)''' ''Comments:'' She made the depth of two peak notes but in light touching way. I thought also that she did this Pop-piano's musicality in a poet's Night-thinking... She is suitable for the performance of this-type-'songs' and improvising (still at little age and need to prepare in time). Let us listen to her and feel her expression. ==== The Cowherd's Flute played by a little girl piano pupil - Guo Guo (Nickname: Happy fruits) ==== This edition has already been her best attempt, regarding with its landscape-painting style, lovely Cowherd's Flute we can refer to Wikipedia introduction (Seeking key words 'The Cowboy's Flute' in). It's recorded as a beautiful memory of her piano-learning and her Childhood. Let us listen to her: [[File:The Cowboy&#039;s Flute - Yuxuan, Lu (Guo Guo).ogg|thumb|The Cowherd's Flute - Yuxuan, Lu (Guo Guo)]] ==== Clementi Sonata Op35 No. 5 (Movement 1st), played by Piano pupil Yixuan, Qiao ==== Clementi's Sonata-Op35 No. 5 was a so long and difficult piece for students around their ninth year. Therefore, we have divided it into many small sections and taught. She learnt in progress. Meanwhile, this little and beautiful girl (She was beautifully good at dancing, somehow rather than piano.) has already attempted her best in exercising and recording. I thought it recorded her good piano-learning experiences and those memories of childhood. Regarding with further information about this work, please refer to the educational portal: https://en.wikiversity.org/wiki/Portal:Sonatinas_from_Kids%27_corner_near_heaven#Muzio_Clementi . Let us listen to her: [[File:Clementi Sonatina Op35. No 5 Movement 1st Piano pupil YIxuan, Qiao.ogg|thumb|Clementi Sonatina Op35. No 5 Movement 1st Piano pupil YIxuan, Qiao]] [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 02:42, 18 October 2019 (UTC) ====Little and Little, Twinkling Stars - Little Piano-Kids' Playground==== Say 'Hello' to our 'Little Goldman' [[File:Vienna Trip - The Little Goldman- Strauss Family.jpg|thumb|Hand-making my own picture of Strauss Family's Little Goldman]] * [[Portal:Little and Little, Twinkling Stars - Little Piano-Kids' Playground| Little and Little, Twinkling Stars - Little Piano-Kids' Playground]] cp8kanrhiqyq3sbisopwkh5oq2aogtj 2690255 2690251 2024-12-04T13:26:13Z Jason M. C., Han 2696291 Adding new life picture as musicality of piano 2690255 wikitext text/x-wiki '''No Orders, Beautiful flowers were blooming from Childhood! Let us pick up your most beautiful moments...''' [[File:Salzburg Cathedral (Inside) - Mozart Baptized place 1. - Mozart-Complexes World under Water (Modernist View).jpg|thumb|Piano Kids, on that day, modern colours, lights and technologies made Mozart's birthplace (Washing) -Salzburg Cathedral like a world (palace) under the biggest water - Blue Sea]] [[File:Beautiful landscapes surrounding Mozart's birthplace - Heavenly Spring flowing from Alps Moutain.jpg|thumb|Heavenly Ice-spring from Alps Mountain]] [[File:Mozart from Green Nature 1.jpg|thumb|Mozart comes from Green Mountain and Nature]] [[File:Models for Chaoshan Family Tourism - Taifo Hall.jpg|thumb|Uncle Han -the piano teacher and his wife Yang, Gao (Cynthia) were volunteeringly pictured outside the door of Taifo Hall in Chaoshan. They are promoting it for Family Tourism there. It's a beautiful time in all family members' memories, as a piano song told in life story...]] [[File:Beautiful Sunset View of Oceanic City from top building during COVID-19 period.jpeg|thumb|Beautiful Sunset View of Oceanic City during COVID-19 period]] [[File:Small items in Commons Life of 2019 Spring 2.jpg|thumb|Flaming Katy and piano-peers as Micro-landscape in classroom ]] [[File:An Oil Painting watched since born.jpeg|thumb|Oil Painting: Forest Cabin in front of Fuji(or Helan) Mountain 'hanging on the wall']] Oh, dear piano pupils, we have seen so many beautiful landscapes, across the times and spaces. Now, firstly, getting into East- your nature and self-meditation, let us try to take the Pentatonic Scales Exercises for relaxing you 'busy heart' and focusing your attention back to piano... *[[Portal:Pentatonic Impressionism (China Wu Sheng) in the view of Neo-classical Piano Techniques-training]] '''Afterwards, we can take further great life-tasks in piano...''' [[File:Piano Lecture of Dalian Library (Music) delivered by Jason (Jixun) on October 13th, 2024.jpg|thumb|Piano Lecture of Dalian Library (Music) Delivered by Jason (Jixun), Han and His Piano Pupils Team Associated by Family members and Library workers]] '''New Time and Fresh Air from 2024''' *[[Portal:Fresh Air of Piano after COVID-19 and beginning up a Recovering New Time of Openness from 2024]] ===='''Firstly, passing by a Skyline of Citylife Night-Skyline'''==== [[File:Flowing City (Lin Hai), Player and Teacher JMC,Han (Jason).ogg|thumb| Flowing City from 'Forest Sea', JMC,Han [[File:Dalian Urban Night Skyline from Xiao Ping Island Mountain-top No.2.jpg|thumb|Dalian Urban Night Skyline from Xiao Ping Island Mountain-top No.2]] ]] ===='''New Era is coming...:(New Teaching List)'''==== [[File:Skiing Statue made during my rollers-skating.jpg|thumb|Skiing Statue shows Spirit (like piano)]] [[File:The new situation of bubbles-fish statue.jpg|thumb|Ice & Snow Bubbles-fish are welcoming]] [[File:Mmexport1614470203508.jpg|thumb|Picture of piano lessons still under Covid-19 situation]] Etude in the Advanced level - Sir Chopin's famous Black-keys (Flatten G Major) Etude: [[File:Chopin Black-keys Flat G Etude (Exercise), player JMC, Han.ogg|thumb| Black-keys: An Etude your teacher has no chance in childhood, but still hurries up to be better]] [[File:Chopin Impromptus Op.29, JMC, Han.ogg|thumb|Chopin Impromptus Op.29, JMC, Han - Deep Fast Waltz-thinking in Quiet Gentleman's Self-Expression (Improvising)]] [[File:Sogdian whirl with large pipa.jpg|thumb| (Public Domain Work: See original page: https://commons.m.wikimedia.org/wiki/File:Sogdian_whirl_with_large_pipa.jpg#mw-jump-to-license): Could you find some musicality similarly with Chopin's Black-keys' Pentatonic?]] True Fairland - a piano melody of The Nutcracker (Dance of Sugar Plum Fairy): the advanture from childhood realizes your piano dream-"Fingering Ballet" [[File:The Nutcracker (Dance of Sugar Plum Fairy), Piano performer JMC, Han.ogg|thumb|The piano performance of <Dance of the Sugar Plum Fairy> in 'The Nutcracker' (Ballet Re-edited Piano Melody)]] Czerny 299 etudes were designed for the smart&strong fingering&modelling of hands and a pair of Vienna school's ears for the harmony...such as No.23... [[File:Czerny 299 No. 23 JMC,Han.ogg|thumb|Czerny 299 No. 23, player JMC,Han]] A peacefully praying Sinfonia of Sir Bach is freshly added in the "New exam book's list' during this COVID-19 period, to which you can have a relaxing hear and try (Don't worry, listening, it's enough time, you knew, 'Andatino'-Peacefully walking, and to sing by hands, in a small Baroque place : [[File:Sinfonia No.11 - 3Ps Invention, Bach, JMC,Han.ogg|thumb|Sinfonia No.11 Andatino- 3Ps Invention, Bach, Teacher JMC,Han (COVID-19 Protection Time)]] Encouragement in Italy Smart Fashion, but needs the very carefully fingering-techniques training (Long time, advanced), feet-edges' staccato, sentence-Pizzicato, flowing streams...Italian artist techniques always attract our eyeballs... Let us attempt to... make out your own Italy style! Cheer up! [[File:Domenico Scarlatti G Major Sonata, JMC,Han.ogg|thumb| Domenico Scarlatti G Major Sonata (In COVID-19 Pandemic Period), JMC,Han.ogg]] '''A good teaching video result played by Yixuan, Wang:''' http://m.kugou.com/mv/?hash=f00b36624f27b091b79e3f30e158aa03&sruserid=640650901 Baroque staccato techniques were always in a reasonable, confident, relaxing(wrists), fluent and vivid - 'Enough manners' of the Era, which needs us apply very careful fingering trainings. In a view of the whole structure, according to ears' musical suitable habits(psychological), I gave 1st section a twice repetitions, and then a throughout 2nd section to the Code. Hopefully, French Suite would make us brave, confident and relaxed. (but it also need years' accumulation of hand working to let those out and better) Have a try? Good luck! [[File:BWV.816 Gigue-French Suites No.5 Bach, Player JMC, Han.ogg|thumb|BWV.816 Gigue-French Suites No.5 Bach, Player JMC, Han]] (By turning to Wikimedia Commons, you will find two versions under its 'historical tree', currently. They are showing different stages we can reach. The first version was kept because it's relatively slower and more stable that in the basic stage we can make notes heard staccato and clear. After feeling suitable in this stage, we need to improve its tempo and get Gigue Dances' happy, vivid,,dialoguing with moods, jumping and wrists' breathing naturally. It needs time to train your hands frequently, untill relaxing but accurately. Main Difficulties: Stiffen wrists, Cramp and Tiredness... Now, it's the time of yourself...) Hi! We, piano kids: Imaging a scene, let us hands-dances with the good manners and a earnest mind in a beautiful Baroque palace. It's easy and natural... [[File:Primary Bach No. 16 March, Player JMC, Han.ogg|thumb|'Primary Bach No. 16 March' -A peaceful Bach-melody for all 'Piano Kids'...]] Sir Debussy's Arabesque Suites (The second suite) is also in the list. The musicality in my world is: It's the legend of Butterfly in birds' chorus... it took us to a Life-mountain behind our living garden facing a quiet sea... Watching, in some time of one section, you can also hear Monet's 'Quiet Morning' upon the sea... Alongside Butterfly's dance-suite, imagination is beginning. Oh, listening... (Main meaning referenced from my main page) What about it in your world and imagination? '''Sound teaching demonstration:''' [[File:Claude Debussy - 2nd Arabic Suite (Arabesque) - Spring Butterfly, Performer JMC, Han.wav|thumb|2nd Arabic Suite- Spring Butterfly (Impressionism Singing -Main Natural Lines&Breathes from Sir Debussy)]] '''Good video teaching result from a 12 years' old little girl piano-pupil Mo Zhou:''' http://m.kugou.com/mv/?hash=b50e133a360fa8d30cdcd9fca4163e73&sruserid=640650901 (Photographer: Ms. Yang, Gao) Listening! boys and girls, Dvorak's Humor-jumping and Homesick-expressing: A true Czech-homeland heart, but Dancing... somewhere in American Countryside [[File:Flatten G Humoresque Dvorak, Player JMC,Han.ogg|thumb|Flatten G Humoresque, A. Dvorak, Player JMC,Han]] [[File:One Town-view from Cesky Krumlov Castle.jpeg|thumb| Krumlov Castle-town's view]] ====Xinran, Yu - a lovely Chinese little piano girl's 'Ink-Mountain & Green Rivers' view of <The Cowherd's Flute> ==== '''Comment:''' "Before taking the national examination and the exhibition competition, we together listened and learned to the net-editions of young master Lang Lang and Yujia Wang...(regarding with this famous little melody of Chinese tradition) I think in this melody, she tried out her best for the techniques-training and the musicality in her age... from a little performer's view. Therefore,I gave the comment-Excellent. Close your eyes, thinking of a little lovely girl happily playing among ink-mountains and the green rivers, with a water buffalo, some birds followed, and her smart flute... let us relax in the Chinese Ink-Landscape and listen to this little melody...(referenced partly from the writing in Wikimedia Commons page) [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 02:28, 27 May 2021 (UTC) ==== Zichen Tao - A little Chinese snowgirl's hardworking of D Major Sonatina==== '''Comment:''' D Major Sonatina is still piano Children's favorites to perform and show... taken to the national grade examination, daily performed with each other, and also to city's piano Competition &Exhibition. That's an educational case lasting for many years in Dalian. This edition is played by a lovely and white Chinese girl - Zichen, Tao. She and her mother took the very responsibility to check the wrong notes, improve the learning progress, and make the performing manners and designs for the stage-show... Therefore, in my view as her piano teacher, this edition is already great in her age...(though hand-running details need to improve for her age). Hopefully, her family can enjoy this piano experience, companying with this melody in her childhood. (Partly referenced from her Wikimedia Commons' page) [[File:D major Sonatina , Piano student Zichen Tao.ogg|thumb| D major Sonatina (Kuhlau's) played by piano pupil Zichen Tao]] ==== Meng's Performance and Comments after learning in the reality from Jason M. C.,Han in Children's Corner: ==== [[File:Children's corner of Meng.ogg|thumb| Meng Meng (nick name)'s edition of Doctorial in Children's Corner: Currently, the Fourth Version was her most beautiful one self-made in classroom before Piano Grade Test Exanimation. Regarding with all editions' comments and reasons, please reference to the original file in Wikimedia Commons]] [[File:MM Good classroom F major 1838 Grande valse brillante.ogg|thumb|MM Good classroom F major 1838 Grande valse brillante]] '''[[Portal: Part of Comments - 'for students' Examination Performance, Piano tutor's teaching self-reflexivity and possible some requirements of Pedalling Sound-effects with Artist Fashion of Post-impressionism' | Part of Comments - 'for students' Examination Performance, Piano tutor's teaching self-reflexivity and possible some requirements of Pedalling Sound-effects with Artist Fashion of Post-impressionism']]''' [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 04:15, 12 October 2018 (UTC) ==== Kim Hui's 'Doctoral Training','Under the sunshine' in Children's Corner: ==== '''Comments for Kim Hui's first draft:''' 1. She did the second theme (associated) well 'much deeper like vocally singing out a better life in New Era under the sunshine, on a piece of small area in a rainforest'... 2. Her Korean dance (Wikipedia introduction: https://en.wikipedia.org/wiki/Korean_dance) has been done well, in which I can hear traditional drum-points in bass-part and crossing hand to tremble part. 3. I can hear Time-travelling and space-shaking to the past through a 'Dark-cave', from..., minutes 1.30-1.40... But, I think: if 'dramatically' and 'significantly' in dynamics (loudness), it would be better to show... 4. I can hear Forest's Evensong in Coda part - 'dim.' to the silence of night and a 'rit.' slowing down to the sleeping dream, and even several night-birds' dreaming voices...But, please make a much gentler taste (not so hurry up and not so strong) of those pictures. Meanwhile, I hope you can get a better & coherent control of the rhythm among different sections. 5. I knew, regarding with 'peak-parts', she had made many attempts 'drumming beats rights and keeping those connections clear'. However, still, in minutes 1.04 and 1.59, I felt it's a little bit 'rough', and needed to be handled in of the solidification... Oh, maybe, I am so severe... sorry, I should give you the encouragement. Main comment: 'Under the sunshine' is suitable to Kim Hui's fashion and can be kept in her performance list. Her first draft and its preparation has given me an enjoyable teaching experience and many beautiful memories of life. It's fluent and vivid, expressive and dedicated. Thanks, Kim Hui! More colours and lights would be added from technique details, from her independent fingering and some traditional piano manners, meanwhile, the rhythm should be balanced well in the future. There are many developing zones of 'this painting' she can better and draw out for her future.[[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 21:22, 17 October 2018 (UTC) ===== Kim Hui's second 'Show-Time' in classroom - Poem of Music (Piano Etude) ===== '''Teacher's Demonstration:''' [[File:Poem of music.ogg|thumb|Poem of music JMC, Han]] '''Student's Second Performance - Poem of Music: ''' [[File:Poem of Music (Piano Etude) - Student Kim Hui.ogg|thumb|Poem of Music (Piano Etude) - Student Kim Hui]] '''Comments for Kim Hui's first draft:''' 1. She has mainly got the technique-points, but a little bit of rough in some details, such as the minutes 0.30-0.31 - 'Tail-closing part' of a sentence - in the progression of 'Diminished Seventh Chord-Arpeggio'... However, as her first draft and the random collection from a normal classroom, I thought it's well-done. we can wish its further 'Developing Zone', in the view of piano education. 2. In 'Coda Part' of Poem of Music (Piano Etude), she was able to show a great controllability of the 'Legato' between two hands, as the pieces of falling leaves slowly flying-upon the surface of water. Sometimes, it was evenly better than mine. I hoped she could manage it in a better way. 3. She showed some thoughts of musicality... However, 'Techniques-points' still wasted much of her energy. I think the total Dynamics in physics will be improved soon. (waiting more) [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 14:53, 23 May 2019 (UTC) ===== Kim Hui can reach the advanced level of piano performance in this 'Little Children's Corner' ===== Though It's still Covid-19 Health-protection Time, after Home-self-Training Time, some classroom face-mask covering & health-good-protection lessons and an examination of China Culture&Tourism Ministry, She can play this 'Doctoral Etude' which is dreamed many piano pupils, with impressionism style. As her teacher, I didn't think It's a simple Etude which was expressed in many scenes, but with the big universe imagination and impression. Therefore, we have trained it as Meng's approach and further developed it. Indeed, I think she performed far greater than this edition, right in that online national examination. She got it, Congratulation! Let us listen to her ... [[File:Debussy-Children's Corner-Doctorial Etude, Piano pupil Jinghui Jin (Kim).ogg|thumb|Debussy-Children's Corner-Doctorial Etude, Piano pupil Jinghui Jin (Kim)]] ==== 'Colourful Clouds Running After Moon' impressed into the Heart of Xinyi, Hua (Hua family's Heart-sweet girl from 'Painting Imagery')==== '''Comments for the first draft of Xinyi, Hua:''' 1. I like her treatment of the prelude part in 'Colourful Clouds Running After Moon'. It's light and soft like silky clouds up-bridging alongside moonlight towards a round moon above the dark-blue sea. However, please try to link each silky pentatonic-arpeggio weave as a smooth whole from the bottom to the top, and from the left hand to the right hand. If so, her progress will be enlarged; 2. I can hear the situation 'Colourful Clouds Running After Moon' appeared in many linking parts before and later. She was attempting to give an acceleration imitating this procedure from a slower speed to a fast one, and between two hands' echo-following from a loose density to a tight one... However, if obviously, it will be better; 3. Like 0.58 to 1.03 minutes, I can hear that in some parts, she would like to make a returning sound-boomerang (Wikipedia introduction: https://en.wikipedia.org/wiki/Boomerang) up-rising 'to the moon' and down-landing upon the sea-surface. If a small time of middle reaction was canceled out by her proficiency, we will appreciate the musical beauty in a much more advanced situation; 4. In 1.22 to 1.30 minutes, I know she would like to make a silky veil, with the colourful clouds as material, upon moon's beautiful face by her right hand. It's a little bit of pity that the controlling ability of relative loudness made she carried this willing but harder to realize. Meanwhile, this veil needed to be smooth. Oh, sorry, I am so critical... indeed, she did not bad; 5. There is a hard hurt in 1.36 minute - it's still a repetition of bass-chord though she has already attempted her best to grasp the bass large chord through left hand's opening degrees (Little girl, I knew you had tried your best. Though the momentum was great, I still need to point it out.); 6. 2.12 - 2.40 minutes is the part - 'Bright Moon up-rising above the harvest sea'. This is a grand scene which needs great forces from students' forearms and a fast reactions for some flexible connections to arpeggio-parts... Congratulations, little girl, she have got it, though it's a little bit slower. She have given out a great momentum; 7. In the Moon-tail part, she has expressed her great musicality to make moon disappear in the dawn of sea; 8. Many ornamentations she has done well, though still some need to be gently breezing in the impressionism of Chinese landscape painting. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 12:22, 24 October 2018 (UTC) ===== 'Pure White Dove' in Young Teenager Xinyi's Eyes ===== [[File:Dove in the eyes of Young Teenager girl Xinyi, Hua.ogg|thumb|'Pure White Dove' ('La Paloma' - 'No More' in English) in the eyes of Young Teenager girl Xinyi, Hua]] '''Congratulation to your beautifully singing of the 'Melodic lines' behind the right hand's octaves-grasping!''' '''Comments:''' 1. Four biggest designs appeared: around B50 (Minutes 1.36), B55 (Minutes 1.48), B58 (Minutes 1.54) and B62 (Minutes 2.01) - four Peak-currents, we'd like to throw (rit.) the 'missing notes' into the air and rotate them a little bit more slowly - like to send, wait and feel Dove's messages across the ocean in a self-holding & self-releasing intoxication. She tried her best to make them out, but not quite clearly and still need much time to grow up... 2. I liked her coda part (from B65 Minutes 2.08 to the end): She was so sure about two hands' March-doubling, as a confirmation of future and belief; or to say, she transformed her 'missing' in the melody to be a true hope of tomorrow, or someday... Evenly, I thought it's better than mine... 3. For more than half years, we have worked hard to help her link all octave-grasping pearls out of melodic lines in singing breaths. She almost got it successfully, through small breaks... 4. Some 'Spanish Dotted notes' and 'Triples-wandering', with the rhythm of Spanish Dance Habanera-Andante (Wikipedia introduction: https://en.wikipedia.org/wiki/Contradanza) can be fulfilled, but some not really... I am happy she recognized them and paid more attentions to... It's waiting time that she could perform much better. 5. Yes, I had to say: still some small faults there... The good usage of pedalling almost hide some, but... also a little bit rough... Oh, I didn't want to be a so severe teacher. Rather than, much more good wishes of her growth should be given. Okay, hopefully, she enjoyed it. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 12:57, 4 April 2019 (UTC) '''Teacher's Demonstration for standards above and Further Progress (Advanced Level):''' [[File:Dove With Spanish Sense in Piano JMC.Han.ogg|thumb|Dove With Spanish Sense in Piano JMC.Han (It was once used by the photography gallery of music friend User:PigeonIP - https://commons.wikimedia.org/wiki/User:PigeonIP/Tauben/2019_April_1-10 and the main page La Paloma in wikipedia )]] '''More information and reading - articles (list) about 'Dove', please watch:''' https://en.wikipedia.org/wiki/La_Paloma ==== Malagueña Dream from a little Chinese girl - Yinuo's heart (A promise ... to piano) in a small beach-side classroom ==== [[File:Student edition - Malagueña Suite (modified for piano) played by Yinuo, Liu.ogg|thumb|Student edition - Malagueña Suite (modified for piano) played by Yinuo, Liu]] '''Comments for the first draft of Yinuo, Liu:''' 1. We can hear the impression of Flamenco rhythmic pattern (Compás) (Wikipedia introduction: https://en.wikipedia.org/wiki/Flamenco) diffusing from some simple rhythm-components of a parts in a total ABA structure of Malagueña Suite. This is what I - the tutor and the little girl -learner would like to express through three more different accompaniment bass-forms, including pizzicatos, small slurring breathes and some opposite weights..., which imitated some of Classical Guitar's handling ways. Thanks to the little girl Yinuo, you have realized most of our designs! Congratulations! 2. I really like her beautiful Cante jondo - associated 'vocal' lines by right hand which was balanced & flying above the flamenco accompaniment of the left hand when the second thematic melody began. It's a deep, profound and emotions-rich singing, almost from a beautiful Spanish girl's natural expression for the missing, the reasoning of life & Universe when facing a 'deep and far' sea. Though if the dynamics would be dramatically and the singing would be much deeper, the emotional atmosphere would be better: I thought to only a girl of her 11 years' old age, she has already attempted her best to understand those across cultures; 3. I like the middle B's fantastical view of holiday beach under the sunshine, which was almost formed by white waves from blue sea. It's relaxable, dreamful and graceful, like a girl poet's walk alongside a small sand bay... (Yinuo, you knew, if you can make the 'rit. - A Tempo' much more nature like the real tides of sea and the speed tiny faster, the progress zone will be enlarged); 4. I know in two middle long 'vocal' ornaments, she would like to show us ' the blackbirds or the nightingales of its gardens...' However, if making it much more smoothly, expressively, and flexibly, even a little bit down-slowed, her Spanish 'tasteful' fashion will be more beautiful; 5. Repeated A part is better to be different in small details which can show the ability of hands and the variation of music. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 13:35, 24 October 2018 (UTC) ==== Für Elise in piano boy Zhe's eyes ==== '''Teacher's demonstration in classroom for better:'''[[File:For Elise (Für Elise) Beethoven JMC Han.ogg|thumb|For Elise (Für Elise) Beethoven JMC Han]] '''Student's performance in classroom:''' [[File:Für Elise -Student Performance Zhe,Zhang.ogg|thumb|Für Elise in piano boy Zhe's eyes]] ==== D major Sonatina (Kuhlau) - Piano-pupil girl Mengshuang's 'Strong Willpower and Persistence' ==== '''Teacher's demonstration in classroom for better:''' [[File:D major sonatina 2nd movement Kuhlau (played by Jason).ogg|thumb|D major sonatina 2nd movement Kuhlau - Teacher JMC. Han]] '''Student's performance in classroom:''' [[File:D major Sonatina (Kuhlau) - the version from piano-student Meng Shuang, Wang.ogg|thumb|D major Sonatina (Kuhlau) - the version from piano-student Meng Shuang, Wang. This classroom version has been selected by https://commons.wikimedia.org/wiki/User:Rsteen/Artists_from_Denmark/2019_August_1-10]] '''Comments and Statements:''' 1. Totally to say, the main melody fast-run by the right hand has kept its fluency, transparency and clearance. It's very hard in piano training for herself, owe to that her hands-shape was a little bit of 'frozen'. Thanks for your hard-working in the training. Congratulation! 2. Her musicality in this melody has also been motivated out - unrestrained and natural in the expression. 3. Left hand's accompaniment was in good triplet-treatment, but please light and dedicate a little bit... It's to say: the controllability still needs to improve. 4. Some heads of sentences and smaller phrases need to be match together between two hands in a better way - some parts, because of small ornamentations and dotted notes, weren't quite well... 5. I am very happy that you (in your 12th year of life) were willing to play out the middle 'rit. - A Tempo' in a comparison ('rit.' was slowing down the waiting, then, 'A Tempo' for the Peak expression in return). However, it was still a little bit rough (before its right time). You can try to modify it in a better view. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 03:08, 2 August 2019 (UTC) ==== How to play Mozart's Classics? ==== ===== Work from Mozart's earlier time - Turkish March ===== '''Teacher's Demonstration in classroom:''' '''Video:''' [[File:Turkey March Video-Mozart-Jason Han.webm|thumb|Turkish March (Video) for Mozart's; JMC, Han]] '''Sound:''' [[File:Turkey March for Mozart and Memory JMC, Han.ogg|thumb|Turkish March (Sound) for Mozart's and Memory (many times used in peer Tokfo/Vienna Gallery - such as https://commons.wikimedia.org/wiki/User:Tokfo/Vienna/2019_January_26-28 ); JMC, Han]] ==== Sure, Ma's boy-view of Mozart's 'Turkish March' ==== [[File:Sure, Ma's Version of 'Turkish March' in piano classroom for piano education.ogg|thumb|'Turkish March' - an old mysterious Turkey story in piano student (Jason's piano pupil) boy Sure, Ma's classroom edition. Thanks, this good teaching & Learning result was selected by Tokfo/Vienna Gallery: https://commons.wikimedia.org/wiki/User:Tokfo/Vienna/2019_April_25-27 ]] '''Further Comments for his first draft:''' 1. It's very difficult for a young boy to manage Turkish March's speed in a smooth way... He tried his best to keep it stable and unified, and almost did achieve it. (Turkish March is easily to make people play faster and faster until crazily broken. He tried to solve it by giving a slower beginning ) But, it's a little bit of too serious,afraid to touch wrong. Indeed, I heard his another time, in which he totally open himself and relax from nature... We could give him more hopes. 2. His melodic flow of scale-phrases (legato) are quite fluent and natural, which shows his scale-playing and fingering were quite great. But, a small break occurred around 1.14 to 1.17 minutes could be caused by the stiff right wrist (too tried) and no-good fingering design. He should frequently move second and third fingers in a much smarter way. To a young boy in his age, it should already be 'okay'. 3. When the theme occurred in the second time, it's better to give a dynamics-difference in contrast. My mother-Ms Song said: it's like an old story (sound) heard from a far distance to near somewhere - mysteriously. However, he gave a very tight connection, as if it was linked with the previous section. 4. He tried his best to take the Worldwide difficult challenge - 'Broken-chordal Arpeggiated-octaves' (Around 2.00 to 2.14 minutes). I gave him a 'LIKE' that he had taken this challenge which even many pianists or teachers made some 'faults' as their heart pities - You can hear the edition of Romuald Greiss' in Wikipedia and several my previous times... However, this boy achieved it after many trainings time after time... Though later half one, compared with the beginning, might be in lower distinguishing degree, he didn't make any 'breaks', which comforted my teaching way so much. Thanks, boy Sure! 5. The final problem would occur in 'Alberti Bass' (left hand) of Coda part. Coming to it, you will feel easy to give up, which required more endeavors to control your hands in narrow and elaborate dealing way. He did it good, but lost in the counting of number (B111), and further, the connection with the final 'Square-opening Dance' (a small break). other things, such as the strength, are fine in his age. 6. In addition, I am planning to add a 'Turkish Stop' by a final pedaling. I didn't know whether he could, someday. Overall, I gave him an Excellent Comment. Hopefully, he will play better after better in his growth. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 14:13, 8 March 2019 (UTC) '''Further Comments for his second draft:''' 1. As his teacher, I am very happy to hear several great designs we have made in classrooms can be achieved in the second draft, according to the background knowledge of 'Turkey March' I taught, such as the final 'Turkey Stop' (not really in modern piano, but a little bit similar) and the Bass-points-layer (simulating the military drum) beneath the long fast running scale-phrases of right hand (middle section)... Cool boy, thanks that you can remember your teacher's words! Congratulation! 2. Yes, right after the chance of Music library Report-performance in local we have made and getting back, you improved the edition's speed and fluency. You can evenly save 15 seconds, contrasted with before, which showed that your fingering & running ability of hands had been greatly improved. However, the disadvantage is that it's easier to make some small motives uneven and rough (touching wrong notes) without purpose, which needs more your careful attention and exactness about details. 3. I knew you tried your best to face the peak challenge Mozart made to all people - making broken-progression of octaves message (middle part) and hearing out its hidden melodic lines. Great! However, it's still a little bit beyond your ability that its distinction with chords-effect weren't so clear. No matter, Sure boy, more exercises, it will be better. In all, progressing soon which shows the potential, thanks to your performances1 There is still the developing zone waiting for you. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 15:08, 25 April 2019 (UTC) ==== Clever Girl Jia Xin's Clever view of Sir Bach's 3-parts Invention ==== '''Teacher's Demonstration for Future Development:''' [[File:3 Parts Invention 8th- from Sir Bach.ogg|thumb|3 Parts Invention 8th- from Sir Bach's BWV 794 – Sinfonia No. 8 in F major ]] ''' Jia Xin's Performance:''' [[File:3-parts Invention No. 8 (F major) in piano-girl Jia Xin's view.ogg|thumb|3-parts Invention No. 8 (F major) in piano-girl Jia Xin's view]] '''Comments of her first draft:''' 1. Totally, 3 parts are ranged in Bach's harmony, to a girl in her age - 12 years old. It's not easier to make so clear layers out. I was satisfied with this point, after heard every time; 2. I can hear piano techniques for polyphonic & counterpoint music like Bach's, such as cannon, intimation between two hands, up-climbing shoulder by shoulder, dialogues, long-notes down-pressed for different parts' SHE (sentence-head-enhancement), long-notes kept for parts' division and maintaining, fingering grouping in one hand for 2 parts, parts' continuously melting into one for the summarization, and..., Baroque ornamentation... mentioned for long. For those trainings, and further, the internalization into her own mind-analysis, we had searched information & knowledge through Wikipedia, two more manuscripts, books and other webs online, further, spent classroom time to reason, analyse, train and fix note by note for long time... In this case, I gave her hard-working a 'Like' again; 3. There would also be some problems regarding with recording pressure and her memory...: some big ones - left hand's relatively weak ability in managing two parts, small mistakes (like B18's f note played as sharp f - around 1.13 minute, B21's final g isn't raised there - around 1.23-1.24 minutes, and others...), small ornaments in a little bit of rough view and a much more graceful manner in Coda part. (sorry, to such a 12 years old girl, my suggestions could be so severe. But when listening, they are directly in my ears...) 4. She did really pay her attention to Dynamics, but please better - lighter, smarter and more obviously... 5. The speed of later part is better than the slower earlier part. Overall, I also gave her an excellent comment for her performance (Live) in classroom. For further development, she can listen to my edition and the one in 'Inventions and Sinfonias (Bach)' - Wikipedia article (I thought it's great, but I didn't like too many speed-variations in Bach's works. It's better more reasonable: https://en.wikipedia.org/wiki/Inventions_and_Sinfonias_(Bach) ) [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 07:33, 9 March 2019 (UTC) ==== Yixuan's piano-view and traveling landscapes ==== '''First view (only 9 years old) of Sir Bach's 2-parts Inventions''' [[File:Bach 2-parts Invention, played by Yixuan Wang (Only 9 years old).ogg|thumb|Bach 2-parts Invention, played by Yixuan Wang (Only 9 years old)]] '''Comments from teacher:''' 1. Musical parts and space-dimensions were much clearer than before, which reflected the little girl Yixuan's hard-working continuously after her piano examination... 2. I can hear the heads of musical sentences which were highlighted by each hand when required. I can also hear cannon-following, doubling and countermelody which were clearly shown into her performance. It's quite necessary for students in this age - 9 years old. In certain degree, she is already a good and careful student in piano. 3. Still, the controllability and the stability of hands, especially the turn of her 3rd, 4th and 5th fingers, need to be improved, which caused some small faults, such as 0.49-0.50 minutes (Bar 22) - a recovered B in right 3rd finger, 0.54-0.55 minutes (Bar 25) - 3rd, 4th & 5th fingers of left hand, and a small disharmonic note - flatten B in the right hand - 0.32 (Bar 14)... 4. It's great that I can hear Baroque staccatos were in their graceful manners - like imperfect pearls required by its era. She almost did it... 5. Totally saying: it must be a very hard-job for a student in 9 years old to play Bach's 2-parts Invention. She bravely took up this life-task and successfully completed - this point should be affirmed. Congratulation! You can do more further... [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 02:28, 6 September 2019 (UTC) '''MARIAGE D'AMOUR (Dreaming Wedding Ceremony) and its Educational Story:''' It's only no more than 2 weeks she did get the main techniques, after taking a Covid-19 Protection piano lesson and further test(Face-mask covering and breathe-prevention…).Then, she went back home and made a hard self-working exercise. Afterwards, around 10 days, this edition can come out. Why was she so keen on making it? She told me... One of her family's friends would like a piece of background music for his wedding ceremony, and they knew she was a good piano pupil.They invited her to take this task. She online self found out her long dreaming piece, and felt very happy for them. She thought only hard-working at home can realize this dream in this 'Hard Recovery Time'. She has beautifully taken this life-task for a very short time, and finally I could find a beautiful smile on her face... Though there were still some small faults in teacher's view, such as the biggest 1.02-1.03... ('''Problems:''' Mentioned one is because of the distance of the Tenth-grasping is out of her hand-shape and ability in this age - rolling but touching a wrong note; In addition, the breathing of each sentence's tail somehow is with a longer responded break... Further, the Pedaling for the coherence from natural breathing need to improve; The final departing dropping notes were too noisy... which needs to be quiet,rit. and peaceful...), her hands' ability (especially the big chords-grasping, whole-viewing, locating, and sight-reading) was improved by her own endeavors (Maybe... subjects-divided examination-taking online through self-video-recording,in this special time, motivated her self-management...). This point made me feel happy... Hopefully, the friend of her family enjoyed their wedding ceremony with this own and LIVE background music, luckily as in a fantastic, peaceful and forever-lasting life-dream of happiness. In future, Hoping: Yixuan, you can play this fantastic wedding song of piano (fluently and heart-touchingly) for more families and share their friendship, love and happiness... Little girl pupil, thank you!^_^ [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 01:55, 24 September 2020 (UTC) [[File:CCzerny 299 Etude No. 27, Piano student Yixuan Wang, Tutor Jason Han.oggzerny 299 Etude No. 27, Piano student Yixuan Wang, Tutor Jason Han.ogg|thumb| Czerny 299 Etude No. 27, Piano student Yixuan Wang, Tutor Jason Han.ogg]] I have taught two children this Etude-a girl and also a boy (with outcomes). They played all well in very different musicalities. One is like a fast gym meeting 299's standards. The other-hers is with a good sound effect -light and peaceful after her grade examination. Both I all like. Regarding with how to train this sound effect with pedal, Please see my etudes'platform: https://en.m.wikiversity.org/wiki/Portal:Piano_Etudes_as_Poems '''G major Sonata L.349 - Yixuan Wang's New Attempt of Italian Baroque Style of A. Scarlatti''' A. Scarlatti's Sonatas are quite hard for young students and young teenagers to train and perform.However, Yixuan is fine, I thought. It needs a very fast & light fingering of Scale & Arpeggios and different STACCATOing keyboard-touching way, meanwhile, the exaggerating fluency of simple patterns... I thought she somehow had touched at her own little age. Just, more from nature, more details-care and the flexibility of hands&body could make things better. At her age, it's already fine.Thanks to the recent striving in this still hard time of COVID-19 Recovery. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 02:31, 20 August 2021 (UTC) [[File:G major Sonata L.349, Piano Pupil Yixuan, Wang.ogg|thumb|A, Scarlatti L.349 Sonata, Piano-pupil Yixuan, Wang]] It's very hard to train Scarlatti's Italian Style of technique skills... However, Yixuan, Wang has never given up... during this COVID-19 Time...-Protecting herself with masks, meanwhile, playing times after times... Finally, we can get some senses of fast, airily,lightly and breezily... Yes, there may also be some problems, like- it's very easy to be stressful and breathless on the stage...But to her age, do you think it's already good... Therefore, I recorded it in a video and published it on a musical platform -Kugou, and a educational platform - Youku, as to remember her growth: http://m.kugou.com/mv/?hash=f00b36624f27b091b79e3f30e158aa03&sruserid=640650901 ====Piano Pupil Mo Zhou's Smart Growth and Hard-working learning of Techniques ==== The video of Mo Zhou's most beautiful performance of Debussy's work- 2re Arabesque (I call it 'Butterfly's Dream') : Kugou musical platform - http://m.kugou.com/mv/?hash=547cb2c1e2f57a9e8ec66e8ecf36c269&sruserid=640650901 Youku educational part - http://v.youku.com/v_show/id_XNTgxNzA2Njk2MA==.html?x&sharefrom=android&sharekey=9631de9a76de1af3d601221019590cd26 (Published on the musical platform of Kugou and the educational part of Youku; the classroom volunteering photographer is Ms. Yang Gao) '''Piano Pupil Mo, Zhou's Violin simulation of Cremer's Etude's Art''' Catching the hands' positions (somehow borrowed from voilin's) is almost the hardest point to train.One focuses on left hand's Notes-Slipping; the other regards with the interval Position-switching (2 notes) check of right hand frequently. Though this little girl has a pair of smart&slim hands, she attempted her best. You can hear the most part's effect LIVE in classroom... In this point, I gave a 'LIKE'. [[File:Cramer Etude, Performed by Piano Pupil Mo, Zou.ogg|thumb|Cramer Etude, Performed by Piano Pupil Mo, Zhou]] '''A, Scarlatti L.349 Sonata - A Italian Style Taste of Baroque Music''' ''Comments:'' Mo,Zhou's hands are very smart, regarding with which some very tiny actions she can take, though they aren't quite big. Yes, she has been always willing to enlarge her hands. This point, but somehow, associate her to take this Italian Baroque Style (Rocca) quite easy. Yes, I thought she was fine regarding with much more details.(though it's LIVE that very few unexpected faults could be caused by the stress of the recording). I thought: to her performance, my teaching is working well. She did many requirements... Let us listen to her.[[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 03:25, 20 August 2021 (UTC) [[File:G major Sonata L.349, Piano pupil Mo, Zhou.ogg|thumb|File:G major Sonata L.349, Piano pupil Mo, Zhou.ogg]] ==== Brilliant Snow-ball boy (Yu) of Zhang family is praying for his father working in New Zealand ==== [[File:Pupil Yu,Zhang's edition - e minor sonata of Sir Haydn.ogg|thumb|E minor sonata of Sir Haydn was played by Piano student Yu, Zhang in classroom]] '''(Waiting better)''' Could you understand how hard Sir Haydn's & Mozart's mature sonata-structuralism and Classical Countermelody (from String Quartet: https://en.wikipedia.org/wiki/Wolfgang_Amadeus_Mozart) were for the training of such a young boy or some students around this aget? Oh, looking back, I, myself, also did feel hard... However, this boy and another older piano sister did really insist on doing so. Today, they can give their own editions - very different with own personalities and natures. Another point I would like to say: It's only one week's time that this boy was fighting for 'a good hearing' of his father. Afterwards, a modified recording edition soon got out, which showed his proficiency and quality...Good boy! To be honest, reviewing the past year, in order to train Sir Haydn's melody, we researched many ways together, including mathematics... Sometimes, evenly felt hopeless... Playing from childhood, Haydn's style is quite simple to me - models, switching, sonata structure..., but to students, they didn't quite like the sense of thinking being structured... And at the very beginning, I even didn't understand why they felt difficult... Recognizing something, We began to make many games, and evenly counting out some scores for the achievement of his 'fried chicken legs'... Here, from rhythm to notation, and from melodic interaction to parts-division, I felt it's much clearer, more fluent and stable, than before... His ability of coordination has also been improved, though still some problems. I dared and felt confidential to say: it's a great edition of himself. Hopefully, he can progress further. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 16:08, 4 April 2019 (UTC) Now, he made it much more fluent and accurate, and also played out his own fashion, though some details still need to be modified. Honestly to say, I thought somehow he got his progress in this period which we can hear...[[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 03:28, 2 August 2019 (UTC) '''Comments''' It's the second time of piano boy Yu Zhang's teaching result show. In this time, he chose the melody <Under the Sunshine> - a Chinese fork teaching melody as one subject of national examination and also a performance of one piano competition in Dalian. In my view as his teacher, he gave a very different view of this melody, compared with girls'. He paid more attentions to the whole view of melody's energy, strength, fluency and the joyfulness in Under the Sunshine, but didn't too much care about the details of some parts. However, on the stage, it showed a very great expression as from boy's situation... Good luck and happy experiment. After some practices, in the classroom we together recorded it and submit it up... (Referenced Partly from his page in Wikimedia Commons) ==== How to play Chinese Folk tune - 'Kids' Dance' with Chinese kids' fashion? Listen to little girl Kunlu's performance ==== '''Teacher's Demonstration in classroom:''' [[File:Kid's Dance Chinese Folk Piano Player Jason M. C.,Han.ogg|thumb|Kid's Dance - 'Kid's Dance', from a folk piano-tune in China National Grading Book, was personally performed here, as a gift for all piano-kids' 'Happy 2019 Lucky Pig Year']] '''Student Kunlu's Performance in classroom:''' [[File:Kid's Dance (Chinese) - Student Kunlu, Han.ogg|thumb|Student Kunlu, Han's (Han family's girl born as bright as dewdrop in Kun - Saturn of Wuxing) good performance of Kid's Dance (Chinese)]] '''Teacher's Comments:''' Totally to say: Though She can play better editions (many better ones, last winter), in this sound file, she showed the coherence, fluency, flexibility and stability ( as Chinese fork-tune required). Hearing such a smart Chinese girl playing such a fugue-cannoning song, you will feel: it's a right song designing for a right girl... I think that's one meaning of piano-performance. Though spending much time, We did also research special 'Chinese supplemental positions & Dialogues' in polyphony together, which gave us many beautiful memories... Further more, in this age, her staccatos, slurs and Tenuto have been performed quite well, which helped her to keep a unified speed to the end. Taking back a step, there must still be some small faults in classroom (without purposes) that I have to point out: such as B18's #C blowing to D a little bit, the attention didn't get back in B41 head A which made a small break, and a small mistake of 'Recovered C' rather than #C... In order to dream of its accuracy and pentatonic harmony, it's a hard-working that we have already come over many problems and mistakes... Therefore, I think she fulfilled herself and achieved many things from 'Kid's Dance'. Hopefully, she enjoys the procedure of music-carving. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 15:07, 4 April 2019 (UTC) ===== Kunlu's Crystal Heart on International Children's Day - Kleine Kinder Kleine Sorgen (Little Child) ===== '''Teacher's Comments:''' 1. The degrees of proficiency, fluency (and internal speed) have been improved, right on International Children's Day. 2. I preferred her treble part very much - so cool, pure, clean and refreshing, which reflected her crystal heart in childhood. 3. The grasping of big chords - stronger, that's great - but needs to be more accurately and deep (The word 'deep' wasn't always 'loud' and 'heavy'). Please try to understand this point. Yes, it needs to show the hardness of growth (to young teenager), but also the achievement 'to be stronger and more confident of yourself...' [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 03:32, 28 June 2019 (UTC) [[File:Internationl Children&#039;s Day Gifts - 2. Kleine Kinder Kleine Sorgen (Little Child, Piano-modification of Germany Song).ogg|thumb| Kleine Kinder Kleine Sorgen (Little Child, Piano-modification of Germany Song), played by Kunlu, Han]] ==== Teacher's Shares of his own home-works from childhood (Open) - Jason, Han==== ====='''M. Moszkowsky Etude (Op.72 No.5) - "C major's Fluency, Clarification, Sunshine and Love'''===== New Beginning with...: M. Moszkowsky Etude (Op.72 No.5) played by Jixun Han (Jason) for piano teaching. It's long time that my piano classroom on the cloud in wikiversity hasn't update its situation. After so many things, now I can partly return to English writing world. The first Etude I would like to upload is still MM Edute which gave me so fluent and clean mind in my childhood. Oh, 38 years old, and after a wedding ceremony with my real lover, my fingers would not be so great as around 15s'... However, I would like to update its situation and new editions untill great someday. Now, let's began with this new melody. It's taught to my good Chinese boy pupil named Guoguo (fruit zeyu, Cui) when I grasped up and recorded. Yes, this little boy will also play well. Let's listen to my version, firstly. Thanks Jason M. C., Han (talk) 13:26, 20 November 2024 (UTC) More information, please see https://en.wikipedia.org/wiki/List_of_compositions_by_Moritz_Moszkowski Homework Requirements (challenges): 1. B23-B24(B stands for Musical Bar): By right hand, heads of every 4-notes group make a down-going semi-notes scale, which needs a very careful&exact arpegio-fingering with a whole—palm holded and also thumb-measuring ability. Meanwhile, the left hand is making a whole-tenth measure, but arranged upon every two chords' link. The semi-notes scale is also its fixed channel accordingly. This point is very different to follow and be made accurately and perfectly, which needs long-time training. 2. B49-50 It's almost a two-hands doubling for playing arpeggio-phrase.But not really! You can watch the second phrase- fingering! Your left hand need a smallish shape. Meanwhile, the little finger's head of last phrase need to jump out a minor third distance down. It's very hard to control and also not a doubling. (Hard for playing, but good for sounding, if out. Therefore, dears, have a try like mine...) Yours little uncle Han [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 14:15, 25 November 2024 (UTC) [[File:M. Moszkowsky Etude (Op.72 No.5) played by Jixun Han (Jason) for piano teaching.wav|thumb|M. Moszkowsky Etude (Op.72 No.5) played by Jixun Han (Jason) for piano teaching]] ===== '''Beethoven's Moonlight Sonata''' ===== Indeed, Beethoven left a historical challenge (difficulty) but the continuously creative inspiration to understand the techniques & musicality of all his movements, equally to all people. We can attempt different approaches and own personal life-experiences to understand them, and discuss out some possible results. [[File:Moonlight Sonata - 3rd Movement of Sharp c minor Sonata Beethoven.ogg|thumb|Moonlight Sonata - 3rd Movement of Sharp c minor Sonata after a library presentation of Beethoven (Further, thanks to the April-collections of Tokfo Gallery (Vienna: https://commons.wikimedia.org/wiki/User:Tokfo/Vienna/2019_April_28-30) and Sir James Gallery (Bonn:https://commons.wikimedia.org/wiki/User:Sir_James/Bonn/2019_April_29) - great encouragements!)]] '''3rd Movement- 'Moonlights Storming' - Techniques Analysis from Notation-reading ('Presto agitato' of Breitkopf & Hartel Company and Berlin Arts Collage also compared with Old New York Edition - as the remembrance of one monitor):''' '''Musicality:''' In a grandly general view, it's like...in a crazily running (very fast) race, viewed from the window, moonlights have been dismembered upon deep Lake Lucerne (many fragmental sections composed together). [[File:Vienna Beethoven Monument (with angels and children surrounding).jpg|thumb| Beethoven's Monument in Vienna]] [[File:Beside Beethoven's Musicality.jpg|thumb| A third-person's Watching of Beethoven's Musicality]] For its musicality cultivation, I could give a similar sense of its situation, like in Picasso's works- such as Picasso's Guernica (Ceridwen's Creative Commons Attribution-Share Alike 2.0 Generic license) For achieving it, a little bit of dark-moods anger and sadness faced from the unfairness and out of control could be inputted, after all technique points were trained in the dexterity. Therefore, from emotion to say, I thought the video right after getting back from UK and the lost one in Newcastle central station were better than this time. [[File:3rd movement of Sonata &#039;Moonlight&#039; Rocking Video JMC, Han (Jason).webm|thumb|3rd movement of moonlight sonata; Rocking Video JMC, Han (Jason)]] However, I satisfied with it, right like in life and after the presentation. From this point, we can see: Beethoven, as a piano master, has super-reached too much before the time - even abstractionism and postmodernism (deconstructionism). '''2nd Movement - 'a little Fantasy Moonflower blooming between two rocky layers' - Techniques Analysis from Notation-reading (Allegretto of Breitkopf & Hartel Company and Berlin Arts Collage):''' 1. Parts-distinguishing way can be applied to pick up the main melodic points from its background and legato them into lines. '''( Notice: Here, from the historical observation, a thing needs to be clarified: Baroque-regression (back-reasoning) was usually made by classical composers (in Vienna school: Mozart, Haydn and Beethoven etc.), especially in their later years of life for calming down the dramatical emotions, and keeping Life's Reasonability. Meanwhile, from Haydn, they discussed and created classical counterpoints from symphony and string quartet together, to modify creative inspirations. Beethoven also inherited it. Therefore, when we play some in piano, we need to analyse and apply some special techniques, commonly used in classical polyphony, to pick up the main from the background, sentence by sentence, as an era-responding.)[[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 03:38, 15 May 2019 (UTC)''' 2. Octaves-bridging and chords-connections for big hands, into their hidden melodies, are the most difficulties, which need your frequent exercises, sentence by sentence. (Painful but worthful! Finally, flexible and skillful... ) 3. Long keeping-notes, in certain parts, are important for the continuity of the tune and the texture, without broken. 4. It's better in light and tender keyboard-touching way to make melodic lines clear and 'the little flower' smile lightly. 5. 'Rondo' (ABA) formation can be applied to understand its repetitions, responding and structure. [[File:Moonlight Sonata (Sharp c minor Sonata) 2nd Movement Beethoven JMC,Han.ogg|thumb|Moonlight Sonata - a little fantasy flower between two rocky layers]][[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 14:51, 14 May 2019 (UTC) ===== Blue Danube is always flowing from heart and life, with the vitality as spring: ===== Children and young teenagers, let us swim in this life-long river, to see some beautiful landscapes! [[File:Blue Denube in my heart.jpg|thumb|Blue Denube in my heart]] [[File:Blue Danube (Exercise and variations-collections in piano) JMC, Han.ogg|thumb|Blue Danube (Exercise and variations-collections in piano) JMC, Han]] ===== Pachelbel's Canon is always the canon (polyphonic technique) since Baroque Era, but in '''Modern Piano's Pop Variations''' ===== [[File:Pachelbel&#039;s Canon in Pop Variations (Geoge Winston Notation) Player Jason, Han.ogg|thumb| Pachelbel's Canon in Pop Variations (George Winston Notation) Player Jason, Han]] '''Story of teaching & learning (from Wikimedia Commons):''' Regarding with this piano melody, there is a long story in my heart... Oh, did you hear Mag-pie's singing (I like 'pie' in the tail of this word) in the first draft? Yes, it was attracted and landing on the tree outside my balcony... You can clearly hear it at the beginning and in the tail in my first draft... Almost, it would like to share my memory... Long long ago, my old brother on my mother's side used to be one hero of my life and fashion... On each holiday, he was always able to find great music pieces, MTVs, transcripts , and scientific fictions, from foreign countries, such as American and Japan (Summer)... and brought & shared with me... Then, I attempted my best to exercise them into the reality, which included this song - Canon Variations from pianist George Winston... Those memories have never faded out, but in my deep sea. To now, evenly did I think Canon was from US and a POP song... After seeking the exact information in Wikipedia, I found it's Pachelbel's Canon in D and Baroque Era and German, rather than C and Modern and Pop in American... and with a 'Gigue for Violins and Basso Continuo', it's not only for piano in many parts than our 3 parts in original piano edition. However either, I still like it very much and would like call it American POP in my music world... Then after, a male colleague in my working college said to me: Jason, on my wedding ceremony, I would like to play it for a girl... Could you give me a simple one? Then, searching online, I found a simple (middle level) notation and an original (advanced level) notation, I downloaded both, and chose the simple one for him Three months, he was able to play it from 0 level (he wasn't able to read the notation)... I thought piano would have give him a good memory of wedding... Following, I found a girl felt bored about her piano examination... Then, by choosing the simple transcript and inserting into her lessons... it made my tutoring classrooms really beautiful, relaxable, magical and peaceful... Now, I have time to play the original edition out... One long dream of my heart is going to be fullfilled... Though my hands in several points didn't make my perfectionism satisfied contrasted with before, especially the tenth-cross design between the left hand and the right hand, I knew it's my life, and fate?... I prefered to update its situations for bettering continuously... if having time... Compared with the firstly draft, I thought the second was much down-calmed and peaceful...Somehow, I preferred the first draft, but a little bit of 'fast'... I cannot make the decision...then, kept two. However either, I still felt very happy the little natural friend - mag-pie can join... For this reason, I kept it. Hopefully, you will enjoy... [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 04:50, 16 September 2019 (UTC) ===== KV.265 12 Variations on Ah vous dirai-je, Maman - 'Twinkle Twinkle Little Star' ===== Analysis (Waiting) [[File:KV.265 12 Variations on Ah vous dirai-je, Maman Mozart JMC, Han.ogg|thumb|KV.265 12 Variations on Ah vous dirai-je, Maman Mozart JMC, Han]] [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 08:43, 7 October 2019 (UTC) =====''' 'The beautiful views of Scotti-highlands' from Thompson's Book (Advanced Level) - Teaching and demonstration '''===== [[File:The beautiful views of Scotti-highlands (Teaching demonstration - video; Jason, Han).webm|thumb| After the performance in local Crossing-year 2019-2020 Library Concert and many further exercises, a good edition in classroom got out - Piano kids, always, you knew: No pains, No gains]] For its musicality and academic thoughts, please visit Wikiverty's Portal of Piano education(The Section: The beautiful views of Scotti-highlands' in a Far-land Home (Academic thoughts, musicality, literature-writing and case-realization)): *[[Portal:Green Sleeves (Impressionist Visualization)]] *[[Portal: Sir J.S.BACH and His contributions to Piano Kids' Reasonable Life]] [[File:C minor Prelude Bach (BWV847), Performer JMC, Han.wav|thumb|C minor Prelude Bach (BWV847)]] ==== 'Swan's Dream Upon the lake' - Little Girl 'Wenxin's' (Brilliant & Sweet literatures in arts ) Performance==== '''Teacher's Demonstration:''' [[File:The Dying Swan - black angel JMC Han.ogg|thumb| Musicality from watching 'The Dying Swan - the black angel', performed by JMC Han]] '''Wenxin's Performance:''' 1. Techniques-recovery: The Arpeggio-training of left hand in the accompaniment was the biggest challenge to not only a piano-child at her age - no more than 12 (In Chinese culture, Kid's first year was in mother's womb. Thereby, I asked her - how old are you, and she gave the number '13'...), to me and evenly some expertise pianists. (Camille Saint-Saëns's 'The Swan' on wikipedia or other social editions). The arpeggio-accompaniment is travelling in rich variations of tunes, which caused left hand much harder to expand, shrink and positions-change. Therefore, it spent us more than half a year to train and recover her hand's dexterity from a small failure of her piano life in the Grade Test, just like 'Princess Swan's' experience. Now, totally to say, she got an excellent situation in which children at her age can perform. Thanks to your hard-working! 2. Musicality-cultivation: Usually, she showed a very great musicality in the first page - to the minute (Approximately 1.05) - tender, expending, lyrical and expressive... However, it's really a hardness to keep it throughout the second section - a shading & wandering heart-road in the growth. The attention has to be paid too much on the exactness of left hand's arpeggio-travelling. With a pity, still, some notes were beaten wrongly. But oppositely again, we can see: Princess Swan, in her period of Darkness growth - facing Satan, turning into a dark angel and only appearing in night... She really faced a hardness and the difficulty of life, right as beating wrong notes, getting out some noises and travelling a little bit slowly and roughly in a channel. In this view, perhaps that the difficulties can be transformed- in the musical needs and with a small fashion. Congratulation, more exercises, haha! 3. Together, we gave two great designs: one is the 'Big Brightness' began from the minute (Approximately 2.03) when the main theme happens again; and the other is 'Swan's Departure like Sound of Fall-Leaves rotating upon Lake's Surface' (from minute 2.49 to the end)... She almost achieved some - the mood calmed down very much and stably progressed to further with a confidence. However, a little bit of disfluency made the impression fade, somehow. Meanwhile, a 'rit. to a tempo' turned inversely - what a pity. Totally to say, musicality, at her age, was preciously showing in this time's performance. The hard-working of recovery and exercises, during many classes, touched my heart very much. (I knew that...) More trainings of Arpeggio-running (dominate sevenths) and its fluency can help her achieve more in the future. Wenxin, thanks to you for letting us appreciate this world-famous melody in piano. [[File:Growth of Swan in eyes of the little girl - Wenxi, Zhang.ogg|thumb|Growth of Swan in the eyes of the little girl - Wenxi, Zhang]] [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 16:06, 5 May 2019 (UTC) ==== Listen to Mother's Old Story! - A beautiful and quiet little girl's Good Wish ==== '''Teacher's Demonstration:''' [[File:Mother&#039;s Old story - China Impression.ogg|thumb| Listen to Mother's Old story - China Impression (JMC, Han - teacher's domenstration)]] '''Student's Performance:''' [[File:Listen to Mother&#039;s Old Story - Piano Pupil Yiwen, Cui.ogg|thumb| Listen to Mother's Old Story- Piano Pupil Yiwen, Cui]] '''Teacher's Comments:''' Yiwen, Cui (Direct translation of her Chinese name - A beautiful girl who is good at the translation of art and literature, from Cui family), at the age of 10, is a quite and beautiful girl. She got a good life effect from this Chinese piano-kid's song - 'Listen to Mother's Old Story': making her family and parents happy, getting some confidences through this piano song from the examination, showing her fashion in my library concert held for piano kids... After those more above, frequent exercises, and getting her permission, I can submit this classroom-recording edition. Though in the tail I found a note lost... and some parts of her left hand might run much more fluently... , I think her emotional background of this music reached to a good level, and those polyphonic parts can be clearly heard two layers, their cannoning, and so on... Congratulation! '''(Words from the description in Wikimedia Commons page)''' [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 01:04, 8 August 2019 (UTC) ==== Moonlight upon Lotus Pool (Summer) - Letters-Accompaniment Improvising Chinese Pop-song with both Classical Tradition and Pentatonic Scale ==== '''Teacher's Comment:''' 1. I am very excited that you (only 10-years-old) understood Letter-To-Accompaniment Improvising sheet and its approach in a very fast way. 2. It's great you can use both Pentatonic Arpeggios and Tenth-Rolling-Bass-dropping in your accompaniment (You can make Tenth-rolling Bass in a more fluent view, I thought) 3. We can feel the musical scene from your musicality - In a beautiful summer night, Walking along a lotus pool, you and your family members were enjoying the moonlight and a breeze of cool wind... 4. In future, hopefully, you can improve your 'new learns' to a higher level. [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 01:04, 8 August 2019 (UTC) ''' Xuan, Lee's Second Attempt - Pearl of the Orient''' ==== 'Mother in the candlelight' - A little girl (Siqi)'s heart-words for her mother's birthday - in the growth and in the dreams... ==== '''Heart-story:''' Regarding with this piano-song, there is a little story about this little commons' girl: Usually, her parents were very busy in the family's restaurant... I and my mother saw she had independently managed herself well and grew up alone for many years... In this year - 2019, time was near her mother's birthday. In the KTV (a place like karaoke bar, but for small single groups of people in rooms, with TV in the middle for singing ), she heard this song - 'Mother in the candlelight' and found her parents enjoyed singing it very much. Then, she decided to play it as a gift to her mother, right on mother's birthday. It's my biggest honour to be together sight-reading the notation, making the re-designs and re-editions of this song into piano - like, Prelude, Introduction-theme 1st, theme 2nd, Development and Peak, a small Repetition and Coda... She learnt in a very fast and hard-working way that merely around one month she played it in this level. And finally, she got her heart-sweet - playing it for her mother, as a birthday gift.(Wikimedia Commons' original page, 2019) '''Comments from teacher:''' 1. The musical emotions were very rich and expressive, especially the 'Peak-Calling for mother' (2.53 minutes - 3.53 minutes). I almost can hear 'Mum...' (or Mumu...) for many times in a kid's tear-drops and in the candlelight... by your right hand's touchable singing... 2. I liked our 'Flanger tr. Ornaments' very much (I thought it's from Mozartian). I am very happy you can put it in for soon time... 3. I am very happy in the Coda-tail, you can get my suggestion - ending by a Major Seventh Progression-Arpeggio. This point should give the thanks to my mentor - Ray. I quite enjoy its special colour... 4. Your strong and mixed left hand accompaniment must have been trained for many times. I knew it's a hard-working job, but tender and flexible a little bit... better? 5. The singing of right hand and its 'breathing' were quite natural and fine, sentence by sentence..., but the total speed is too slower than normal, which reflect the running ability of the left hand needed to improve. I knew: to your 9-years-old hands, it's a very hard requirement... However, waiting the up-grown, I have the confidence you can hands-sing it in a much more fluent way... [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 14:10, 5 September 2019 (UTC) '''Night's Piano-Song - The depth of China Pop-Piano (and Siqi's Heart-Try)''' ''Comments:'' She made the depth of two peak notes but in light touching way. I thought also that she did this Pop-piano's musicality in a poet's Night-thinking... She is suitable for the performance of this-type-'songs' and improvising (still at little age and need to prepare in time). Let us listen to her and feel her expression. ==== The Cowherd's Flute played by a little girl piano pupil - Guo Guo (Nickname: Happy fruits) ==== This edition has already been her best attempt, regarding with its landscape-painting style, lovely Cowherd's Flute we can refer to Wikipedia introduction (Seeking key words 'The Cowboy's Flute' in). It's recorded as a beautiful memory of her piano-learning and her Childhood. Let us listen to her: [[File:The Cowboy&#039;s Flute - Yuxuan, Lu (Guo Guo).ogg|thumb|The Cowherd's Flute - Yuxuan, Lu (Guo Guo)]] ==== Clementi Sonata Op35 No. 5 (Movement 1st), played by Piano pupil Yixuan, Qiao ==== Clementi's Sonata-Op35 No. 5 was a so long and difficult piece for students around their ninth year. Therefore, we have divided it into many small sections and taught. She learnt in progress. Meanwhile, this little and beautiful girl (She was beautifully good at dancing, somehow rather than piano.) has already attempted her best in exercising and recording. I thought it recorded her good piano-learning experiences and those memories of childhood. Regarding with further information about this work, please refer to the educational portal: https://en.wikiversity.org/wiki/Portal:Sonatinas_from_Kids%27_corner_near_heaven#Muzio_Clementi . Let us listen to her: [[File:Clementi Sonatina Op35. No 5 Movement 1st Piano pupil YIxuan, Qiao.ogg|thumb|Clementi Sonatina Op35. No 5 Movement 1st Piano pupil YIxuan, Qiao]] [[User:Jason M. C., Han|Jason M. C., Han]] ([[User talk:Jason M. C., Han|discuss]] • [[Special:Contributions/Jason M. C., Han|contribs]]) 02:42, 18 October 2019 (UTC) ====Little and Little, Twinkling Stars - Little Piano-Kids' Playground==== Say 'Hello' to our 'Little Goldman' [[File:Vienna Trip - The Little Goldman- Strauss Family.jpg|thumb|Hand-making my own picture of Strauss Family's Little Goldman]] * [[Portal:Little and Little, Twinkling Stars - Little Piano-Kids' Playground| Little and Little, Twinkling Stars - Little Piano-Kids' Playground]] rp55p1ja7tqwaqxue1mry9nf11ahcbq 10 242359 2690310 2636848 2024-12-04T21:19:23Z OhanaUnited 18921 clarifying the electorate due to confusion over who's eligible to vote 2690310 wikitext text/x-wiki <div style=" min-height:50px; padding:5px 15px; background-color:#EEEEFF;color:#202122;"> '''This is a page to discuss the proposed current applications to join the [[{{ROOTPAGENAME}}/Editorial_board|editorial board]] or as an [[{{ROOTPAGENAME}}/Associate_editors|associate editor]]''' {{#ifeq:{{ROOTPAGENAME}}|WikiJournal User Group |'''of a journal in the [[{{ROOTPAGENAME}}]].''' |'''of the ''[[{{ROOTPAGENAME}}]]''.'''<br>To see open applications across all WikiJournals, see '''[[Talk:WikiJournal User Group/Editors|this page]]'''. }} </div> {{{archive box|}}} {{TOC right}} {{#ifeq:{{ROOTPAGENAME}}|WikiJournal User Group |To submit an application, or view an archive of all precious applications, see the individual journal pages at:<br> [[Talk:WikiJournal_of_Medicine/Editors|WikiJMed]], [[Talk:WikiJournal_of_Science/Editors|WikiJSci]], and [[Talk:WikiJournal_of_Humanities/Editors|WikiJHum]]. |New applications should be made below. Please see the [[{{ROOTPAGENAME}}/Bylaws#ARTICLE_III_-_EDITORIAL_BOARD|relevant bylaws]], [[{{ROOTPAGENAME}}/Ethics statement#Duties_of_the_editorial_board|ethics requirements for editorial board members]], and [[{{ROOTPAGENAME}}/Ethics statement#Duties_of_the_associate_editors|ethics requirements for associate editors]] for eligibility, powers, duties and voting. {{Clickable button 2 |Apply to the editorial Board |url=https://en.wikiversity.org/w/index.php?title=Talk:{{ROOTPAGENAMEE}}/Editors&action=edit&section=new&preload={{ROOTPAGENAMEE}}%2FEditorial_board%2FApplication&summary=Editorial+board+application |class=mw-ui-progressive }} {{Clickable button 2 |Apply to be an associate editor |url=https://en.wikiversity.org/w/index.php?title=Talk:{{ROOTPAGENAMEE}}/Editors&action=edit&section=new&preload={{ROOTPAGENAMEE}}%2FAssociate_editors%2FApplication&summary=Associate+editor+application |class=mw-ui-progressive }} }} As per the [[{{ROOTPAGENAME}}/Bylaws|journal bylaws]], an editorial board member or associate editor is elected after a majority vote, counting after 7 days of voting, by the following electorate: *Current editorial board members, or *Peer reviewers who have completed at least one peer review of {{ROOTPAGENAME}}, or *Editors with at least 30 edits to {{ROOTPAGENAME}} pages, and *Not an individual voting for herself/himself. After fulfillment of the above, any current editorial board member can execute the acceptance, by copy-pasting <code><nowiki>{{subst:</nowiki>[[Template:WikiJournal accepted board member|WikiJournal accepted board member]]<nowiki>}}</nowiki></code> or <code><nowiki>{{subst:</nowiki>[[Template:WikiJournal accepted associate editor|WikiJournal accepted associate editor]]<nowiki>}}</nowiki></code> underneath the votes. or8cw17m205s5pco1gi8hj69gqsqutk 0 268969 2690268 2685482 2024-12-04T14:48:05Z Lbeaumont 278565 Added the German tank problem 2690268 wikitext text/x-wiki This is a collection of math-oriented activities, games, and puzzles that are fun and instructive. Math adventure can help answer the question, [[Why study math?|Why study math]]? *[[/Roll the dice/]] *[[/Wheat and the Chessboard/]] **Boo who? *Same Birthdays *The area of a circle *[[/Fibonacci and the Golden Ratio/]] *[[/Triangular numbers/]] *[[/Pascal's triangle in wiki-latex/]] *[[/Palindromes/]] *[[/Peanut Butter Power/]] *Seven Bridges of Königsberg *Pythagorean Theorem *[[/Square Roots using Newton’s Method/]] *[[/Volume of a rotating rectangle/]] *[[/Tetrahedron in a Cube/]] *[[/Multiplying Negative Numbers/]] *Binary Numbers *Decoding mighty things *Benford's law *Prime Numbers *The square root of 2 is irrational *Braess's paradox *Arithmetic with [[w:Kaktovik_numerals|Kaktovik]] numerals *Logistic map *Feigenbaum constants *Three-body problem *First 10-digit prime found in consecutive digits of e *The Pigeonhole Principle<ref>{{Cite web|url=https://www.cantorsparadise.com/the-pigeonhole-principle-and-its-surprisingly-powerful-applications-531d8d7539ce|title=The Pigeonhole Principle and Its Surprisingly Powerful Applications|last=Müller|first=Kasper|date=2024-02-02|website=Medium|language=en|access-date=2024-02-04}}</ref> * [[/Find the Light Gumball/]] * The Peano axioms * [[/Triangle Test Cases/]] * [[/German Tank Problem/]] {{CourseCat}} [[Category:Mathematics/Activities]] f933isszvz0y0szgvr3lep4ziwzo9il 0 271408 2690277 2687832 2024-12-04T15:42:54Z 82.219.7.128 /* Prokaryotic (bacterial) Cell */ 2690277 wikitext text/x-wiki Cells are the basis of all life on Earth, producing every living organism, from a single cellular bacteria, through to trillions of cells making up a human. All living species on Earth have cells. == Sub-Cellular Structures == === Animal Cells === [[File:Endomembrane system diagram en.svg|thumb|A diagram showing the organelles of an animal cell]] Nucleus — contains the cells' genetic material. Visible as an opaque area through a light microscope. The nucleus controls all activities in the cell. Cell membrane — the membrane encasing the cell. Flexible, allowing movement. Ensures the cell stays together Mitochondria — release energy, through respiration, typically from glucose. Allows the cell to carry out metabolic functions Ribosomes — carry out protein synthesis and the site of DNA transcription === Plant Cell === [[File:Plant cell structure-en.svg|thumb|A diagram showing organelles of a plant cell]] Nucleus — contains the cell's genetic material. Visible as an opaque area through a light microscope Cell membrane — the membrane encasing the cell. Flexible, allowing movement. Ensures the cell's organelles do not spill out. Cell wall — a strong layer of cellulose and other polysaccharides, making the cell rigid. Ensures that plants can stand upright Chloroplasts — allow sunlight, water, and carbon dioxide to be converted to glucose and oxygen. Provide energy for the cell through photosynthesis. Mitochondria — release energy, through respiration, typically from glucose (made by photosynthesis). Allows the cell to carry out metabolic functions Vacuole — a fluid-filled sac within the cell containing cell sap, mineral ions, and other chemicals essential for the cell to function Ribosomes — carry out protein synthesis and the site of DNA transcription === Prokaryotic (bacterial) Cell === Prokaryotic cells do not have a nucleus, unlike eukaryotic cells which have a nucleus.[[File:Prokaryote cell.svg|thumb|A diagram showing the organelles of a prokaryotic cell]] Chromosomal DNA — DNA arranged in chromosomes. Contains most of a prokaryotic cells’ genetic material Plasmid DNA — small rings of DNA within the cytoplasm of the cell. Allows bacteria to share genetic material throughout a population Cell membrane — the membrane encasing the cell. Flexible, allowing movement. Ensures the cells organelles do not spill out. Ribosomes — carry out protein synthesis and are the site of DNA transcription Flagella — whip-like structures on bacteria, allowing them to move == Some Cells are Specialised to their Function == === Sperm Cells === Sperm cells are the male gamete, used during sexual reproduction. [[File:Prefertilization Sperm cell.PNG|thumb|A diagram showing a sperm cell]] Acrosome — the region in the head of the sperm cell containing digestive enzymes. Used to ‘burrow’ through the cell membrane of the egg during fertilisation Haploid nucleus — the nucleus of a sperm cell contains 23 chromosomes, as it is made by the process of meiosis. This creates haploid cells, which have ½ the number of chromosomes of the parent cell. Mitochondria — sperm cells contain many mitochondrion, to release energy to move the flagellum. These are located in the mid piece Tail — sperm cells have a tail (flagellum), a whip-like structure, allowing the cell to propel itself during fertilisation === Egg Cells === Egg cells are the female gamete, used during sexual reproduction. [[File:Egg cell fertilization - Zygote.png|thumb|A diagram showing the fertilisation of an egg cell with a sperm cell]] Nutrients in the Cytoplasm — egg cells contain many nutrients in the cytoplasm to provide sufficient nutrition should the egg be fertilised. Allows the cell division and specialisation immediately after fertilisation to begin Haploid nucleus — the nucleus of an egg cell contains 23 chromosomes (along with the sperm cell), as it is made through the process of meiosis Changes in the Cell Membrane after Fertilisation — when an egg cell is fertilised with a sperm cell, the cell membrane changes, releasing chemicals that prevent any further sperm from fertilising the cell === Ciliated Epithelial Cell === [[File:Bronchiolar epithelium 4 - SEM.jpg|thumb|A scanning electron micrograph of respiratory epithelial cells]] Ciliated epithelial cells have many, small, hair-like projections coming out of their cell membrane. These projections are used to ‘waft’ waste products out of the trachea (wind pipe) and bronchioles. Mucus is transported out of the lungs with cilia, allowing it to enter the stomach, where it is digested. These cells have a large proportion of mitochondria, as they use active methods (energy using) to ‘waft’ their hair-like projections. === Corona radiarta === The corona radiarta is a layer of cells covering an unfertilised egg or egg cell. During fertilisation, sperm must push through the layer to reach the outer layer of the egg, the zona pellucida.<ref>Information given by science based company Nucleus.</ref> == Cell death == === Cell decomposition === When somebody has died, the cells in the human body stop receiving oxygen, and continue to function only for a few minutes. [[Carbon dioxide]] begins to build up, rupturing sacs inside the cells. The sacs contain enzymes that begin to digest the cells from the inside out. Cell death also occurs because of Necrosis, along with the work of bacteria in the body. A third example of cell death: The cell membranes break, releasing the inside fluids into the surrounding tissue.<ref>Information given by scientific company Seeker, also known as D News.</ref> === Original explanation of cell death === A person’s cells die every day, but the body is protecting itself. In a process called apoptosis, cells commit controlled suicide, if infected by viruses or harmful cancerous mutations. == References == [[Category:Biology]] 7nm9cn7kfx58hozkmfwxj2nwnrjqxqn 0 285380 2690270 2690085 2024-12-04T14:53:53Z Young1lim 21186 /* Applications */ 2690270 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20241204.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> jom25i0hjhjn3tva6mempvpp7ytipuw 0 285723 2690287 2689994 2024-12-04T17:44:46Z Scogdill 1331941 2690287 wikitext text/x-wiki Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. The definitions on this page are to provide a sense of what someone in the late 19th century might have meant by the term rather than a definition of what we might mean by it today. In the absence of a specialized glossary from the end of the 19th century in the U.K., we use the ''Oxford English Dictionary'' because the senses of a word are illustrated with examples that have dates so we can be sure that the senses we pick are appropriate for when they are used in the quotations we have. We also sometimes use the French ''Wikipédia'' to define a word because many technical terms of fashion were borrowings from the French. Also, often the French ''Wikipédia'' provides historical context for the uses of a word similar to the way the OED does. == Articles or Parts of Clothing: Non-gender-specific == === Mantle, Cloak, Cape === In 19th-century newspaper accounts, these terms are sometimes used without precision as synonyms. These are all outer garments. '''Mantle''' A mantle — often a long outer garment — might have elements like a train, sleeves, collars, revers, fur, and a cape. A late-19th-century writer making a distinction between a mantle and a cloak might use ''mantle'' if the garment is more voluminous. '''Cloak''' '''Cape''' === Peplum === According to the French ''Wiktionnaire'', a peplum is a "Short skirt or flared flounce layered at the waist of a jacket, blouse or dress" [translation by Google Translate].<ref>{{Cite journal|date=2021-07-02|title=péplum|url=https://fr.wiktionary.org/w/index.php?title=p%C3%A9plum&oldid=29547727|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/p%C3%A9plum.</ref> The ''Oxford English Dictionary'' has a fuller definition, although, it focuses on women's clothing because the sense is written for the present day:<blockquote>''Fashion''. ... a kind of overskirt resembling the ancient peplos (''obsolete''). Hence (now usually) in modern use: a short flared, gathered, or pleated strip of fabric attached at the waist of a woman's jacket, dress, or blouse to create a hanging frill or flounce.<ref name=":5">“peplum, n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1832614702>.</ref></blockquote>Men haven't worn peplums since the 18th century, except when wearing costumes based on historical portraits. The ''Daily News'' reported in 1896 that peplums had been revived as a fashion item for women.<ref name=":5" /> === Revers === According to the ''Oxford English Dictionary'', ''revers'' are the "edge[s] of a garment turned back to reveal the undersurface (often at the lapel or cuff) (chiefly in ''plural''); the material covering such an edge."<ref>"revers, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/164777. Accessed 17 April 2023.</ref> The term is French and was used this way in the 19th century (according to the ''Wiktionnaire'').<ref>{{Cite journal|date=2023-03-07|title=revers|url=https://fr.wiktionary.org/w/index.php?title=revers&oldid=31706560|journal=Wiktionnaire|language=fr}} https://fr.wiktionary.org/wiki/revers.</ref> == Articles or Parts of Clothing: Men's == [[Social Victorians/Terminology#Military|Men's military uniforms]] are discussed below. === À la Romaine === [[File:Johann Baptist Straub - Mars um 1772-1.jpg|thumb|left|alt=Old and damaged marble statue of a Roman god of war with flowing cloak, big helmet with a plume on top, and armor|Johann Baptist Straub's 1772 ''à la romaine'' ''Mars'']] A few people who attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball in 1897]] personated Roman gods or people. They were dressed not as Romans, however, but ''à la romaine'', which was a standardized style of depicting Roman figures that was used in paintings, sculpture and the theatre for historical dress from the 17th until the 20th century. The codification of the style was developed in France in the 17th century for theatre and ballet, when it became popular for masked balls. Women as well as men could be dressed ''à la romaine'', but much sculpture, portraiture and theatre offered opportunities for men to dress in Roman style — with armor and helmets — and so it was most common for men. In large part because of the codification of the style as well as the painting and sculpture, the style persisted and remained influential into the 20th century and can be found in museums and galleries and on monuments. For example, Johann Baptist Straub's 1772 statue of Mars (left), now in the Bayerisches Nationalmuseum, Munich, missing part of an arm, shows Mars ''à la romaine''. In London, an early 17th-century example of a figure of Mars ''à la romaine'', with a helmet, '''was''' "at the foot of the Buckingham tomb in Henry VII's Chapel at Westminster Abbey."<ref>Webb, Geoffrey. “Notes on Hubert Le Sueur-II.” ''The Burlington Magazine for Connoisseurs'' 52, no. 299 (1928): 81–89. http://www.jstor.org/stable/863535.</ref>{{rp|81, Col. 2c}} === Cavalier === [[File:Sir-Anthony-van-Dyck-Lord-John-Stuart-and-His-Brother-Lord-Bernard-Stuart.jpg|thumb|alt=Old painting of 2 men flamboyantly and stylishly dressed in colorful silk, with white lace, high-heeled boots and long hair|Van Dyck's c. 1638 painting of cavaliers Lord John Stuart and his brother Lord Bernard Stuart]] As a signifier in the form of clothing of a royalist political and social ideology begun in France in the early 17th century, the cavalier established France as the leader in fashion and taste. Adopted by [[Social Victorians/Terminology#Military|wealthy royalist British military officers]] during the time of the Restoration, the style signified a political and social position, both because of the loyalty to Charles I and II as well the wealth required to achieve the cavalier look. The style spread beyond the political, however, to become associated generally with dress as well as a style of poetry.<ref>{{Cite journal|date=2023-04-25|title=Cavalier poet|url=https://en.wikipedia.org/w/index.php?title=Cavalier_poet&oldid=1151690299|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier_poet.</ref> Van Dyck's 1638 painting of two brothers (right) emphasizes the cavalier style of dress. === Coats === ==== Doublet ==== * In the 19th-century newspaper accounts we have seen that use this word, doublet seems always to refer to a garment worn by a man, but historically women may have worn doublets. In fact, a doublet worn by Queen Elizabeth I exists and '''is somewhere'''. * Technically doublets were long sleeved, although we cannot be certain what this or that Victorian tailor would have done for a costume. For example, the [[Social Victorians/People/Spencer Compton Cavendish#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Duke of Devonshire's costume as Charles V]] shows long sleeves that may be part of the surcoat but should be the long sleeves of the doublet. ==== Pourpoint ==== A padded doublet worn under armor to protect the warrior from the metal chafing. A pourpoint could also be worn without the armor. ==== Surcoat ==== Sometimes just called ''coat''. [[File:Oscar Wilde by Sarony 1882 18.jpg|thumb|alt=Old photograph of a young man wearing a velvet jacket, knee breeches, silk hose and shiny pointed shoes with bows, seated on a sofa and leaning on his left hand and holding a book in his right| Oscar Wilde, 1882, by Napoleon Sarony]] === Hose, Stockings and Tights === Newspaper accounts from the late 19th century of men's clothing use the term ''hose'' for what we might call stockings or tights. In fact, the terminology is specific. ''Stockings'' is the more general term and could refer to hose or tights. With knee breeches men wore hose, which ended above the knee, and women wore hose under their dresses. The ''Oxford English Dictionary'' defines tights as "Tight-fitting breeches, worn by men in the 18th and early 19th centuries, and still forming part of court-dress."<ref>“Tights, N.” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2693287467.</ref> By 1897, the term was in use for women's stockings, which may have come up only to the knee. Tights were also worn by dancers and acrobats. This general sense of ''tights'' does not assume that they were knitted. ''Clocking'' is decorative embroidery on hose, usually, at the ankles on either the inside or the outside of the leg. It started at the ankle and went up the leg, sometimes as far as the knee. On women's hose, the clocking could be quite colorful and elaborate, while the clocking on men's hose was more inconspicuous. In many photographs men's hose are wrinkled, especially at the ankles and the knees, because they were shaped from woven fabric. Silk hose were knitted instead of woven, which gave them elasticity and reduced the wrinkling. The famous Sarony carte de visite photograph of Oscar Wilde (right) shows him in 1882 wearing knee breeches and silk hose, which are shiny and quite smoothly fitted although they show a few wrinkles at the ankles and knees. In the portraits of people in costume at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the men's hose are sometimes quite smooth, which means they were made of knitted silk and may have been smoothed for the portrait. In painted portraits the hose are almost always depicted as smooth, part of the artist's improvement of the appearance of the subject. === Shoes and Boots === == Articles or Parts of Clothing: Women's == === '''Chérusque''' === According to the French ''Wikipedia'', ''chérusque'' is a 19th-century term for the kind of standing collar like the ones worn by ladies in the Renaissance.<ref>{{Cite journal|date=2021-06-26|title=Collerette (costume)|url=https://fr.wikipedia.org/w/index.php?title=Collerette_(costume)&oldid=184136746|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Collerette_(costume)#Au+xixe+siècle+:+la+Chérusque.</ref> === Corsage === According to the ''Oxford English Dictionary'', the corsage is the "'body' of a woman's dress; a bodice."<ref>"corsage, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/42056. Accessed 7 February 2023.</ref> This sense is well documented in the ''OED'' for the mid and late 19th-century, used this way in fiction as well as in a publication like ''Godey's Lady's Book'', which would be expected to use appropriate terminology associated with fashion and dress making. The sense of "a bouquet worn on the bodice" is, according to the ''OED'', American. === Décolletage === === Girdle === === Mancheron === According to the ''Oxford English Dictionary'', a ''mancheron'' is a "historical" word for "A piece of trimming on the upper part of a sleeve on a woman's dress."<ref>"mancheron, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/113251. Accessed 17 April 2023.</ref> At the present, in French, a ''mancheron'' is a cap sleeve "cut directly on the bodice."<ref>{{Cite journal|date=2022-11-28|title=Manche (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manche_(v%C3%AAtement)&oldid=199054843|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Manche_(v%C3%AAtement).</ref> === Petticoat === According to the ''O.E.D.'', a petticoat is a <blockquote>skirt, as distinguished from a bodice, worn either externally or showing beneath a dress as part of the costume (often trimmed or ornamented); an outer skirt; a decorative underskirt. Frequently in ''plural'': a woman's or girl's upper skirts and underskirts collectively. Now ''archaic'' or ''historical''.<ref>“petticoat, n., sense 2.b”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1021034245></ref> </blockquote>This sense is, according to the ''O.E.D.'', "The usual sense between the 17th and 19th centuries." However, while petticoats belong in both outer- and undergarments — that is, meant to be seen or hidden, like underwear — they were always under another garment, for example, underneath an open overskirt. The primary sense seems to have shifted through the 19th century so that, by the end, petticoats were underwear and the term ''underskirt'' was used to describe what showed under an open overskirt. === Stomacher === According to the ''O.E.D.'', a stomacher is "An ornamental covering for the chest (often covered with jewels) worn by women under the lacing of the bodice,"<ref>“stomacher, n.¹, sense 3.a”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1169498955></ref> although by the end of the 19th century, the bodice did not often have visible laces. Some stomachers were so decorated that they were thought of as part of the jewelry. === Train === A train is The Length of the Train '''For the monarch [or a royal?]''' According to Debrett's,<blockquote>A peeress's coronation robe is a long-trained crimson velvet mantle, edged with miniver pure, with a miniver pure cape. The length of the train varies with the rank of the wearer: * Duchess: for rows of ermine; train to be six feet * Marchioness: three and a half rows of ermine; train to be three and three-quarters feet * Countess: three rows of ermine; train to be three and a half feet * Viscountess: two and a half rows of ermine; train to be three and a quarter feet * Baroness: two rows of ermine; train to be three feet<ref name=":2">{{Cite web|url=https://debretts.com/royal-family/dress-codes/|title=Dress Codes|website=debretts.com|language=en-US|access-date=2023-07-27}} https://debretts.com/royal-family/dress-codes/.</ref> </blockquote>The pattern on the coronet worn was also quite specific, similar but not exactly the same for peers and peeresses. Debrett's also distinguishes between coronets and tiaras, which were classified more like jewelry, which was regulated only in very general terms. Peeresses put on their coronets after the Queen or Queen Consort has been crowned. ['''peers?'''] === Foundation Garments === Unlike undergarments, Victorian women's foundation garments created the distinctive silhouette. Victorian undergarments included the chemise, the bloomers, the corset cover — articles that are not structural. The corset was an important element of the understructure of foundation garments — hoops, bustles, petticoats and so on — but it has never been the only important element. === Corset === [[File:Corset - MET 1972.209.49a, b.jpg|thumb|alt=Photograph of an old silk corset on a mannequin, showing the closure down the front, similar to a button, and channels in the fabric for the boning. It is wider at the top and bottom, creating smooth curves from the bust to the compressed waist to the hips, with a long point below the waist in front.|French 1890s corset, now in the Metropolitan Museum of Art, NYC]] The understructure of Victorian women's clothing is what makes the costumes worn by the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] so distinctly Victorian in appearance. An example of a corset that has the kind of structure often worn by fashionably dressed women in 1897 is the one at right. This corset exaggerated the shape of the women's bodies and made possible a bodice that looked and was fitted in the way that is so distinctive of the time — very controlled and smooth. And, as a structural element, this foundation garment carried the weight of all those layers and all that fabric and decoration on the gowns, trains and mantles. (The trains and mantles could be attached directly to the corset itself.) * This foundation emphasizes the waist and the bust in particular, in part because of the contrast between the very small waist and the rounded fullness of the bust and hips. * The idealized waist is defined by its small span and the sexualizing point at the center-bottom of the bodice, which directs the eye downwards. Interestingly, the pointed waistline worn by Elizabethan men has become level in the Victorian age. Highly fashionable Victorian women wearing the traditional style, however, had extremely pointed waists. * The busk (a kind of boning in the front of a corset that is less flexible than the rest) smoothed the bodice, flattened the abdomen and prevented the point on the bodice from curling up. * The sharp definition of the waist was caused by ** length of the corset (especially on the sides) ** the stiffness of the boning ** the layers of fabric ** the lacing (especially if the woman used tightlacing) ** the over-all shape, which was so much wider at the top and the bottom ** the contrast between the waist and the wider top and bottom * The late-19th-century corset was long, ending below the waist even on the sides and back. * The boning and the top edge of the late 19th-century fashion corset pushed up the bust, rounding (rather than flattening, as in earlier styles) the breasts, drawing attention to their exposed curves and creating cleavage. * The exaggerated bust was larger than the hips, whenever possible, an impression reinforced by the A-line of the skirt and the inverted Vs in the decorative trim near the waist and on the skirt. * This corset made the bodice very smooth with a very precise fit, that had no wrinkles, folds or loose drapery. The bodice was also trimmed or decorated, but the base was always a smooth bodice. More formal gowns would still have the fitted bodice and more elaborate trim made from lace, embroidery, appliqué, beading and possibly even jewels. The advantages and disadvantages of corseting and especially tight lacing were the subject of thousands of articles and opinions in the periodical press for a great part of the century, but the fetishistic and politicized tight lacing was practiced by very few women. And no single approach to corsetry was practiced by all women all the time. Most of the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 ball]] were not tightly laced, but the progressive style does not dominate either, even though all the costumes are technically historical dress. Part of what gives most of the costumes their distinctive 19th-century "look" is the more traditional corset beneath them. Even though this highly fashionable look was widely present in the historical costumes at the ball, some women's waists were obviously very small and others were hardly '''emphasized''' at all. Women's waists are never mentioned in the newspaper coverage of the ball — or, indeed, of any of the social events attended by the network at the ball — so it is only in photographs that we can see the effects of how they used their corsets. === Hoops === Striking for how long they lasted and '''how much''' they evolved, hoops were the foundation undergarment for a skirt and petticoat. Women wore hoops from the '''15th century''', around the time of Katherine of Aragon, through the bustle of the late 19th century. The cage caused the silhouette of skirts to change shape over time and enabled the extreme distortions of panniers and the bustle. Hoops circled the body symmetrically in a cone or drum shape, were moved to the sides with panniers, ballooned around the body like the top half of a sphere, and were pulled to the rear with a bustle. That is, like corsets, the distorted shapes of high fashion were made possible by hoops. High fashion demanded these shapes, which disguised women's bodies, '''especially below the waist'''. Besides the shape, the structure used to construct hoops changed — cane, wood, whalebone, steel or wire. Add fabric structural stuff ''Hoops'' is a mid-19th-century term for a cage-like structure worn under a skirt to hold it away from the body. ==== 15th Century ==== [[File:Pedro García de Benabarre St John Retable Detail.jpg|thumb|Pedro García de Benabarre St John Retable Detail]] Hoops began in Spain in the 15th century and influenced European fashion for many years:<blockquote>Katherine of Aragon is reputed to have introduced the Spanish farthingale (a hooped underskirt) into England early in the century. The result was to convert the columnar skirt of the fifteenth century into the cone shape of the sixteenth. ...<p> Spanish influence had introduced the hoop-supported skirt, smooth in contour, which was quite generally worn.<ref name=":11" /> (291)</blockquote>Illustrations ==== 16th Century ==== [[File:Farthingale 2 (PSF).png|thumb|Farthingale 2 (PSF)]] [[File:Alonso Sánchez Coello 011.jpg|thumb|Alonso Sánchez Coello 011]] In the 16th century, the garment we call ''hoops'' was called a farthingale. ''Farthingale'' is the term in English; in French, it's ''vertugadin'', and in Spanish ''vertugado''. ''Vertugadin'' is a French term for ''farthingale'', a cage made of hoops supporting a skirt — "un élément essentiel de la mode Tudor en Angleterre [an essential element of Tudor fashion in England]."<ref name=":0">{{Cite journal|date=2022-03-12|title=Vertugadin|url=https://fr.wikipedia.org/w/index.php?title=Vertugadin&oldid=191825729|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Vertugadin.</ref> In fact, "The Spanish princess Catherine of Aragon brought the fashion to England for her marriage to Prince Arthur, eldest son of Henry VII in 1501 [La princesse espagnole Catherine d'Aragon amena la mode en Angleterre pour son mariage avec le prince Arthur, fils aîné d'Henri VII en 1501]."<ref name=":0" /> Catherine of Aragon, of course, married Henry VIII after Arthur's death. The French and Spanish farthingales were not identical by the end of the 16th century. The Spanish farthingale shaped the skirt into an A-line with a graduated series of hoops sewn to an undergarment. The French farthingale was a flattish "cartwheel" or platter of hoops worn below the waist and above the hips held the skirt out more or less horizontally. Once past the vertugadin, the skirt then fell straight to the floor, shaping the skirt into a kind of drum. The shoes show in the portraits of women wearing the French farthingale.<blockquote>"FARTHINGALE:  Renaissance (1450-1550 C.E. to Elizabethan (1550-1625 C.E.). Linen underskirt with wire supports which, when shaped, produced a variety of dome, bell, and oblong shapes."<ref name=":7" /> (105)</blockquote>By the 18th century, it was called hoops, which were made of wood. ==== 17th Century ==== Generally, panniers were a kind of undergarment worn in the 17th and 18th centuries. Their design evolved during the century. Made of hoops of wood, they were "baskets" or cages worn on either side of the waist to broaden the skirts to the sides. Illustration ==== 18th Century ==== Blanche Payne outlines the evolution of hoops, and thus the shape of the skirt, in the 18th century:<blockquote>SKIRT FASHIONS. Since skirts experienced the greatest alterations, a brief summary of the successive silhouettes should help to place individual costumes in their proper niches. Six basic forms appeared during the century, in the following order: # The bustle was a continuation of the 1690 mode. # The bell or dome shape resulted from the reintroduction of hoops; in England by 1710, in France by 1720. # The ellipse, the second phase of the hoop skirt, was achieved by broadening the support from side to side and compressing it from front to back. It had a long run of popularity, from 1740 to 1770, the extreme width being retained in court costumes. In France it persisted until the revolution, except that skirts were allowed to curve outward in [the] back again. English court costume followed this fashion well into the nineteenth century. # The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, which were used to form the three great ‘poufs’ known as the polonaise (see Fig. 452). These diversions appeared in the late sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period. # The return of the bustle in the 1780s. # The tubular form, drawn from classic art, in the 1790s.<ref name=":11" /> (411–413) </blockquote>While we think of the bustle as a 19th-century look, it can be found in the 18th century, as Payne says. Polonaise may need its own section. ==== 19th Century ==== [[File:Cutaway sketch of crinoline.gif|thumb|Cutaway sketch of crinoline]] [[File:Paris voulant englober la banlieue.JPG|thumb|Paris voulant englober la banlieue]] [[File:Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723.jpg|thumb|Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723]] In the 19th century, the hoops were made of wire and became lighter. By the 1860s, hoops made for huge round skirts. In Laura Ingalls Wilder's 1941 ''Little Town on the Prairie'', the '''16-year-old Laura wears hoops. Laura Ingalls Wilder and thus the character Laura were born in 1862, so this moment is set in 1883'''.<ref>Hill, Pamela Smith, ed. ''Pioneer Girl: The Annotated Autobiography''.</ref> She makes the comment that she wants to be in style, but that would be on the prairie and not necessarily the latest Parisian style.<blockquote>“Well,” Laura began; then she stopped and spun round and round, for the strong wind blowing against her always made the wires of her hoop skirt creep slowly upward under her skirts until they bunched around her knees. Then she must whirl around and around until the wires shook loose and spiraled down to the bottom of her skirts where they should be. “As she and Carrie hurried '''ob''' she began again. “I think it was silly, the way they dressed when Ma was a girl, don’t you? Drat this wind!” she exclaimed as the hoops began creeping upward again. “Quietly Carrie stood by while Laura whirled. “I’m glad I’m not old enough to have to wear hoops,” she said. “They’d make me dizzy.” “They are rather a nuisance,” Laura admitted. “But they are stylish, and when you’re my age you’ll want to be in style.”<ref>Wilder, Laura Ingalls. ''Little Town on the Prairie.'' Harper and Row, 1941. Pp. 272–273.</ref></blockquote>This very unusual description of the way the wind could make hoops creep and the solution. It must have been happening to other women wearing hoops at the time.[[File:Panniers 1.jpg|thumb|alt=Photograph of the wooden and fabric skeleton of an 18th-century women's foundation garment|Panniers 1]] == '''Traditional vs Progressive Style''' == === Progressive Style === The terms ''artistic dress'' and ''aesthetic dress'' are not synonymous and were in use at different times to refer to different groups of people in different contexts, but we recognize them as referring to a similar kind of personal style in clothing, a style we call progressive dress or the progressive style. Used in a very precise way, ''artistic dress'' is associated with the Pre-Raphaelite artists and the women in their circle beginning in the 1860s. Similarly, ''aesthetic dress'' is associated with the 1880s and 1890s and dress reform movements. In general, the progressive style is characterized by its resistance to the highly structured fashion of its day, especially corseting, aniline dyes and an extremely close fit. === Traditional Style === By the end of the century designs from the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]] (or Maison Worth) define what we think of as the traditional Victorian look, which was very stylish and expensive. Blanche Payne describes an example of the 1895 "high style" in a gown by Worth with "the idiosyncrasies of the [1890s] full blown":<blockquote>The dress is white silk with wine-red stripes. Sleeves, collars, bows, bag, hat, and hem border match the stripes. The sleeve has reached its maximum volume; the bosom full and emphasized with added lace; the waistline is elongated, pointed, and laced to the point of distress; the skirt is smooth over the hips, gradually swinging out to sweep the floor. This is the much vaunted hourglass figure.<ref name=":11">Payne, Blanche. ''History of Costume from the Ancient Egyptians to the Twentieth Century''. Harper & Row, 1965.</ref>{{rp|530}}</blockquote> The Victorian-looking gowns at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] are stylish in a way that recalls the designs of the House of Worth. The elements that make their look so Victorian are anachronisms on the costumes representing fashion of earlier eras. The women wearing these gowns preferred the standards of beauty from their own day to a more-or-less historically accurate look. The style competing at the very end of the century with the Worth look was not the historical, however, but a progressive style called at the time ''artistic'' or ''aesthetic''. William Powell Frith's 1883 painting ''A Private View at the Royal Academy, 1881'' (discussion below) pits this kind of traditional style against the progressive or artistic style. === The Styles === [[File:Frith A Private View.jpg|thumb|William Powell Frith, ''A Private View at the Royal Academy, 1881'']] We typically think of the late-Victorian silhouette as universal but, in the periods in which corsets dominated women's dress, not all women wore corsets and not all corsets were the same, as William Powell Frith's 1883 ''A Private View at the Royal Academy, 1881'' (right) illustrates. Frith is clear in his memoir that this painting — "recording for posterity the aesthetic craze as regards dress" — deliberately contrasts what he calls the "folly" of the Artistic Dress movement and the look of the traditional corseted waist.<ref>Frith, William Powell. ''My Autobiography and Reminiscences''. 1887.</ref> Frith considered the Artistic Movement and Artistic Dress "ephemeral," but its rejection of corsetry looks far more consequential to us in hindsight than it did in the 19th century. As Frith sees it, his painting critiques the "craze" associated with the women in this set of identifiable portraits who are not corseted, but his commitment to realism shows us a spectrum, a range, of conservatism and if not political then at least stylistic progressivism among the women. The progressives, oddly, are the women wearing artistic (that is, somewhat historical) dress, because they’re not corseted. It is a misreading to see the presentation of the women’s fashion as a simple opposition. Constance, Countess of Lonsdale — situated at the center of this painting with Frederick Leighton, president of the Royal Academy of Art — is the most conservatively dressed of the women depicted, with her narrow sleeves, tight waist and almost perfectly smooth bodice, which tells us that her corset has eyelets so that it can be laced precisely and tightly, and it has stays (or "bones") to prevent wrinkles or natural folds in the overclothing. Lillie Langtry, in the white dress, with her stylish narrow sleeves, does not have such a tightly bound waist or smooth bodice, suggesting she may not be corseted at all, as we know she sometimes was not.['''citation'''] Jenny Trip, a painter’s model, is the woman in the green dress in the aesthetic group being inspected by Anthony Trollope, who may be taking notes. She looks like she is not wearing a corset. Both Langtry and Trip are toward the middle of this spectrum: neither is dressed in the more extreme artistic dress of, say, the two figures between Trip and Trollope. A lot has been written about the late-Victorian attraction to historical dress, especially in the context of fancy-dress balls and the Gothic revival in social events as well as art and music. Part of the appeal has to have been the way those costumes could just be beautiful clothing beautifully made. Historical dress provided an opportunity for some elite women to wear less-structured but still beautiful and influential clothing. ['''Calvert'''<ref>Calvert, Robyne Erica. ''Fashioning the Artist: Artistic Dress in Victorian Britain 1848-1900''. Ph.D. thesis, University of Glasgow, 2012. <nowiki>https://theses.gla.ac.uk/3279/</nowiki></ref>] The standards for beauty, then, with historical dress were Victorian, with the added benefit of possibly less structure. So, at the Duchess of Devonshire's ball, "while some attendees tried to hew closely to historical precedent, many rendered their historical or mythological personage in the sartorial vocabulary they knew best. The [photographs of people in their costumes at the ball offer] a glimpse into how Victorians understood history, not a glimpse into the costume of an authentic historical past."<ref>Mitchell, Rebecca N. "The Victorian Fancy Dress Ball, 1870–1900." ''Fashion Theory'' 2017 (21: 3): 291–315. DOI: 10.1080/1362704X.2016.1172817.</ref> (294) * historical dress: beautiful clothing. * the range at the ball, from Minnie Paget to Gwladys * "In light of such efforts, the ball remains to this day one of the best documented outings of the period, and a quick glance at the album shows that ..." Women had more choices about their waists than the simple opposition between no corset and tightlacing can accommodate. The range of choices is illustrated in Frith's painting, with a woman locating herself on it at a particular moment for particular reasons. Much analysis of 19th-century corsetry focuses on its sexualizing effects — corsets dominated Victorian photographic pornography ['''citations'''] and at the same time, the absence of a corset was sexual because it suggested nudity.['''citations'''] A great deal of analysis of 19th-century corsetry, on the other hand, assumes that women wore corsets for the male gaze ['''citations'''] or that they tightened their waists to compete with other women.['''citations'''] But as we can see in Frith's painting, the sexualizing effect was not universal or sweeping, and these analyses do not account for the choices women had in which corset to wear or how tightly to lace it. Especially given the way that some photographic portraits were mechanically altered to make the waist appear smaller, the size of a woman's waist had to do with how she was presenting herself to the world. That is, the fact that women made choices about the size of or emphasis on their waists suggests that they had agency that needs to be taken into account. As they navigated the complex social world, women's fashion choices had meaning. Society or political hostesses had agency not only in their clothing but generally in that complex social world. They had roles managing social events of the upper classes, especially of the upper aristocracy and oligarchy, like the Duchess of Devonshire's ball. Their class and rank, then, were essential to their agency, including to some degree their freedom to choose what kind of corset to wear and how to wear it. Also, by the end of the century lots of different kinds of corsets were available for lots of different purposes. Special corsets existed for pregnancy, sports (like tennis, bicycling, horseback riding, golf, fencing, archery, stalking and hunting), theatre and dance and, of course, for these women corsets could be made to support the special dress worn over it. Women's choices in how they presented themselves to the world included more than just their foundation garments, of course. "Every cap, bow, streamer, ruffle, fringe, bustle, glove," that is, the trim and decorations on their garments, their jewelry and accessories — which Davidoff calls "elaborations"<ref name=":1">Davidoff, Leonore. ''The Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig 1973).</ref>{{rp|93}} — pointed to a host of status categories, like class, rank, wealth, age, marital status, engagement with the empire, how sexual they wanted to seem, political alignment and purpose at the social event. For example, when women were being presented to the monarch, they were expected to wear three ostrich plumes, often called the [[Social Victorians/Terminology#Prince of Wales's Feathers or White Plumes|Prince of Wales's feathers]]. Like all fashions, the corset, which was quite long-lasting in all its various forms, eventually went out of style. Of the many factors that might have influenced its demise, perhaps most important was the women's movement, in which women's rights, freedom, employment and access to their own money and children were less slogan-worthy but at least as essential as votes for women. The activities of the animal-rights movements drew attention not only to the profligate use of the bodies and feathers of birds but also to the looming extinction of the baleen whale, which made whale bone scarce and expensive. Perhaps the century's debates over corseting and especially tightlacing were relevant to some decisions not to be corseted. And, of course, perhaps no other reason is required than that the nature of fashion is to change. == Cinque Cento == According to the ''Oxford English Dictionary'', ''Cinque Cento'' is a shortening of ''mil cinque cento'', or 1500.<ref>"cinquecento, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/33143. Accessed 7 February 2023.</ref> The term, then would refer, perhaps informally, to the sixteenth century. == Crevé == ''Creve'', without the accent, is an old word in English (c. 1450) for burst or split.<ref>"creve, v." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44339. Accessed 8 February 2023.</ref> ['''With the acute accent, it looks like a past participle in French.'''] == Elastic == Elastic had been invented and was in use by the end of the 19th century. For the sense of "Elastic cord or string, usually woven with india-rubber,"<ref name=":6">“elastic, adj. & n.”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1199670313>.</ref> the ''Oxford English Dictionary'' has usage examples beginning in 1847. The example for 1886 is vivid: "The thorough-going prim man will always place a circle of elastic round his hair previous to putting on his college cap."<ref name=":6" /> == Elaborations == In her 1973 ''The Best Circles: Society, Etiquette and the Season'', Leonore Davidoff notes that women’s status was indicated by dress and especially ornament: “Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration,” she says, “symbolised some status category for the female wearer.”<ref name=":1" />{{rp|93}} Looking at these elaborations as meaningful rather than dismissing them as failed attempts at "historical accuracy" reveals a great deal about the individual women who wore or carried them — and about the society women and political hostesses in their roles as managers of the social world. In her review of ''The House of Worth: Portrait of an Archive'', Mary Frances Gormally says,<blockquote>In a socially regulated year, garments custom made with a Worth label provided women with total reassurance, whatever the season, time of day or occasion, setting them apart as members of the “Best Circles” dressed in luxurious, fashionable and always appropriate attire (Davidoff 1973). The woman with a Worth wardrobe was a woman of elegance, lineage, status, extreme wealth and faultless taste.<ref>Gormally, Mary Frances. Review essay of ''The House of Worth: Portrait of an Archive'', by Amy de la Haye and Valerie D. Mendes (V&A Publishing, 2014). ''Fashion Theory'' 2017 (21, 1): 109–126. DOI: 10.1080/1362704X.2016.1179400.</ref> (117)</blockquote> === Aglet, Aiglet === Historically, an aglet is a "point or metal piece that capped a string [or ribbon] used to attach two pieces of the garment together, i.e., sleeve and bodice."<ref name=":7" />{{rp|4}} Although they were decorative, they were not always visible on the outside of the clothing. They were often stuffed inside the layers at the waist (for example, attaching the bodice to a skirt or breeches). === Frou-frou === In French, ''frou-frou'' or, spelled as ''froufrou'', is the sound of the rustling of silk or sometimes of fabrics in general.<ref>{{Cite journal|date=2023-07-25|title=frou-frou|url=https://fr.wiktionary.org/w/index.php?title=frou-frou&oldid=32508509|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/frou-frou.</ref> The first use the French ''Wiktionnaire'' lists is Honoré Balzac, ''La Cousine Bette'', 1846.<ref>{{Cite journal|date=2023-06-03|title=froufrou|url=https://fr.wiktionary.org/w/index.php?title=froufrou&oldid=32330124|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/froufrou.</ref> ''Frou-frou'' is a term clothing historians use to describe decorative additions to an article of clothing; often the term has a slight negative connotation, suggesting that the additions are superficial. === Pouf, Puff, Poof === According to the French ''Wikipédia'', a pouf was, beginning in 1744, a "kind of women's hairstyle":<blockquote>The hairstyle in question, known as the “pouf”, had launched the reputation of the enterprising Rose Bertin, owner of the Grand Mogol, a very prominent fashion accessories boutique on Rue Saint-Honoré in Paris in 1774. Created in collaboration with the famous hairdresser, Monsieur Léonard, the pouf was built on a scaffolding of wire, fabric, gauze, horsehair, fake hair, and the client's own hair held up in an almost vertical position. — (Marie-Antoinette, ''Queen of Fashion'', translated from the American by Sylvie Lévy, in ''The Rules of the Game'', n° 40, 2009)</blockquote>''Puff'' and ''poof'' are used to describe clothing. === Shirring === ''Shirring'' is the gathering of fabric to make poufs or puffs. The 19th century is known for its use of this decorative technique. Even men's clothing had shirring: at the shoulder seam. === Sequins === Sequins, paillettes, spangles Sequins — or paillettes — are "small, scalelike glittering disks."<ref name=":7" />(216) The French ''Wiktionnaire'' defines ''paillette'' as "Lamelle de métal, brillante, mince, percée au milieu, ordinairement ronde, et qu’on applique sur une étoffe pour l’orner [A strip of metal, shiny, thin, pierced in the middle, usually round, and which is applied to a fabric in order to decorate it.]"<ref name=":8">{{Cite journal|date=2024-03-18|title=paillette|url=https://fr.wiktionary.org/w/index.php?title=paillette&oldid=33809572|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/paillette.</ref> According to the ''OED'', the use of ''sequin'' as a decorative device for clothing (as opposed to gold coins minted and used for international trade) goes back to the 1850s.<ref>“Sequin, N.” ''Oxford English Dictionary'', Oxford UP, September 2023, https://doi.org/10.1093/OED/4074851670.</ref> The first instance of ''spangle'' as "A small round thin piece of glittering metal (usually brass) with a hole in the centre to pass a thread through, used for the decoration of textile fabrics and other materials of various sorts" is from c. 1420.<ref>“Spangle, N. (1).” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/4727197141.</ref> The first use of ''paillette'' listed in the French ''Wiktionnaire'' is in Jules Verne in 1873 to describe colored spots on icy walls.<ref name=":8" /> Currently many distinguish between sequins (which are smaller) and paillettes (which are larger). Before the 20th century, sequins were metal discs or foil leaves, and so of course if they were silver or copper, they tarnished. It is not until well into the 20th century that plastics were invented and used for sequins. === Trim and Lace === ''A History of Feminine Fashion'', published sometime before 1927 and probably commissioned by [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|the Maison Worth]], describes Charles Frederick Worth's contributions to the development of embroidery and [[Social Victorians/Terminology#Passementerie|passementerie]] (trim) from about the middle of the 19th century:<blockquote>For it must be remembered that one of M. Worth's most important and lasting contributions to the prosperity of those who cater for women's needs, as well as to the variety and elegance of his clients' garments, was his insistence on new fabrics, new trimmings, new materials of every description. In his endeavours to restore in Paris the splendours of the days of La Pompadour, and of Marie Antoinette, he found himself confronted at the outset with a grave difficulty, which would have proved unsurmountable to a man of less energy, resource and initiative. The magnificent materials of those days were no longer to be had! The Revolution had destroyed the market for beautiful materials of this, type, and the Restoration and regime of Louis Philippe had left a dour aspect in the City of Light. ... On parallel lines [to his development of better [[Social Victorians/Terminology#Satin|satin]]], [Worth] stimulated also the manufacture of embroidery and ''passementerie''. It was he who first started the manufacture of laces copied from the designs of the real old laces. He was the / first dressmaker to use fur in the trimming of light materials — but he employed only the richer furs, such as sable and ermine, and had no use whatever for the inferior varieties of skins.<ref name=":9" />{{rp|6–7}}</blockquote> ==== Gold and Silver Fabric and Lace ==== The ''Encyclopaedia Britannica'' (9th edition) has an article on gold and silver fabric, threads and lace attached to the article on gold. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>GOLD AND SILVER LACE. Under this heading a general account may be given of the use of the precious metals in textiles of all descriptions into which they enter. That these metals were used largely in the sumptuous textiles of the earliest periods of civilization there is abundant testimony; and to this day, in the Oriental centres whence a knowledge and the use of fabrics inwoven, ornamented, and embroidered with gold and silver first spread, the passion for such brilliant and costly textiles is still most strongly and generally prevalent. The earliest mention of the use of gold in a woven fabric occurs in the description of the ephod made for Aaron (Exod. xxxix. 2, 3) — "And he made the ephod of gold, blue, and purple, and scarlet, and fine twined linen. And they did beat the gold into thin plates, and cut it into wires (strips), to work it in the blue, and in the purple, and in the scarlet, and in the fine linen, with cunning work." In both the ''Iliad'' and the ''Odyssey'' distinct allusion is frequently made to inwoven and embroidered golden textiles. Many circumstances point to the conclusion that the art of weaving and embroidering with gold and silver originated in India, where it is still principally prosecuted, and that from one great city to another the practice travelled westward, — Babylon, Tarsus, Baghdad, Damascus, the islands of Cyprus and Sicily, Con- / stantinople and Venice, all in the process of time becoming famous centres of these much prized manufactures. Alexander the Great found Indian kings and princes arrayed in robes of gold and purple; and the Persian monarch Darius, we are told, wore a war mantle of cloth of gold, on which were figured two golden hawks as if pecking at each other. There is reason, according to Josephus, to believe that the “royal apparel" worn by Herod on the day of his death (Acts xii. 21) was a tissue of silver. Agrippina, the wife of the emperor Claudius, had a robe woven entirely of gold, and from that period downwards royal personages and high ecclesiastical dignitaries used cloth and tissues of gold and silver for their state and ceremonial robes, as well as for costly hangings and decorations. In England, at different periods, various names were applied to cloths of gold, as ciclatoun, tartarium, naques or nac, baudekiu or baldachin, Cyprus damask, and twssewys or tissue. The thin flimsy paper known as tissue paper, is so called because it originally was placed between the folds of gold "tissue" to prevent the contiguous surfaces from fraying each other. At what time the drawing of gold wire for the preparation of these textiles was first practised is not accurately known. The art was probably introduced and applied in different localities at widely different dates, but down till mediaeval times the method graphically described in the Pentateuch continued to be practised with both gold and silver. Fabrics woven with gold and silver continue to be used on the largest scale to this day in India; and there the preparation of the varieties of wire, and the working of the various forms of lace, brocade, and embroidery, is at once an important and peculiar art. The basis of all modern fabrics of this kind is wire, the "gold wire" of the manufacturer being in all cases silver gilt wire, and silver wire being, of course, composed of pure silver. In India the wire is drawn by means of simple draw-plates, with rude and simple appliances, from rounded bars of silver, or gold-plated silver, as the case may be. The wire is flattened into the strip or ribbon-like form it generally assumes by passing it, fourteen or fifteen strands simultaneously, over a fine, smooth, round-topped anvil, and beating it as it passes with a heavy hammer having a slightly convex surface. From wire so flattened there is made in India soniri, a tissue or cloth of gold, the web or warp being composed entirely of golden strips, and ruperi, a similar tissue of silver. Gold lace is also made on a warp of thick yellow silk with a weft of flat wire, and in the case of ribbons the warp or web is composed of the metal. The flattened wires are twisted around orange (in the case of silver, white) coloured silk thread, so as completely to cover the thread and present the appearance of a continuous wire; and in this form it is chiefly employed for weaving into the rich brocades known as kincobs or kinkhábs. Wires flattened, or partially flattened, are also twisted into exceedingly fine spirals, and in this form they are the basis of numerous ornamental applications. Such spirals drawn out till they present a waved appearance, and in that state flattened, are much used for rich heavy embroideries termed karchobs. Spangles for embroideries, &c., are made from spirals of comparatively stout wire, by cutting them down ring by ring, laying each C-like ring on an anvil, and by a smart blow with a hammer flattening it out into a thin round disk with a slit extending from the centre to one edge. Fine spirals are also used for general embroidery purposes. The demand for various kinds of loom-woven and embroidered gold and silver work in India is immense; and the variety of textiles so ornamented is also very great. "Gold and silver," says Dr Birdwood in his ''Handbook to the British-Indian Section, Paris Exhibition'', 1878, "are worked into the decoration of all the more costly loom-made garments and Indian piece goods, either on the borders only, or in stripes throughout, or in diapered figures. The gold-bordered loom embroideries are made chiefly at Sattara, and the gold or silver striped at Tanjore; the gold figured ''mashrus'' at Tanjore, Trichinopoly, and Hyderabad in the Deccau; and the highly ornamented gold-figured silks and gold and silver tissues principally at Ahmedabad, Benares, Murshedabad, and Trichinopoly." Among the Western communities the demand for gold and silver lace and embroideries arises chiefly in connexion with naval and military uniforms, court costumes, public and private liveries, ecclesiastical robes and draperies, theatrical dresses, and the badges and insignia of various orders. To a limited extent there is a trade in gold wire and lace to India and China. The metallic basis of the various fabrics is wire round and flattened, the wire being of three kinds — 1st, gold wire, which is invariably silver gilt wire; 2d, copper gilt wire, used for common liveries and theatrical purposes; and 3d, silver wire. These wires are drawn by the ordinary processes, and the flattening, when done, is accomplished by passing the wire between a pair of revolving rollers of fine polished steel. The various qualities of wire are prepared and used in precisely the same way as in India, — round wire, flat wire, thread made of flat gold wire twisted round orange-coloured silk or cotton, known in the trade as "orris," fine spirals and spangles, all being in use in the West as in the East. The lace is woven in the same manner as ribbons, and there are very numerous varieties in richness, pattern, and quality. Cloth of gold, and brocades rich in gold and silver, are woven for ecclesiastical vestments and draperies. The proportions of gold and silver in the gold thread for the lace trade varies, but in all cases the proportion of gold is exceedingly small. An ordinary gold lace wire is drawn from a bar containing 90 parts of silver and 7 of copper, coated with 3 parts of gold. On an average each ounce troy of a bar so plated is drawn into 1500 yards of wire; and therefore about 16 grains of gold cover a mile of wire. It is estimated that about 250,000 ounces of gold wire are made annually in Great Britain, of which about 20 per cent, is used for the headings of calico, muslin, &c., and the remainder is worked up in the gold lace trade.<ref>William Chandler Roberts-Austen and H. Bauerman [W.C.R. — H.B.]. "Gold and Silver Lace." In "Gold." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. 10 (X). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2010%20%28G-GOT%29%20193592738.23/page/753/mode/1up (accessed January 2023): 753, Col. 2c – 754, Cols. 1a–b – 2a–b.</ref></blockquote> ==== Honiton Lace ==== Kate Stradsin says,<blockquote>Honiton lace was the finest English equivalent of Brussels bobbin lace and was constructed in small ‘sprigs, in the cottages of lacemakers[.'] These sprigs were then joined together and bleached to form the large white flounces that were so sought after in the mid-nineteenth century.<ref>Strasdin, Kate. "Rediscovering Queen Alexandra’s Wardrobe: The Challenges and Rewards of Object-Based Research." ''The Court Historian'' 24.2 (2019): 181-196. Rpt http://repository.falmouth.ac.uk/3762/15/Rediscovering%20Queen%20Alexandra%27s%20Wardrobe.pdf: 13, and (for the little quotation) n. 37, which reads "Margaret Tomlinson, ''Three Generations in the Honiton Lace Trade: A Family History'', self-published, 1983."</ref></blockquote> [[File:Strook in Alençon naaldkant, 1750-1775.jpg|thumb|alt=A long piece of complex white lace with garlands, flowers and bows|Point d'Alençon lace, 1750-1775]] ==== Passementerie ==== ''Passementerie'' is the French term for trim on clothing or furniture. The 19th century (especially during the First and Second Empire) was a time of great "''exubérance''" in passementerie in French design, including the development and widespread use of the Jacquard loom.<ref>{{Cite journal|date=2023-06-10|title=Passementerie|url=https://fr.wikipedia.org/w/index.php?title=Passementerie&oldid=205068926|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Passementerie.</ref> ==== Point d'Alençon Lace ==== A lace made by hand using a number of complex steps and layers. The lacemakers build the point d'Alençon design on some kind of mesh and sometimes leave some of the mesh in as part of the lace and perhaps to provide structure. Elizabeth Lewandowski defines point d'Alençon lace and Alençon lace separately. Point lace is needlepoint lace,<ref name=":7">Lewandowski, Elizabeth J. ''The Complete Costume Dictionary''. Scarecrow Press, 2011.</ref>{{rp|233}} so Alençon point is "a two thread [needlepoint] lace."<ref name=":7" />{{rp|7}} Alençon lace has a "floral design on [a] fine net ground [and is] referred to as [the] queen of French handmade needlepoint laces. The original handmade Alençon was a fine needlepoint lace made of linen thread."<ref name=":7" />{{rp|7}} The sample of point d'Alençon lace (right), from 1750–1775, shows the linen mesh that the lace was constructed on.<ref>{{Cite web|url=http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689|title=MoMu - Open Fashion|website=openfashion.momu.be|access-date=2024-02-26}} ModeMuseum Antwerpen. http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689.</ref> The consistency in this sample suggests it may have been made by machine. == Fabric == === Brocatelle === Brocatelle is a kind of brocade, more simple than most brocades because it uses fewer warp and weft threads and fewer colors to form the design. The article in the French ''Wikipédia'' defines it like this:<blockquote>La '''brocatelle''' est un type de tissu datant du <abbr>xvi^e</abbr> siècle qui comporte deux chaînes et deux trames, au minimum. Il est composé pour que le dessin ressorte avec un relief prononcé, grâce à la chaîne sur un fond en sergé. Les brocatelles les plus anciennes sont toujours fabriquées avec une des trames en lin.<ref>{{Cite journal|date=2023-06-01|title=Brocatelle|url=https://fr.wikipedia.org/w/index.php?title=Brocatelle&oldid=204796410|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Brocatelle.</ref></blockquote>Which translates to this:<blockquote>Brocatelle is a type of fabric dating from the 16th century that has two warps and two wefts, at a minimum. It is composed so that the design stands out with a pronounced relief, thanks to the weft threads on a twill background. The oldest brocades were always made with one of the wefts being linen.</blockquote>The ''Oxford English Dictionary'' says, brocatelle is an "imitation of brocade, usually made of silk or wool, used for tapestry, upholstery, etc., now also for dresses. Both the nature and the use of the stuff have changed" between the late 17th century and 1888, the last time this definition was revised.<ref>"brocatelle, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/23550. Accessed 4 July 2023.</ref> === Broché === === Ciselé === === Crépe de Chine === The ''Oxford English Dictionary'' distinguishes the use of ''crêpe'' (using a circumflex rather than an acute accent over the first ''e'') from ''crape'' in textiles, saying ''crêpe'' is "often borrowed [from the French] as a term for all crapy fabrics other than ordinary black mourning crape,"<ref>"crêpe, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44242. Accessed 10 February 2023.</ref> with usage examples ranging from 1797 to the mid 20th century. Crêpe de chine, it says is "a white or other coloured crape made of raw silk." === Épinglé Velvet === Often spelled ''épingle'' rather than ''épinglé'', this term appears to have been used for a fabric made of wool, or at least wool along with linen or cotton, that was heavier and stiffer than silk velvet. It was associated with outer garments and men's clothing. Nowadays, épinglé velvet is an upholstery fabric in which the pile is cut into designs and patterns, and the portrait of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary, Duchess of Hamilton]] shows a mantle described as épinglé velvet that does seem to be a velvet with a woven pattern perhaps cut into the pile. === Lace === While lace also functioned sometimes as fabric — at the décolletage, for example, on the stomacher or as a veil — here we organize it as a [[Social Victorians/Terminology#Trim and Lace|part of the elaboration of clothing]]. === Liberty Fabrics === === Lisse === According to the ''Oxford English Dictionary'', the term ''lisse'' as a "kind of silk gauze" was used in the 19th-century UK and US.<ref>"lisse, n.1." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/108978. Accessed 4 July 2023.</ref> === Satin === The pre-1927 ''History of Feminine Fashion'', probably commissioned by Charles Frederick Worth's sons, describes Worth's "insistence on new fabrics, new trimmings, new materials of every description" at the beginning of his career in the mid 19th century:<blockquote>When Worth first entered the business of dressmaking, the only materials of the richer sort used for woman's dress were velvet, faille, and watered silk. Satin, for example, was never used. M. Worth desired to use satin very extensively in the gowns he designed, but he was not satisfied with what could be had at the time; he wanted something very much richer than was produced by the mills at Lyons. That his requirements entailed the reconstruction of mills mattered little — the mills were reconstructed under his directions, and the Lyons looms turned out a richer satin than ever, and the manufacturers prospered accordingly.<ref name=":9">[Worth, House of.] {{Cite book|url=http://archive.org/details/AHistoryOfFeminineFashion|title=A History Of Feminine Fashion (1800s to 1920s)}} Before 1927. [Likely commissioned by Worth. Link is to Archive.org; info from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Worth_Biarritz_salon.jpg.]</ref>{{rp|6 in printed, 26 in digital book}}</blockquote> === Selesia === According to the ''Oxford English Dictionary'', ''silesia'' is "A fine linen or cotton fabric originally manufactured in Silesia in what is now Germany (''Schlesien'').<ref>"Silesia, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/179664. Accessed 9 February 2023.</ref> It may have been used as a lining — for pockets, for example — in garments made of more luxurious or more expensive cloth. The word ''sleazy'' — "Of textile fabrics or materials: Thin or flimsy in texture; having little substance or body."<ref>"sleazy, adj." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/181563. Accessed 9 February 2023.</ref> — may be related. === Shot Fabric === According to the ''Oxford English Dictionary'', "Of a textile fabric: Woven with warp-threads of one colour and weft-threads of another, so that the fabric (usually silk) changes in tint when viewed from different points."<ref>“Shot, ''Adj.''”  ''Oxford English Dictionary'', Oxford UP,  July 2023, https://doi.org/10.1093/OED/2977164390.</ref> A shot fabric might also be made of silk and cotton fibers. === Tissue === A lightly woven fabric like gauze or chiffon. The light weave can make the fabric translucent and make pleating and gathering flatter and less bulky. Tissue can be woven to be shot, sheer, stiff or soft. Historically, the term in English was used for a "rich kind of cloth, often interwoven with gold or silver" or "various rich or fine fabrics of delicate or gauzy texture."<ref>“Tissue, N.” ''Oxford English Dictionary'', Oxford UP, March 2024, https://doi.org/10.1093/OED/5896731814.</ref> == Fan == The ''Encyclopaedia Britannica'' (9th edition) has an article on the fan. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>FAN (Latin, ''vannus''; French, ''éventail''), a light implement used for giving motion to the air. ''Ventilabrum'' and ''flabellum'' are names under which ecclesiastical fans are mentioned in old inventories. Fans for cooling the face have been in use in hot climates from remote ages. A bas-relief in the British Museum represents Sennacherib with female figures carrying feather fans. They were attributes of royalty along with horse-hair fly-flappers and umbrellas. Examples may be seen in plates of the Egyptian sculptures at Thebes and other places, and also in the ruins of Persepolis. In the museum of Boulak, near Cairo, a wooden fan handle showing holes for feathers is still preserved. It is from the tomb of Amen-hotep, of the 18th dynasty, 17th century <small>B</small>.<small>C</small>. In India fans were also attributes of men in authority, and sometimes sacred emblems. A heartshaped fan, with an ivory handle, of unknown age, and held in great veneration by the Hindus, was given to the prince of Wales. Large punkahs or screens, moved by a servant who does nothing else, are in common use by Europeans in India at this day. Fans were used in the early Middle Ages to keep flies from the sacred elements during the celebrations of the Christian mysteries. Sometimes they were round, with bells attached — of silver, or silver gilt. Notices of such fans in the ancient records of St Paul’s, London, Salisbury cathedral, and many other churches, exist still. For these purposes they are no longer used in the Western church, though they are retained in some Oriental rites. The large feather fans, however, are still carried in the state processions of the supreme pontiff in Rome, though not used during the celebration of the mass. The fan of Queen Theodolinda (7th century) is still preserved in the treasury of the cathedral of Monza. Fans made part of the bridal outfit, or ''mundus muliebris'', of ancient Roman ladies. Folding fans had their origin in Japan, and were imported thence to China. They were in the shape still used—a segment of a circle of paper pasted on a light radiating frame-work of bamboo, and variously decorated, some in colours, others of white paper on which verses or sentences are written. It is a compliment in China to invite a friend or distinguished guest to write some sentiment on your fan as a memento of any special occasion, and this practice has continued. A fan that has some celebrity in France was presented by the Chinese ambassador to the Comtesse de Clauzel at the coronation of Napoleon I. in 1804. When a site was given in 1635, on an artificial island, for the settlement of Portuguese merchants in Nippo in Japan, the space was laid out in the form of a fan as emblematic of an object agreeable for general use. Men and women of every rank both in China and Japan carry fans, even artisans using them with one hand while working with the other. In China they are often made of carved ivory, the sticks being plates very thin and sometimes carved on both sides, the intervals between the carved parts pierced with astonishing delicacy, and the plates held together by a ribbon. The Japanese make the two outer guards of the stick, which cover the others, occasionally of beaten iron, extremely thin and light, damascened with gold and other metals. Fans were used by Portuguese ladies in the 14th century, and were well known in England before the close of the reign of Richard II. In France the inventory of Charles V. at the end of the 14th century mentions a folding ivory fan. They were brought into general use in that country by Catherine de’ Medici, probably from Italy, then in advance of other countries in all matters of personal luxury. The court ladies of Henry VIII.’s reign in England were used to handling fans, A lady in the Dance of Death by Holbein holds a fan. Queen Elizabeth is painted with a round leather fan in her portrait at Gorhambury; and as many as twenty-seven are enumerated in her inventory (1606). Coryat, an English traveller, in 1608 describes them as common in Italy. They also became of general use from that time in Spain. In Italy, France, and Spain fans had special conventional uses, and various actions in handling them grew into a code of signals, by which ladies were supposed to convey hints or signals to admirers or to rivals in society. A paper in the ''Spectator'' humorously proposes to establish a regular drill for these purposes. The chief seat of the European manufacture of fans during the 17th century was Paris, where the sticks or frames, whether of wood or ivory, were made, and the decorations painted on mounts of very carefully prepared vellum (called latterly ''chicken skin'', but not correctly), — a material stronger and tougher than paper, which breaks at the folds. Paris makers exported fans unpainted to Madrid and other Spanish cities, where they were decorated by native artists. Many were exported complete; of old fans called Spanish a great number were in fact made in France. Louis XIV. issued edicts at various times to regulate the manufacture. Besides fans mounted with parchment, Dutch fans of ivory were imported into Paris, and decorated by the heraldic painters in the process called “Vernis Martin,” after a famous carriage painter and inventor of colourless lac varnish. Fans of this kind belonging to the Queen and to the late baroness de Rothschild were exhibited in 1870 at Kensington. A fan of the date of 1660, representing sacred subjects, is attributed to Philippe de Champagne, another to Peter Oliver in England in the / 17th century. Cano de Arevalo, a Spanish painter of the 17th century devoted himself to fan painting. Some harsh expressions of Queen Christina to the young ladies of the French court are said to have caused an increased ostentation in the splendour of their fans, which were set with jewels and mounted in gold. Rosalba Carriera was the name of a fan painter of celebrity in the 17th century. Lebrun and Romanelli were much employed during the same period. Klingstet, a Dutch artist, enjoyed a considerable reputation for his fans from the latter part of the 17th and the first thirty years of the 18th century. The revocation of the edict of Nantes drove many fan-makers out of France to Holland and England. The trade in England was well established under the Stuart sovereigns. Petitions were addressed by the fan-makers to Charles II. against the importation of fans from India, and a duty was levied upon such fans in consequence. This importation of Indian fans, according to Savary, extended also to France. During the reign of Louis XV. carved Indian and China fans displaced to some extent those formerly imported from Italy, which had been painted on swanskin parchment prepared with various perfumes. During the 18th century all the luxurious ornamentation of the day was bestowed on fans as far as they could display it. The sticks were made of mother-of-pearl or ivory, carved with extraordinary skill in France, Italy, England, and other countries. They were painted from designs of Boucher, Watteau, Lancret, and other "genre" painters, Hébert, Rau, Chevalier, Jean Boquet, Mad. Verité, are known as fan painters. These fashions were followed in most countries of Europe, with certain national differences. Taffeta and silk, as well as fine parchment, were used for the mounts. Little circles of glass were let into the stick to be looked through, and small telescopic glasses were sometimes contrived at the pivot of the stick. They were occasionally mounted with the finest point lace. An interesting fan (belonging to Madame de Thiac in France), the work of Le Flamand, was presented by the municipality of Dieppe to Marie Antoinette on the birth of her son the dauphin. From the time of the Revolution the old luxury expended on fans died out. Fine examples ceased to be exported to England and other countries. The painting on them represented scenes or personages connected with political events. At a later period fan mounts were often prints coloured by hand. The events of the day mark the date of many examples found in modern collections. Amongst the fanmakers of the present time the names of Alexandre, Duvelleroy, Fayet, Vanier, may be mentioned as well known in Paris. The sticks are chiefly made in the department of Oise, at Le Déluge, Crèvecœur, Méry, Ste Geneviève, and other villages, where whole families are engaged in preparing them; ivory sticks are carved at Dieppe. Water-colour painters of distinction often design and paint the mounts, the best designs being figure subjects. A great impulse has been given to the manufacture and painting of fans in England since the exhibition which took place at South Kensington in 1870. Other exhibitions have since been held, and competitive prizes offered, one of which was gained by the Princess Louise. Modern collections of fans take their date from the emigration of many noble families from France at the time of the Revolution. Such objects were given as souvenirs and occasionally sold by families in straitened circumstances. A large number of fans of all sorts, principally those of the 18th century, French, English, German, Italian Spanish, &c., have been lately bequeathed to the South Kensington Museum. Regarding the different parts of folding fans it may be well to state that the sticks are called in French ''brins'', the two outer guards ''panaches'', and the mount ''feuille''.<ref>J. H. Pollen [J.H.P.]. "Fan." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. '''10''' ('''X'''). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%209%20%28FAL-FYZ%29%20193323016.23/page/26/mode/2up (accessed January 2023): 27, Col. 1b – 28, Col. 1c.</ref></blockquote> == Fancy-dress Ball == Fancy-dress (or costume) balls were popular and frequent in the U.K. and France as well as the rest of Europe during the 19th century. The themes and styles of the fancy-dress balls influenced those that followed. At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the guests came dressed in costume from times before 1820, as instructed on '''the invitation''', but their clothing was much more about late-Victorian standards of beauty and fashion than the standards of whatever time period the portraits they were copying or basing their costumes on. ''The Queen'' published dress and fashion information and advice under the byline of Ardern Holt, who regularly answered questions from readers about fashion as well as about fancy dress. (More about Ardern Holt, which is almost certainly a pseudonym, can be found on the [[Social Victorians/People/Working in Publishing#Journalists|People Working in Publishing]] page.) Holt also ran wrote entire articles with suggestions for what might make an appealing fancy-dress costume as well as pointing readers away from costumes that had been worn too frequently. The suggestions for costumes are based on familiar types or portraits available to readers, similar to Holt's books on fancy dress, which ran through a number of editions in the 1880s and 1890s. Fancy-dress questions sometimes asked for details about costumes worn in theatrical or operatic productions, which Holt provides. In November 1897, Holt refers to the Duchess of Devonshire's 2 July ball: "Since the famous fancy ball, given at Devonshire House during this year, historical fancy dresses have assumed a prominence that they had not hitherto known."<ref>Holt, Ardern. "Fancy Dress a la Mode." The ''Queen'' 27 November 1897, Saturday: 94 [of 145 in BNA; print p. 1026], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971127/459/0094.</ref> Holt goes on to provide a number of ideas for costumes for historical fancy dress, as always with a strong leaning toward Victorian standards of beauty and style and away from any concern for historical accuracy. Ardern Holt published books on fancy dress as well as writing for the ''Queen'' and other periodicals, but not all of them were about fancy dress. # ''Gentlemen's Fancy Dress: How to Choose It''. Wyman & Sons, 1882. (''Google Books'' has this: https://books.google.com/books/about/Gentlemen_s_Fancy_Dress.html?id=ED8CAAAAQAAJ.) Later editions: 1898 (HathiTrust) # ''Fancy Dresses Described; Or, What to Wear at Fancy Balls''. Debenham & Freebody, 1882. Illustr., Lillian Young. (HathiTrust has this.) Later editions: 4th ed — 1884; 1887 (HathiTrust); 6th ed. — 1896 (HathiTrust) As Leonore Davidoff says, "Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration symbolised some status category for the female wearer."<ref name=":1" />{{rp|93}} [handled under Elaborations] === Historical Accuracy === Many of the costumes at the ball were based on portraits, especially when the guest was dressed as a historical figure. If possible, we have found the portraits likely to have been the originals, or we have found, if possible, portraits that show the subjects from the two time periods at similar ages. The way clothing was cut changed quite a bit between the 18th and 19th centuries. We think of Victorian clothing — particularly women's clothing, and particularly at the end of the century — as inflexible and restrictive, especially compared to 20th- and 21st-century customs permitting freedom of movement. The difference is generally evolutionary rather than absolute — that is, as time has passed since the 18th century, clothing has allowed an increasingly greater range of movement, especially for people who did not do manual labor. By the end of the 19th century, garments like women's bodices and men's coats were made fitted and smooth by attention to the grain of the fabric and by the use of darts (rather than techniques that assembled many small, individual pieces of fabric). * clothing construction and flat-pattern techniques * Generally, the further back in time we go, the more 2-dimensional the clothing itself was. ==== Women's Versions of Historical Accuracy at the Ball ==== As always with this ball, whatever historical accuracy might be present in a woman's costume is altered so that the wearer is still a fashionable Victorian lady. What makes the costumes look "Victorian" to our eyes is the line of the silhouette caused by the foundation undergarments as well as the many "elaborations"<ref name=":1" />{{rp|93}}, mostly in the decorations, trim and accessories. Also, the clothing hangs and drapes differently because the fabric was cut on grain and the shoulders were freed by the way the sleeves were set in. ==== Men's Versions of Historical Accuracy at the Ball ==== Because men were not wearing a Victorian foundation garment at the end of the century, the men's costumes at the ball are more historically accurate in some ways. * Trim * Mixing neck treatments * Hair * Breeches * Shoes and boots * Military uniforms, arms, gloves, boots == Feathers and Plumes == === Aigrette === Elizabeth Lewandowski defines ''aigrette'' as "France. Feather or plume from an egret or heron."<ref name=":7" />(5) Sometimes the newspapers use the term to refer to an accessory (like a fan or ornament on a hat) that includes such a feather or plume. The straight and tapered feathers in an aigrette are in a bundle. === Prince of Wales's Feathers or White Plumes === The feathers in an aigrette came from egrets and herons; Prince of Wales's feathers came from ostriches. A fuller discussion of Prince of Wales's feathers and the white ostrich plumes worn at court appears on [[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|Victorian Things]]. For much of the late 18th and 19th centuries, white ostrich plumes were central to fashion at court, and at a certain point in the late 18th century they became required for women being presented to the monarch and for their sponsors. Our purpose here is to understand why women were wearing plumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] as part of their costumes. First published in 1893, [[Social Victorians/People/Lady Colin Campbell|Lady Colin Campbell]]'s ''Manners and Rules of Good Society'' (1911 edition) says that<blockquote>It was compulsory for both Married and Unmarried Ladies to Wear Plumes. The married lady’s Court plume consisted of three white feathers. An unmarried lady’s of two white feathers. The three white feathers should be mounted as a Prince of Wales plume and worn towards the left hand side of the head. Colored feathers may not be worn. In deep mourning, white feathers must be worn, black feathers are inadmissible.<p> White veils or lace lappets must be worn with the feathers. The veils should not be longer than 45 inches.<ref>{{Cite web|url=https://www.edwardianpromenade.com/etiquette/the-court-presentation/|title=The Court Presentation|last=Holl|first=Evangeline|date=2007-12-07|website=Edwardian Promenade|language=en-US|access-date=2022-12-18}} https://www.edwardianpromenade.com/etiquette/the-court-presentation/.</ref></blockquote>[[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|This fashion was imported from France]] in the mid 1770s.<ref>"Abstract" for Blackwell, Caitlin. "'<nowiki/>''The Feather'd Fair in a Fright''': The Emblem of the Feather in Graphic Satire of 1776." ''Journal for Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. ''Wiley Online'' DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x (accessed November 2022).</ref> Separately, a secondary heraldic emblem of the Prince of Wales has been a specific arrangement of 3 ostrich feathers in a gold coronet<ref>{{Cite journal|date=2022-11-07|title=Prince of Wales's feathers|url=https://en.wikipedia.org/w/index.php?title=Prince_of_Wales%27s_feathers&oldid=1120556015|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Prince_of_Wales's_feathers.</ref> since King Edward III (1312–1377<ref>{{Cite journal|date=2022-12-14|title=Edward III of England|url=https://en.wikipedia.org/w/index.php?title=Edward_III_of_England&oldid=1127343221|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_III_of_England.</ref>). Some women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] wore white ostrich feathers in their hair, but most of them are not Prince of Wales's feathers. Most of the plumes in these portraits are arrangements of some kind of headdress to accompany the costume. A few, wearing what looks like the Princes of Wales's feathers, might be signaling that their character is royal or has royal ancestry. '''One of the women [which one?] was presented to the royals at this ball?''' Here is the list of women who are wearing white ostrich plumes in their portraits in the [[Social Victorians/1897 Fancy Dress Ball/Photographs|''Diamond Jubilee Fancy Dress Ball'' album of 286 photogravure portraits]]: # Kathleen Pelham-Clinton, the [[Social Victorians/People/Newcastle|Duchess of Newcastle]] # [[Social Victorians/People/Louisa Montagu Cavendish|Luise Cavendish]], the Duchess of Devonshire # Jesusa Murrieta del Campo Mello y Urritio (née Bellido), [[Social Victorians/People/Santurce|Marquisa de Santurce]] # Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]] # Princess (Laura Williamina Seymour) Victor of  [[Social Victorians/People/Gleichen#Laura%20Williamina%20Seymour%20of%20Hohenlohe-Langenburg|Hohenlohe Langenburg]] # Louisa Acheson, [[Social Victorians/People/Gosford|Lady Gosford]] # Alice Emily White Coke, [[Social Victorians/People/Leicester|Viscountess Coke]] # Lady Mary Stewart, Helen Mary Theresa [[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]] #[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], Duchess of [[Social Victorians/People/Marlborough|Marlborough]], dressed as the wife of the French Ambassador at the Court of Catherine of Russia (not white, but some color that reads dark in the black-and-white photograph) #Mrs. Mary [[Social Victorians/People/Chamberlain|Chamberlain]] (at 491), wearing white plumes, as Madame d'Epinay #Lady Clementine [[Social Victorians/People/Tweeddale|Hay]] (at 629), wearing white plumes, as St. Bris (''Les Huguenots'') #[[Social Victorians/People/Meysey-Thompson|Lady Meysey-Thompson]] (at 391), wearing white plumes, as Elizabeth, Queen of Bohemia #Mrs. [[Social Victorians/People/Grosvenor|Algernon (Catherine) Grosvenor]] (at 510), wearing white plumes, as Marie Louise #Lady [[Social Victorians/People/Ancaster|Evelyn Ewart]], at 401), wearing white plumes, as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson #[[Social Victorians/People/Lyttelton|Edith Sophy Balfour Lyttelton]] (at 580), wearing what might be white plumes on a large-brimmed white hat, after a picture by Romney #[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), wearing what might be white plumes, as Cydalise of the Comedie Italienne from the time of Louis XV #Lady [[Social Victorians/People/Ilchester|Muriel Fox Strangways]] (at 403), wearing what might be two smallish white plumes, as Lady Sarah Lennox, one of the bridesmaids of Queen Charlotte A.D. 1761 #Lady [[Social Victorians/People/Lucan|Violet Bingham]] (at 586), wearing perhaps one white plume in a headdress not related to the Prince of Wales's feathers #Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), wearing a headdress that includes some white plumes, as Lady Burleigh #[[Social Victorians/People/Dupplin|Agnes Blanche Marie Hay-Drummond]] (at 682), in a big headdress topped with white plumes, as Mademoiselle Andrée de Taverney A.D. 1775 #Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]] (at 336), wearing what looks like Prince of Wales's plumes #[[Social Victorians/People/Suffolk|Marguerite Hyde "Daisy" Leiter]] (at 684), wearing what looks like Prince of Wales's plumes #Lady [[Social Victorians/People/Spicer|Margaret Spicer]] (at 281), wearing one smallish white and one black plume, as Countess Zinotriff, Lady-in-Waiting to the Empress Catherine of Russia #Mrs. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 318), wearing what looks like Prince of Wales's plumes, as Elizabeth Cavendish, daughter of Bess of Hardwick #Nellie, [[Social Victorians/People/Kilmorey|Countess of Kilmorey]] (at 207), wearing three tall plumes, 2 white and one dark, as Comtesse du Barri #Daisy, [[Social Victorians/People/Warwick|Countess of Warwick]] (at 53), wearing at least 1 white plume, as Marie Antoinette More men than women were wearing plumes reminiscent of the Prince of Wales's feathers: * ==== Bibliography for Plumes and Prince of Wales's Feathers ==== * Blackwell, Caitlin. "'''The Feather'd Fair in a Fright'<nowiki/>'': The Emblem of the Feather in Graphic Satire of 1776." Journal for ''Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. Wiley Online DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x. * "Prince of Wales's feathers." ''Wikipedia'' https://en.wikipedia.org/wiki/Prince_of_Wales%27s_feathers (accessed November 2022). ['''Add women to this page'''] * Simpson, William. "On the Origin of the Prince of Wales' Feathers." ''Fraser's magazine'' 617 (1881): 637-649. Hathi Trust https://babel.hathitrust.org/cgi/pt?id=chi.79253140&view=1up&seq=643&q1=feathers (accessed December 2022). Deals mostly with use of feathers in other cultures and in antiquity; makes brief mention of feathers and plumes in signs and pub names that may not be associated with the Prince of Wales. No mention of the use of plumes in women's headdresses or court dress. == Honors == === The Bath === The Most Honourable Order of the Bath (GCB, Knight or Dame Grand Cross; KCB or DCB, Knight or Dame Commander; CB, Companion) === The Garter === The Most Noble Order of the Knights of the Garter (KG, Knight Companion; LG, Lady Companion) [[File:The Golden Fleece - collar exhibited at MET, NYC.jpg|thumb|The Golden Fleece collar and pendant for the 2019 "Last Knight" exhibition at the MET, NYC.|alt=Recent photograph of a gold necklace on a wide band, with a gold skin of a sheep hanging from it as a pendant]] === The Golden Fleece === To wear the golden fleece is to wear the insignia of the Order of the Golden Fleece, said to be "the most prestigious and historic order of chivalry in the world" because of its long history and strict limitations on membership.<ref name=":10">{{Cite journal|date=2020-09-25|title=Order of the Golden Fleece|url=https://en.wikipedia.org/w/index.php?title=Order_of_the_Golden_Fleece&oldid=980340875|journal=Wikipedia|language=en}}</ref> The monarchs of the U.K. were members of the originally Spanish order, as were others who could afford it, like the Duke of Wellington,<ref name=":12">Thompson, R[obert]. H[ugh]. "The Golden Fleece in Britain." Publication of the ''British Numismatic Society''. 2009 https://www.britnumsoc.org/publications/Digital%20BNJ/pdfs/2009_BNJ_79_8.pdf (accessed January 2023).</ref> the first Protestant to be admitted to the order.<ref name=":10" /> Founded in 1429/30 by Philip the Good, Duke of Burgundy, the order separated into two branches in 1714, one Spanish and the other Austrian, still led by the House of Habsburg.<ref name=":10" /> [[File:Prince Albert - Franz Xaver Winterhalter 1842.jpg|thumb|1842 Winterhalter portrait of Prince Albert wearing the insignia of the Order of the Golden Fleece, 1842|left|alt=1842 Portrait of Prince Albert by Winterhalter, wearing the insignia of the Golden Fleece]] The photograph (upper right) is of a Polish badge dating from the "turn of the XV and XVI centuries."<ref>{{Citation|title=Polski: Kolana orderowa orderu Złotego Runa, przełom XV i XVI wieku.|url=https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg|date=2019-11-10|accessdate=2023-01-10|last=Wulfstan}}. https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg.</ref> The collar to this Golden Fleece might be similar to the one the [[Social Victorians/People/Spencer Compton Cavendish#The Insignia of the Order of the Golden Fleece|Duke of Devonshire is wearing in the 1897 Lafayette portrait]]. The badges and collars that Knights of the Order actually wore vary quite a bit. The 1842 Franz Xaver Winterhalter portrait (left) of Prince Consort Albert, Victoria's husband and father of the Prince of Wales, shows him wearing the Golden Fleece on a red ribbon around his neck and the star of the Garter on the front of his coat.<ref>Winterhalter, Franz Xaver. ''Prince Albert''. {{Cite web|url=https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61|title=Explore the Royal Collection Online|website=www.rct.uk|access-date=2023-01-16}} https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61.</ref> === Royal Victorian Order === (GCVO, Knight or Dame Grand Cross; KCVO or DCVO, Knight or Dame Commander; CVO, Commander; LVO, Lieutenant; MVO, Member) === St. John === The Order of the Knights of St. John === Star of India === Most Exalted Order of the Star of India (GCSI, Knight Grand Commander; KCSI, Knight Commander; CSI, Companion) === Thistle === The Most Ancient and Most Noble Order of the Thistle == Jewelry and Stones == === Cabochon === This term describes both the treatment and shape of a precious or semiprecious stone. A cabochon treatment does not facet the stone but merely polishes it, removing "the rough parts" and the parts that are not the right stone.<ref>"cabochon, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/25778. Accessed 7 February 2023.</ref> A cabochon shape is often flat on one side and oval or round, forming a mound in the setting. === Jet === === ''Orfèvrerie'' === Sometimes misspelled in the newspapers as ''orvfèvrerie''. ''Orfèvrerie'' is the artistic work of a goldsmith, silversmith, or jeweler. === Turquoises === == Military == Several men from the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]] were dressed in military uniforms, some historical and some, possibly, not. === Baldric === According to the ''Oxford English Dictionary'', the primary sense of ''baldric'' is "A belt or girdle, usually of leather and richly ornamented, worn pendent from one shoulder across the breast and under the opposite arm, and used to support the wearer's sword, bugle, etc."<ref>"baldric, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/14849. Accessed 17 May 2023.</ref> This sense has been in existence since c. 1300. === Cuirass === According to the ''Oxford English Dictionary'', the primary sense of ''cuirass'' is "A piece of armour for the body (originally of leather); ''spec.'' a piece reaching down to the waist, and consisting of a breast-plate and a back-plate, buckled or otherwise fastened together ...."<ref>"cuirass, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/45604. Accessed 17 May 2023.</ref> [[File:Knötel IV, 04.jpg|thumb|alt=An Old drawing in color of British soldiers on horses brandishing swords in 1815.|1890 illustration of the Household Cavalry (Life Guard, left; Horse Guard, right) at the Battle of Waterloo, 1815]] === Household Cavalry === The Royal Household contains the Household Cavalry, a corps of British Army units assigned to the monarch. It is made up of 2 regiments, the Life Guards and what is now called The Blues and Royals, which were formed around the time of "the Restoration of the Monarchy in 1660."<ref name=":3">Joll, Christopher. "Tales of the Household Cavalry, No. 1. Roles." The Household Cavalry Museum, https://householdcavalry.co.uk/app/uploads/sites/2/2021/06/Household-Cavalry-Museum-video-series-large-print-text-Tales-episode-01.pdf.</ref>{{rp|1}} Regimental Historian Christopher Joll says, "the original Life Guards were formed as a mounted bodyguard for the exiled King Charles II, The Blues were raised as Cromwellian cavalry and The Royals were established to defend Tangier."<ref name=":3" />{{rp|1–2}} The 1st and 2nd Life Guards were formed from "the Troops of Horse and Horse Grenadier Guards ... in 1788."<ref name=":3" />{{rp|3}} The Life Guards were and are still official bodyguards of the queen or king, but through history they have been required to do quite a bit more than serve as bodyguards for the monarch. The Household Cavalry fought in the Battle of Waterloo on Sunday, 18 June 1815 as heavy cavalry.<ref name=":3" />{{rp|3}} Besides arresting the Cato Steet conspirators in 1820 "and guarding their subsequent execution," the Household Cavalry contributed to the "the expedition to rescue General Gordon, who was trapped in Khartoum by The Mahdi and his army of insurgents" in 1884.<ref name=":3" />{{rp|3}} In 1887 they "were involved ... in the suppression of rioters in Trafalgar Square on Bloody Sunday."<ref name=":3" />{{rp|3}} ==== Grenadier Guards ==== Three men — [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Lord Algernon Gordon-Lennox]], [[Social Victorians/People/Stanley#Edward George Villiers Stanley, Lord Stanley|Lord Stanley]], and [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|Hon. F. C. Stanley]] — attended the ball as officers of the Grenadier Guards, wearing "scarlet tunics, ... full blue breeches, scarlet hose and shoes, lappet wigs" as well as items associated with weapons and armor.<ref name=":14">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 34, Col. 2a}} Founded in England in 1656 as Foot Guards, this infantry regiment "was granted the 'Grenadier' designation by a Royal Proclamation" at the end of the Napoleonic Wars.<ref>{{Cite journal|date=2023-04-22|title=Grenadier Guards|url=https://en.wikipedia.org/w/index.php?title=Grenadier_Guards&oldid=1151238350|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Grenadier_Guards.</ref> They were not called Grenadier Guards, then, before about 1815. In 1660, the Stuart Restoration, they were called Lord Wentworth's Regiment, because they were under the command of Thomas Wentworth, 5th Baron Wentworth.<ref>{{Cite journal|date=2022-07-24|title=Lord Wentworth's Regiment|url=https://en.wikipedia.org/w/index.php?title=Lord_Wentworth%27s_Regiment&oldid=1100069077|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Wentworth%27s_Regiment.</ref> At the time of Lord Wentworth's Regiment, the style of the French cavalier had begun to influence wealthy British royalists. In the British military, a Cavalier was a wealthy follower of Charles I and Charles II — a commander, perhaps, or a field officer, but probably not a soldier.<ref>{{Cite journal|date=2023-04-22|title=Cavalier|url=https://en.wikipedia.org/w/index.php?title=Cavalier&oldid=1151166569|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier.</ref> The Guards were busy as infantry in the 17th century, engaging in a number of armed conflicts for Great Britain, but they also served the sovereign. According to the Guards Museum,<blockquote>In 1678 the Guards were ordered to form Grenadier Companies, these men were the strongest and tallest of the regiment, they carried axes, hatches and grenades, they were the shock troops of their day. Instead of wearing tri-corn hats they wore a mitre shaped cap.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/|title=Service to the Crown|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/.</ref></blockquote>The name comes from ''grenades'', then, and we are accustomed to seeing them in front of Buckingham Palace, with their tall mitre hats. The Guard fought in the American Revolution, and in the 19th century, the Grenadier Guards fought in the Crimean War, Sudan and the Boer War. They have roles as front-line troops and as ceremonial for the sovereign, which makes them elite:<blockquote>Queen Victoria decreed that she did not want to see a single chevron soldier within her Guards. Other then [sic] the two senior Warrant Officers of the British Army, the senior Warrant Officers of the Foot Guards wear a large Sovereigns personal coat of arms badge on their upper arm. No other regiments of the British Army are allowed to do so; all the others wear a small coat of arms of their lower arms. Up until 1871 all officers in the Foot Guards had the privilege of having double rankings. An Ensign was ranked as an Ensign and Lieutenant, a Lieutenant as Lieutenant and Captain and a Captain as Captain and Lieutenant Colonel. This was because at the time officers purchased their own ranks and it cost more to purchase a commission in the Foot Guards than any other regiments in the British Army. For example if it cost an officer in the Foot Guards £1,000 for his first rank, in the rest of the Army it would be £500 so if he transferred to another regiment he would loose [sic] £500, hence the higher rank, if he was an Ensign in the Guards and he transferred to a Line Regiment he went in at the higher rank of Lieutenant.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/|title=Formation and role of the Regiments|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/.</ref></blockquote> ==== Life Guards ==== [[Social Victorians/People/Shrewsbury#Reginald Talbot's Costume|General the Hon. Reginald Talbot]], a member of the 1st Life Guards, attended the Duchess of Devonshire's ball dressed in the uniform of his regiment during the Battle of Waterloo.<ref name=":14" />{{rp|p. 36, Col. 3b}} At the Battle of Waterloo the 1st Life Guards were part of the 1st Brigade — the Household Brigade — and were commanded by Major-General Lord Edward Somerset.<ref name=":4">{{Cite journal|date=2023-09-30|title=Battle of Waterloo|url=https://en.wikipedia.org/w/index.php?title=Battle_of_Waterloo&oldid=1177893566|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Battle_of_Waterloo.</ref> The 1st Life Guards were on "the extreme right" of a French countercharge and "kept their cohesion and consequently suffered significantly fewer casualties."<ref name=":4" /> == Photography == == Footnotes == {{reflist}} 0xfxi5ug8t59cqpimv08tcvluy3lp8t 2690288 2690287 2024-12-04T17:47:00Z Scogdill 1331941 /* Hoops */ 2690288 wikitext text/x-wiki Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. The definitions on this page are to provide a sense of what someone in the late 19th century might have meant by the term rather than a definition of what we might mean by it today. In the absence of a specialized glossary from the end of the 19th century in the U.K., we use the ''Oxford English Dictionary'' because the senses of a word are illustrated with examples that have dates so we can be sure that the senses we pick are appropriate for when they are used in the quotations we have. We also sometimes use the French ''Wikipédia'' to define a word because many technical terms of fashion were borrowings from the French. Also, often the French ''Wikipédia'' provides historical context for the uses of a word similar to the way the OED does. == Articles or Parts of Clothing: Non-gender-specific == === Mantle, Cloak, Cape === In 19th-century newspaper accounts, these terms are sometimes used without precision as synonyms. These are all outer garments. '''Mantle''' A mantle — often a long outer garment — might have elements like a train, sleeves, collars, revers, fur, and a cape. A late-19th-century writer making a distinction between a mantle and a cloak might use ''mantle'' if the garment is more voluminous. '''Cloak''' '''Cape''' === Peplum === According to the French ''Wiktionnaire'', a peplum is a "Short skirt or flared flounce layered at the waist of a jacket, blouse or dress" [translation by Google Translate].<ref>{{Cite journal|date=2021-07-02|title=péplum|url=https://fr.wiktionary.org/w/index.php?title=p%C3%A9plum&oldid=29547727|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/p%C3%A9plum.</ref> The ''Oxford English Dictionary'' has a fuller definition, although, it focuses on women's clothing because the sense is written for the present day:<blockquote>''Fashion''. ... a kind of overskirt resembling the ancient peplos (''obsolete''). Hence (now usually) in modern use: a short flared, gathered, or pleated strip of fabric attached at the waist of a woman's jacket, dress, or blouse to create a hanging frill or flounce.<ref name=":5">“peplum, n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1832614702>.</ref></blockquote>Men haven't worn peplums since the 18th century, except when wearing costumes based on historical portraits. The ''Daily News'' reported in 1896 that peplums had been revived as a fashion item for women.<ref name=":5" /> === Revers === According to the ''Oxford English Dictionary'', ''revers'' are the "edge[s] of a garment turned back to reveal the undersurface (often at the lapel or cuff) (chiefly in ''plural''); the material covering such an edge."<ref>"revers, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/164777. Accessed 17 April 2023.</ref> The term is French and was used this way in the 19th century (according to the ''Wiktionnaire'').<ref>{{Cite journal|date=2023-03-07|title=revers|url=https://fr.wiktionary.org/w/index.php?title=revers&oldid=31706560|journal=Wiktionnaire|language=fr}} https://fr.wiktionary.org/wiki/revers.</ref> == Articles or Parts of Clothing: Men's == [[Social Victorians/Terminology#Military|Men's military uniforms]] are discussed below. === À la Romaine === [[File:Johann Baptist Straub - Mars um 1772-1.jpg|thumb|left|alt=Old and damaged marble statue of a Roman god of war with flowing cloak, big helmet with a plume on top, and armor|Johann Baptist Straub's 1772 ''à la romaine'' ''Mars'']] A few people who attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball in 1897]] personated Roman gods or people. They were dressed not as Romans, however, but ''à la romaine'', which was a standardized style of depicting Roman figures that was used in paintings, sculpture and the theatre for historical dress from the 17th until the 20th century. The codification of the style was developed in France in the 17th century for theatre and ballet, when it became popular for masked balls. Women as well as men could be dressed ''à la romaine'', but much sculpture, portraiture and theatre offered opportunities for men to dress in Roman style — with armor and helmets — and so it was most common for men. In large part because of the codification of the style as well as the painting and sculpture, the style persisted and remained influential into the 20th century and can be found in museums and galleries and on monuments. For example, Johann Baptist Straub's 1772 statue of Mars (left), now in the Bayerisches Nationalmuseum, Munich, missing part of an arm, shows Mars ''à la romaine''. In London, an early 17th-century example of a figure of Mars ''à la romaine'', with a helmet, '''was''' "at the foot of the Buckingham tomb in Henry VII's Chapel at Westminster Abbey."<ref>Webb, Geoffrey. “Notes on Hubert Le Sueur-II.” ''The Burlington Magazine for Connoisseurs'' 52, no. 299 (1928): 81–89. http://www.jstor.org/stable/863535.</ref>{{rp|81, Col. 2c}} === Cavalier === [[File:Sir-Anthony-van-Dyck-Lord-John-Stuart-and-His-Brother-Lord-Bernard-Stuart.jpg|thumb|alt=Old painting of 2 men flamboyantly and stylishly dressed in colorful silk, with white lace, high-heeled boots and long hair|Van Dyck's c. 1638 painting of cavaliers Lord John Stuart and his brother Lord Bernard Stuart]] As a signifier in the form of clothing of a royalist political and social ideology begun in France in the early 17th century, the cavalier established France as the leader in fashion and taste. Adopted by [[Social Victorians/Terminology#Military|wealthy royalist British military officers]] during the time of the Restoration, the style signified a political and social position, both because of the loyalty to Charles I and II as well the wealth required to achieve the cavalier look. The style spread beyond the political, however, to become associated generally with dress as well as a style of poetry.<ref>{{Cite journal|date=2023-04-25|title=Cavalier poet|url=https://en.wikipedia.org/w/index.php?title=Cavalier_poet&oldid=1151690299|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier_poet.</ref> Van Dyck's 1638 painting of two brothers (right) emphasizes the cavalier style of dress. === Coats === ==== Doublet ==== * In the 19th-century newspaper accounts we have seen that use this word, doublet seems always to refer to a garment worn by a man, but historically women may have worn doublets. In fact, a doublet worn by Queen Elizabeth I exists and '''is somewhere'''. * Technically doublets were long sleeved, although we cannot be certain what this or that Victorian tailor would have done for a costume. For example, the [[Social Victorians/People/Spencer Compton Cavendish#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Duke of Devonshire's costume as Charles V]] shows long sleeves that may be part of the surcoat but should be the long sleeves of the doublet. ==== Pourpoint ==== A padded doublet worn under armor to protect the warrior from the metal chafing. A pourpoint could also be worn without the armor. ==== Surcoat ==== Sometimes just called ''coat''. [[File:Oscar Wilde by Sarony 1882 18.jpg|thumb|alt=Old photograph of a young man wearing a velvet jacket, knee breeches, silk hose and shiny pointed shoes with bows, seated on a sofa and leaning on his left hand and holding a book in his right| Oscar Wilde, 1882, by Napoleon Sarony]] === Hose, Stockings and Tights === Newspaper accounts from the late 19th century of men's clothing use the term ''hose'' for what we might call stockings or tights. In fact, the terminology is specific. ''Stockings'' is the more general term and could refer to hose or tights. With knee breeches men wore hose, which ended above the knee, and women wore hose under their dresses. The ''Oxford English Dictionary'' defines tights as "Tight-fitting breeches, worn by men in the 18th and early 19th centuries, and still forming part of court-dress."<ref>“Tights, N.” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2693287467.</ref> By 1897, the term was in use for women's stockings, which may have come up only to the knee. Tights were also worn by dancers and acrobats. This general sense of ''tights'' does not assume that they were knitted. ''Clocking'' is decorative embroidery on hose, usually, at the ankles on either the inside or the outside of the leg. It started at the ankle and went up the leg, sometimes as far as the knee. On women's hose, the clocking could be quite colorful and elaborate, while the clocking on men's hose was more inconspicuous. In many photographs men's hose are wrinkled, especially at the ankles and the knees, because they were shaped from woven fabric. Silk hose were knitted instead of woven, which gave them elasticity and reduced the wrinkling. The famous Sarony carte de visite photograph of Oscar Wilde (right) shows him in 1882 wearing knee breeches and silk hose, which are shiny and quite smoothly fitted although they show a few wrinkles at the ankles and knees. In the portraits of people in costume at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the men's hose are sometimes quite smooth, which means they were made of knitted silk and may have been smoothed for the portrait. In painted portraits the hose are almost always depicted as smooth, part of the artist's improvement of the appearance of the subject. === Shoes and Boots === == Articles or Parts of Clothing: Women's == === '''Chérusque''' === According to the French ''Wikipedia'', ''chérusque'' is a 19th-century term for the kind of standing collar like the ones worn by ladies in the Renaissance.<ref>{{Cite journal|date=2021-06-26|title=Collerette (costume)|url=https://fr.wikipedia.org/w/index.php?title=Collerette_(costume)&oldid=184136746|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Collerette_(costume)#Au+xixe+siècle+:+la+Chérusque.</ref> === Corsage === According to the ''Oxford English Dictionary'', the corsage is the "'body' of a woman's dress; a bodice."<ref>"corsage, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/42056. Accessed 7 February 2023.</ref> This sense is well documented in the ''OED'' for the mid and late 19th-century, used this way in fiction as well as in a publication like ''Godey's Lady's Book'', which would be expected to use appropriate terminology associated with fashion and dress making. The sense of "a bouquet worn on the bodice" is, according to the ''OED'', American. === Décolletage === === Girdle === === Mancheron === According to the ''Oxford English Dictionary'', a ''mancheron'' is a "historical" word for "A piece of trimming on the upper part of a sleeve on a woman's dress."<ref>"mancheron, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/113251. Accessed 17 April 2023.</ref> At the present, in French, a ''mancheron'' is a cap sleeve "cut directly on the bodice."<ref>{{Cite journal|date=2022-11-28|title=Manche (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manche_(v%C3%AAtement)&oldid=199054843|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Manche_(v%C3%AAtement).</ref> === Petticoat === According to the ''O.E.D.'', a petticoat is a <blockquote>skirt, as distinguished from a bodice, worn either externally or showing beneath a dress as part of the costume (often trimmed or ornamented); an outer skirt; a decorative underskirt. Frequently in ''plural'': a woman's or girl's upper skirts and underskirts collectively. Now ''archaic'' or ''historical''.<ref>“petticoat, n., sense 2.b”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1021034245></ref> </blockquote>This sense is, according to the ''O.E.D.'', "The usual sense between the 17th and 19th centuries." However, while petticoats belong in both outer- and undergarments — that is, meant to be seen or hidden, like underwear — they were always under another garment, for example, underneath an open overskirt. The primary sense seems to have shifted through the 19th century so that, by the end, petticoats were underwear and the term ''underskirt'' was used to describe what showed under an open overskirt. === Stomacher === According to the ''O.E.D.'', a stomacher is "An ornamental covering for the chest (often covered with jewels) worn by women under the lacing of the bodice,"<ref>“stomacher, n.¹, sense 3.a”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1169498955></ref> although by the end of the 19th century, the bodice did not often have visible laces. Some stomachers were so decorated that they were thought of as part of the jewelry. === Train === A train is The Length of the Train '''For the monarch [or a royal?]''' According to Debrett's,<blockquote>A peeress's coronation robe is a long-trained crimson velvet mantle, edged with miniver pure, with a miniver pure cape. The length of the train varies with the rank of the wearer: * Duchess: for rows of ermine; train to be six feet * Marchioness: three and a half rows of ermine; train to be three and three-quarters feet * Countess: three rows of ermine; train to be three and a half feet * Viscountess: two and a half rows of ermine; train to be three and a quarter feet * Baroness: two rows of ermine; train to be three feet<ref name=":2">{{Cite web|url=https://debretts.com/royal-family/dress-codes/|title=Dress Codes|website=debretts.com|language=en-US|access-date=2023-07-27}} https://debretts.com/royal-family/dress-codes/.</ref> </blockquote>The pattern on the coronet worn was also quite specific, similar but not exactly the same for peers and peeresses. Debrett's also distinguishes between coronets and tiaras, which were classified more like jewelry, which was regulated only in very general terms. Peeresses put on their coronets after the Queen or Queen Consort has been crowned. ['''peers?'''] === Foundation Garments === Unlike undergarments, Victorian women's foundation garments created the distinctive silhouette. Victorian undergarments included the chemise, the bloomers, the corset cover — articles that are not structural. The corset was an important element of the understructure of foundation garments — hoops, bustles, petticoats and so on — but it has never been the only important element. === Corset === [[File:Corset - MET 1972.209.49a, b.jpg|thumb|alt=Photograph of an old silk corset on a mannequin, showing the closure down the front, similar to a button, and channels in the fabric for the boning. It is wider at the top and bottom, creating smooth curves from the bust to the compressed waist to the hips, with a long point below the waist in front.|French 1890s corset, now in the Metropolitan Museum of Art, NYC]] The understructure of Victorian women's clothing is what makes the costumes worn by the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] so distinctly Victorian in appearance. An example of a corset that has the kind of structure often worn by fashionably dressed women in 1897 is the one at right. This corset exaggerated the shape of the women's bodies and made possible a bodice that looked and was fitted in the way that is so distinctive of the time — very controlled and smooth. And, as a structural element, this foundation garment carried the weight of all those layers and all that fabric and decoration on the gowns, trains and mantles. (The trains and mantles could be attached directly to the corset itself.) * This foundation emphasizes the waist and the bust in particular, in part because of the contrast between the very small waist and the rounded fullness of the bust and hips. * The idealized waist is defined by its small span and the sexualizing point at the center-bottom of the bodice, which directs the eye downwards. Interestingly, the pointed waistline worn by Elizabethan men has become level in the Victorian age. Highly fashionable Victorian women wearing the traditional style, however, had extremely pointed waists. * The busk (a kind of boning in the front of a corset that is less flexible than the rest) smoothed the bodice, flattened the abdomen and prevented the point on the bodice from curling up. * The sharp definition of the waist was caused by ** length of the corset (especially on the sides) ** the stiffness of the boning ** the layers of fabric ** the lacing (especially if the woman used tightlacing) ** the over-all shape, which was so much wider at the top and the bottom ** the contrast between the waist and the wider top and bottom * The late-19th-century corset was long, ending below the waist even on the sides and back. * The boning and the top edge of the late 19th-century fashion corset pushed up the bust, rounding (rather than flattening, as in earlier styles) the breasts, drawing attention to their exposed curves and creating cleavage. * The exaggerated bust was larger than the hips, whenever possible, an impression reinforced by the A-line of the skirt and the inverted Vs in the decorative trim near the waist and on the skirt. * This corset made the bodice very smooth with a very precise fit, that had no wrinkles, folds or loose drapery. The bodice was also trimmed or decorated, but the base was always a smooth bodice. More formal gowns would still have the fitted bodice and more elaborate trim made from lace, embroidery, appliqué, beading and possibly even jewels. The advantages and disadvantages of corseting and especially tight lacing were the subject of thousands of articles and opinions in the periodical press for a great part of the century, but the fetishistic and politicized tight lacing was practiced by very few women. And no single approach to corsetry was practiced by all women all the time. Most of the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 ball]] were not tightly laced, but the progressive style does not dominate either, even though all the costumes are technically historical dress. Part of what gives most of the costumes their distinctive 19th-century "look" is the more traditional corset beneath them. Even though this highly fashionable look was widely present in the historical costumes at the ball, some women's waists were obviously very small and others were hardly '''emphasized''' at all. Women's waists are never mentioned in the newspaper coverage of the ball — or, indeed, of any of the social events attended by the network at the ball — so it is only in photographs that we can see the effects of how they used their corsets. === Hoops === Striking for how long they lasted and '''how much''' they evolved, hoops were the foundation undergarment for a skirt and petticoat. Women wore hoops from the '''15th century''', around the time of Katherine of Aragon, through the bustle of the late 19th century. The cage caused the silhouette of skirts to change shape over time and enabled the extreme distortions of panniers and the bustle. Hoops circled the body symmetrically in a cone or drum shape, were moved to the sides with panniers, ballooned around the body like the top half of a sphere, and were pulled to the rear with a bustle. That is, like corsets, the distorted shapes of high fashion were made possible by hoops. High fashion demanded these shapes, which disguised women's bodies, '''especially below the waist'''. Besides the shape, the structure used to construct hoops changed — cane, wood, whalebone, steel or wire. Add fabric structural stuff ''Hoops'' is a mid-19th-century term for a cage-like structure worn under a skirt to hold it away from the body. [[File:Pedro García de Benabarre St John Retable Detail.jpg|thumb|Pedro García de Benabarre St John Retable Detail]] [[File:Alonso Sánchez Coello 011.jpg|thumb|Alonso Sánchez Coello 011]] ==== 15th Century ==== Hoops began in Spain in the 15th century and influenced European fashion for many years:<blockquote>Katherine of Aragon is reputed to have introduced the Spanish farthingale (a hooped underskirt) into England early in the century. The result was to convert the columnar skirt of the fifteenth century into the cone shape of the sixteenth. ...<p> Spanish influence had introduced the hoop-supported skirt, smooth in contour, which was quite generally worn.<ref name=":11" /> (291)</blockquote>Illustrations ==== 16th Century ==== [[File:Farthingale 2 (PSF).png|thumb|Farthingale 2 (PSF)]] In the 16th century, the garment we call ''hoops'' was called a farthingale. ''Farthingale'' is the term in English; in French, it's ''vertugadin'', and in Spanish ''vertugado''. ''Vertugadin'' is a French term for ''farthingale'', a cage made of hoops supporting a skirt — "un élément essentiel de la mode Tudor en Angleterre [an essential element of Tudor fashion in England]."<ref name=":0">{{Cite journal|date=2022-03-12|title=Vertugadin|url=https://fr.wikipedia.org/w/index.php?title=Vertugadin&oldid=191825729|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Vertugadin.</ref> In fact, "The Spanish princess Catherine of Aragon brought the fashion to England for her marriage to Prince Arthur, eldest son of Henry VII in 1501 [La princesse espagnole Catherine d'Aragon amena la mode en Angleterre pour son mariage avec le prince Arthur, fils aîné d'Henri VII en 1501]."<ref name=":0" /> Catherine of Aragon, of course, married Henry VIII after Arthur's death. The French and Spanish farthingales were not identical by the end of the 16th century. The Spanish farthingale shaped the skirt into an A-line with a graduated series of hoops sewn to an undergarment. The French farthingale was a flattish "cartwheel" or platter of hoops worn below the waist and above the hips held the skirt out more or less horizontally. Once past the vertugadin, the skirt then fell straight to the floor, shaping the skirt into a kind of drum. The shoes show in the portraits of women wearing the French farthingale. <blockquote>"FARTHINGALE:  Renaissance (1450-1550 C.E. to Elizabethan (1550-1625 C.E.). Linen underskirt with wire supports which, when shaped, produced a variety of dome, bell, and oblong shapes."<ref name=":7" /> (105)</blockquote>By the 18th century, it was called hoops, which were made of wood. [[File:Panniers 1.jpg|thumb|alt=Photograph of the wooden and fabric skeleton of an 18th-century women's foundation garment|Panniers 1]]==== 17th Century ==== Generally, panniers were a kind of undergarment worn in the 17th and 18th centuries. Their design evolved during the century. Made of hoops of wood, they were "baskets" or cages worn on either side of the waist to broaden the skirts to the sides. Illustration ==== 18th Century ==== Blanche Payne outlines the evolution of hoops, and thus the shape of the skirt, in the 18th century:<blockquote>SKIRT FASHIONS. Since skirts experienced the greatest alterations, a brief summary of the successive silhouettes should help to place individual costumes in their proper niches. Six basic forms appeared during the century, in the following order: # The bustle was a continuation of the 1690 mode. # The bell or dome shape resulted from the reintroduction of hoops; in England by 1710, in France by 1720. # The ellipse, the second phase of the hoop skirt, was achieved by broadening the support from side to side and compressing it from front to back. It had a long run of popularity, from 1740 to 1770, the extreme width being retained in court costumes. In France it persisted until the revolution, except that skirts were allowed to curve outward in [the] back again. English court costume followed this fashion well into the nineteenth century. # The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, which were used to form the three great ‘poufs’ known as the polonaise (see Fig. 452). These diversions appeared in the late sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period. # The return of the bustle in the 1780s. # The tubular form, drawn from classic art, in the 1790s.<ref name=":11" /> (411–413) </blockquote>While we think of the bustle as a 19th-century look, it can be found in the 18th century, as Payne says. Polonaise may need its own section. ==== 19th Century ==== [[File:Cutaway sketch of crinoline.gif|thumb|Cutaway sketch of crinoline]] [[File:Paris voulant englober la banlieue.JPG|thumb|Paris voulant englober la banlieue]] [[File:Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723.jpg|thumb|Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723]] In the 19th century, the hoops were made of wire and became lighter. By the 1860s, hoops made for huge round skirts. In Laura Ingalls Wilder's 1941 ''Little Town on the Prairie'', the '''16-year-old Laura wears hoops. Laura Ingalls Wilder and thus the character Laura were born in 1862, so this moment is set in 1883'''.<ref>Hill, Pamela Smith, ed. ''Pioneer Girl: The Annotated Autobiography''.</ref> She makes the comment that she wants to be in style, but that would be on the prairie and not necessarily the latest Parisian style.<blockquote>“Well,” Laura began; then she stopped and spun round and round, for the strong wind blowing against her always made the wires of her hoop skirt creep slowly upward under her skirts until they bunched around her knees. Then she must whirl around and around until the wires shook loose and spiraled down to the bottom of her skirts where they should be. “As she and Carrie hurried '''ob''' she began again. “I think it was silly, the way they dressed when Ma was a girl, don’t you? Drat this wind!” she exclaimed as the hoops began creeping upward again. “Quietly Carrie stood by while Laura whirled. “I’m glad I’m not old enough to have to wear hoops,” she said. “They’d make me dizzy.” “They are rather a nuisance,” Laura admitted. “But they are stylish, and when you’re my age you’ll want to be in style.”<ref>Wilder, Laura Ingalls. ''Little Town on the Prairie.'' Harper and Row, 1941. Pp. 272–273.</ref></blockquote>This very unusual description of the way the wind could make hoops creep and the solution. It must have been happening to other women wearing hoops at the time. == '''Traditional vs Progressive Style''' == === Progressive Style === The terms ''artistic dress'' and ''aesthetic dress'' are not synonymous and were in use at different times to refer to different groups of people in different contexts, but we recognize them as referring to a similar kind of personal style in clothing, a style we call progressive dress or the progressive style. Used in a very precise way, ''artistic dress'' is associated with the Pre-Raphaelite artists and the women in their circle beginning in the 1860s. Similarly, ''aesthetic dress'' is associated with the 1880s and 1890s and dress reform movements. In general, the progressive style is characterized by its resistance to the highly structured fashion of its day, especially corseting, aniline dyes and an extremely close fit. === Traditional Style === By the end of the century designs from the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]] (or Maison Worth) define what we think of as the traditional Victorian look, which was very stylish and expensive. Blanche Payne describes an example of the 1895 "high style" in a gown by Worth with "the idiosyncrasies of the [1890s] full blown":<blockquote>The dress is white silk with wine-red stripes. Sleeves, collars, bows, bag, hat, and hem border match the stripes. The sleeve has reached its maximum volume; the bosom full and emphasized with added lace; the waistline is elongated, pointed, and laced to the point of distress; the skirt is smooth over the hips, gradually swinging out to sweep the floor. This is the much vaunted hourglass figure.<ref name=":11">Payne, Blanche. ''History of Costume from the Ancient Egyptians to the Twentieth Century''. Harper & Row, 1965.</ref>{{rp|530}}</blockquote> The Victorian-looking gowns at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] are stylish in a way that recalls the designs of the House of Worth. The elements that make their look so Victorian are anachronisms on the costumes representing fashion of earlier eras. The women wearing these gowns preferred the standards of beauty from their own day to a more-or-less historically accurate look. The style competing at the very end of the century with the Worth look was not the historical, however, but a progressive style called at the time ''artistic'' or ''aesthetic''. William Powell Frith's 1883 painting ''A Private View at the Royal Academy, 1881'' (discussion below) pits this kind of traditional style against the progressive or artistic style. === The Styles === [[File:Frith A Private View.jpg|thumb|William Powell Frith, ''A Private View at the Royal Academy, 1881'']] We typically think of the late-Victorian silhouette as universal but, in the periods in which corsets dominated women's dress, not all women wore corsets and not all corsets were the same, as William Powell Frith's 1883 ''A Private View at the Royal Academy, 1881'' (right) illustrates. Frith is clear in his memoir that this painting — "recording for posterity the aesthetic craze as regards dress" — deliberately contrasts what he calls the "folly" of the Artistic Dress movement and the look of the traditional corseted waist.<ref>Frith, William Powell. ''My Autobiography and Reminiscences''. 1887.</ref> Frith considered the Artistic Movement and Artistic Dress "ephemeral," but its rejection of corsetry looks far more consequential to us in hindsight than it did in the 19th century. As Frith sees it, his painting critiques the "craze" associated with the women in this set of identifiable portraits who are not corseted, but his commitment to realism shows us a spectrum, a range, of conservatism and if not political then at least stylistic progressivism among the women. The progressives, oddly, are the women wearing artistic (that is, somewhat historical) dress, because they’re not corseted. It is a misreading to see the presentation of the women’s fashion as a simple opposition. Constance, Countess of Lonsdale — situated at the center of this painting with Frederick Leighton, president of the Royal Academy of Art — is the most conservatively dressed of the women depicted, with her narrow sleeves, tight waist and almost perfectly smooth bodice, which tells us that her corset has eyelets so that it can be laced precisely and tightly, and it has stays (or "bones") to prevent wrinkles or natural folds in the overclothing. Lillie Langtry, in the white dress, with her stylish narrow sleeves, does not have such a tightly bound waist or smooth bodice, suggesting she may not be corseted at all, as we know she sometimes was not.['''citation'''] Jenny Trip, a painter’s model, is the woman in the green dress in the aesthetic group being inspected by Anthony Trollope, who may be taking notes. She looks like she is not wearing a corset. Both Langtry and Trip are toward the middle of this spectrum: neither is dressed in the more extreme artistic dress of, say, the two figures between Trip and Trollope. A lot has been written about the late-Victorian attraction to historical dress, especially in the context of fancy-dress balls and the Gothic revival in social events as well as art and music. Part of the appeal has to have been the way those costumes could just be beautiful clothing beautifully made. Historical dress provided an opportunity for some elite women to wear less-structured but still beautiful and influential clothing. ['''Calvert'''<ref>Calvert, Robyne Erica. ''Fashioning the Artist: Artistic Dress in Victorian Britain 1848-1900''. Ph.D. thesis, University of Glasgow, 2012. <nowiki>https://theses.gla.ac.uk/3279/</nowiki></ref>] The standards for beauty, then, with historical dress were Victorian, with the added benefit of possibly less structure. So, at the Duchess of Devonshire's ball, "while some attendees tried to hew closely to historical precedent, many rendered their historical or mythological personage in the sartorial vocabulary they knew best. The [photographs of people in their costumes at the ball offer] a glimpse into how Victorians understood history, not a glimpse into the costume of an authentic historical past."<ref>Mitchell, Rebecca N. "The Victorian Fancy Dress Ball, 1870–1900." ''Fashion Theory'' 2017 (21: 3): 291–315. DOI: 10.1080/1362704X.2016.1172817.</ref> (294) * historical dress: beautiful clothing. * the range at the ball, from Minnie Paget to Gwladys * "In light of such efforts, the ball remains to this day one of the best documented outings of the period, and a quick glance at the album shows that ..." Women had more choices about their waists than the simple opposition between no corset and tightlacing can accommodate. The range of choices is illustrated in Frith's painting, with a woman locating herself on it at a particular moment for particular reasons. Much analysis of 19th-century corsetry focuses on its sexualizing effects — corsets dominated Victorian photographic pornography ['''citations'''] and at the same time, the absence of a corset was sexual because it suggested nudity.['''citations'''] A great deal of analysis of 19th-century corsetry, on the other hand, assumes that women wore corsets for the male gaze ['''citations'''] or that they tightened their waists to compete with other women.['''citations'''] But as we can see in Frith's painting, the sexualizing effect was not universal or sweeping, and these analyses do not account for the choices women had in which corset to wear or how tightly to lace it. Especially given the way that some photographic portraits were mechanically altered to make the waist appear smaller, the size of a woman's waist had to do with how she was presenting herself to the world. That is, the fact that women made choices about the size of or emphasis on their waists suggests that they had agency that needs to be taken into account. As they navigated the complex social world, women's fashion choices had meaning. Society or political hostesses had agency not only in their clothing but generally in that complex social world. They had roles managing social events of the upper classes, especially of the upper aristocracy and oligarchy, like the Duchess of Devonshire's ball. Their class and rank, then, were essential to their agency, including to some degree their freedom to choose what kind of corset to wear and how to wear it. Also, by the end of the century lots of different kinds of corsets were available for lots of different purposes. Special corsets existed for pregnancy, sports (like tennis, bicycling, horseback riding, golf, fencing, archery, stalking and hunting), theatre and dance and, of course, for these women corsets could be made to support the special dress worn over it. Women's choices in how they presented themselves to the world included more than just their foundation garments, of course. "Every cap, bow, streamer, ruffle, fringe, bustle, glove," that is, the trim and decorations on their garments, their jewelry and accessories — which Davidoff calls "elaborations"<ref name=":1">Davidoff, Leonore. ''The Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig 1973).</ref>{{rp|93}} — pointed to a host of status categories, like class, rank, wealth, age, marital status, engagement with the empire, how sexual they wanted to seem, political alignment and purpose at the social event. For example, when women were being presented to the monarch, they were expected to wear three ostrich plumes, often called the [[Social Victorians/Terminology#Prince of Wales's Feathers or White Plumes|Prince of Wales's feathers]]. Like all fashions, the corset, which was quite long-lasting in all its various forms, eventually went out of style. Of the many factors that might have influenced its demise, perhaps most important was the women's movement, in which women's rights, freedom, employment and access to their own money and children were less slogan-worthy but at least as essential as votes for women. The activities of the animal-rights movements drew attention not only to the profligate use of the bodies and feathers of birds but also to the looming extinction of the baleen whale, which made whale bone scarce and expensive. Perhaps the century's debates over corseting and especially tightlacing were relevant to some decisions not to be corseted. And, of course, perhaps no other reason is required than that the nature of fashion is to change. == Cinque Cento == According to the ''Oxford English Dictionary'', ''Cinque Cento'' is a shortening of ''mil cinque cento'', or 1500.<ref>"cinquecento, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/33143. Accessed 7 February 2023.</ref> The term, then would refer, perhaps informally, to the sixteenth century. == Crevé == ''Creve'', without the accent, is an old word in English (c. 1450) for burst or split.<ref>"creve, v." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44339. Accessed 8 February 2023.</ref> ['''With the acute accent, it looks like a past participle in French.'''] == Elastic == Elastic had been invented and was in use by the end of the 19th century. For the sense of "Elastic cord or string, usually woven with india-rubber,"<ref name=":6">“elastic, adj. & n.”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1199670313>.</ref> the ''Oxford English Dictionary'' has usage examples beginning in 1847. The example for 1886 is vivid: "The thorough-going prim man will always place a circle of elastic round his hair previous to putting on his college cap."<ref name=":6" /> == Elaborations == In her 1973 ''The Best Circles: Society, Etiquette and the Season'', Leonore Davidoff notes that women’s status was indicated by dress and especially ornament: “Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration,” she says, “symbolised some status category for the female wearer.”<ref name=":1" />{{rp|93}} Looking at these elaborations as meaningful rather than dismissing them as failed attempts at "historical accuracy" reveals a great deal about the individual women who wore or carried them — and about the society women and political hostesses in their roles as managers of the social world. In her review of ''The House of Worth: Portrait of an Archive'', Mary Frances Gormally says,<blockquote>In a socially regulated year, garments custom made with a Worth label provided women with total reassurance, whatever the season, time of day or occasion, setting them apart as members of the “Best Circles” dressed in luxurious, fashionable and always appropriate attire (Davidoff 1973). The woman with a Worth wardrobe was a woman of elegance, lineage, status, extreme wealth and faultless taste.<ref>Gormally, Mary Frances. Review essay of ''The House of Worth: Portrait of an Archive'', by Amy de la Haye and Valerie D. Mendes (V&A Publishing, 2014). ''Fashion Theory'' 2017 (21, 1): 109–126. DOI: 10.1080/1362704X.2016.1179400.</ref> (117)</blockquote> === Aglet, Aiglet === Historically, an aglet is a "point or metal piece that capped a string [or ribbon] used to attach two pieces of the garment together, i.e., sleeve and bodice."<ref name=":7" />{{rp|4}} Although they were decorative, they were not always visible on the outside of the clothing. They were often stuffed inside the layers at the waist (for example, attaching the bodice to a skirt or breeches). === Frou-frou === In French, ''frou-frou'' or, spelled as ''froufrou'', is the sound of the rustling of silk or sometimes of fabrics in general.<ref>{{Cite journal|date=2023-07-25|title=frou-frou|url=https://fr.wiktionary.org/w/index.php?title=frou-frou&oldid=32508509|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/frou-frou.</ref> The first use the French ''Wiktionnaire'' lists is Honoré Balzac, ''La Cousine Bette'', 1846.<ref>{{Cite journal|date=2023-06-03|title=froufrou|url=https://fr.wiktionary.org/w/index.php?title=froufrou&oldid=32330124|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/froufrou.</ref> ''Frou-frou'' is a term clothing historians use to describe decorative additions to an article of clothing; often the term has a slight negative connotation, suggesting that the additions are superficial. === Pouf, Puff, Poof === According to the French ''Wikipédia'', a pouf was, beginning in 1744, a "kind of women's hairstyle":<blockquote>The hairstyle in question, known as the “pouf”, had launched the reputation of the enterprising Rose Bertin, owner of the Grand Mogol, a very prominent fashion accessories boutique on Rue Saint-Honoré in Paris in 1774. Created in collaboration with the famous hairdresser, Monsieur Léonard, the pouf was built on a scaffolding of wire, fabric, gauze, horsehair, fake hair, and the client's own hair held up in an almost vertical position. — (Marie-Antoinette, ''Queen of Fashion'', translated from the American by Sylvie Lévy, in ''The Rules of the Game'', n° 40, 2009)</blockquote>''Puff'' and ''poof'' are used to describe clothing. === Shirring === ''Shirring'' is the gathering of fabric to make poufs or puffs. The 19th century is known for its use of this decorative technique. Even men's clothing had shirring: at the shoulder seam. === Sequins === Sequins, paillettes, spangles Sequins — or paillettes — are "small, scalelike glittering disks."<ref name=":7" />(216) The French ''Wiktionnaire'' defines ''paillette'' as "Lamelle de métal, brillante, mince, percée au milieu, ordinairement ronde, et qu’on applique sur une étoffe pour l’orner [A strip of metal, shiny, thin, pierced in the middle, usually round, and which is applied to a fabric in order to decorate it.]"<ref name=":8">{{Cite journal|date=2024-03-18|title=paillette|url=https://fr.wiktionary.org/w/index.php?title=paillette&oldid=33809572|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/paillette.</ref> According to the ''OED'', the use of ''sequin'' as a decorative device for clothing (as opposed to gold coins minted and used for international trade) goes back to the 1850s.<ref>“Sequin, N.” ''Oxford English Dictionary'', Oxford UP, September 2023, https://doi.org/10.1093/OED/4074851670.</ref> The first instance of ''spangle'' as "A small round thin piece of glittering metal (usually brass) with a hole in the centre to pass a thread through, used for the decoration of textile fabrics and other materials of various sorts" is from c. 1420.<ref>“Spangle, N. (1).” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/4727197141.</ref> The first use of ''paillette'' listed in the French ''Wiktionnaire'' is in Jules Verne in 1873 to describe colored spots on icy walls.<ref name=":8" /> Currently many distinguish between sequins (which are smaller) and paillettes (which are larger). Before the 20th century, sequins were metal discs or foil leaves, and so of course if they were silver or copper, they tarnished. It is not until well into the 20th century that plastics were invented and used for sequins. === Trim and Lace === ''A History of Feminine Fashion'', published sometime before 1927 and probably commissioned by [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|the Maison Worth]], describes Charles Frederick Worth's contributions to the development of embroidery and [[Social Victorians/Terminology#Passementerie|passementerie]] (trim) from about the middle of the 19th century:<blockquote>For it must be remembered that one of M. Worth's most important and lasting contributions to the prosperity of those who cater for women's needs, as well as to the variety and elegance of his clients' garments, was his insistence on new fabrics, new trimmings, new materials of every description. In his endeavours to restore in Paris the splendours of the days of La Pompadour, and of Marie Antoinette, he found himself confronted at the outset with a grave difficulty, which would have proved unsurmountable to a man of less energy, resource and initiative. The magnificent materials of those days were no longer to be had! The Revolution had destroyed the market for beautiful materials of this, type, and the Restoration and regime of Louis Philippe had left a dour aspect in the City of Light. ... On parallel lines [to his development of better [[Social Victorians/Terminology#Satin|satin]]], [Worth] stimulated also the manufacture of embroidery and ''passementerie''. It was he who first started the manufacture of laces copied from the designs of the real old laces. He was the / first dressmaker to use fur in the trimming of light materials — but he employed only the richer furs, such as sable and ermine, and had no use whatever for the inferior varieties of skins.<ref name=":9" />{{rp|6–7}}</blockquote> ==== Gold and Silver Fabric and Lace ==== The ''Encyclopaedia Britannica'' (9th edition) has an article on gold and silver fabric, threads and lace attached to the article on gold. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>GOLD AND SILVER LACE. Under this heading a general account may be given of the use of the precious metals in textiles of all descriptions into which they enter. That these metals were used largely in the sumptuous textiles of the earliest periods of civilization there is abundant testimony; and to this day, in the Oriental centres whence a knowledge and the use of fabrics inwoven, ornamented, and embroidered with gold and silver first spread, the passion for such brilliant and costly textiles is still most strongly and generally prevalent. The earliest mention of the use of gold in a woven fabric occurs in the description of the ephod made for Aaron (Exod. xxxix. 2, 3) — "And he made the ephod of gold, blue, and purple, and scarlet, and fine twined linen. And they did beat the gold into thin plates, and cut it into wires (strips), to work it in the blue, and in the purple, and in the scarlet, and in the fine linen, with cunning work." In both the ''Iliad'' and the ''Odyssey'' distinct allusion is frequently made to inwoven and embroidered golden textiles. Many circumstances point to the conclusion that the art of weaving and embroidering with gold and silver originated in India, where it is still principally prosecuted, and that from one great city to another the practice travelled westward, — Babylon, Tarsus, Baghdad, Damascus, the islands of Cyprus and Sicily, Con- / stantinople and Venice, all in the process of time becoming famous centres of these much prized manufactures. Alexander the Great found Indian kings and princes arrayed in robes of gold and purple; and the Persian monarch Darius, we are told, wore a war mantle of cloth of gold, on which were figured two golden hawks as if pecking at each other. There is reason, according to Josephus, to believe that the “royal apparel" worn by Herod on the day of his death (Acts xii. 21) was a tissue of silver. Agrippina, the wife of the emperor Claudius, had a robe woven entirely of gold, and from that period downwards royal personages and high ecclesiastical dignitaries used cloth and tissues of gold and silver for their state and ceremonial robes, as well as for costly hangings and decorations. In England, at different periods, various names were applied to cloths of gold, as ciclatoun, tartarium, naques or nac, baudekiu or baldachin, Cyprus damask, and twssewys or tissue. The thin flimsy paper known as tissue paper, is so called because it originally was placed between the folds of gold "tissue" to prevent the contiguous surfaces from fraying each other. At what time the drawing of gold wire for the preparation of these textiles was first practised is not accurately known. The art was probably introduced and applied in different localities at widely different dates, but down till mediaeval times the method graphically described in the Pentateuch continued to be practised with both gold and silver. Fabrics woven with gold and silver continue to be used on the largest scale to this day in India; and there the preparation of the varieties of wire, and the working of the various forms of lace, brocade, and embroidery, is at once an important and peculiar art. The basis of all modern fabrics of this kind is wire, the "gold wire" of the manufacturer being in all cases silver gilt wire, and silver wire being, of course, composed of pure silver. In India the wire is drawn by means of simple draw-plates, with rude and simple appliances, from rounded bars of silver, or gold-plated silver, as the case may be. The wire is flattened into the strip or ribbon-like form it generally assumes by passing it, fourteen or fifteen strands simultaneously, over a fine, smooth, round-topped anvil, and beating it as it passes with a heavy hammer having a slightly convex surface. From wire so flattened there is made in India soniri, a tissue or cloth of gold, the web or warp being composed entirely of golden strips, and ruperi, a similar tissue of silver. Gold lace is also made on a warp of thick yellow silk with a weft of flat wire, and in the case of ribbons the warp or web is composed of the metal. The flattened wires are twisted around orange (in the case of silver, white) coloured silk thread, so as completely to cover the thread and present the appearance of a continuous wire; and in this form it is chiefly employed for weaving into the rich brocades known as kincobs or kinkhábs. Wires flattened, or partially flattened, are also twisted into exceedingly fine spirals, and in this form they are the basis of numerous ornamental applications. Such spirals drawn out till they present a waved appearance, and in that state flattened, are much used for rich heavy embroideries termed karchobs. Spangles for embroideries, &c., are made from spirals of comparatively stout wire, by cutting them down ring by ring, laying each C-like ring on an anvil, and by a smart blow with a hammer flattening it out into a thin round disk with a slit extending from the centre to one edge. Fine spirals are also used for general embroidery purposes. The demand for various kinds of loom-woven and embroidered gold and silver work in India is immense; and the variety of textiles so ornamented is also very great. "Gold and silver," says Dr Birdwood in his ''Handbook to the British-Indian Section, Paris Exhibition'', 1878, "are worked into the decoration of all the more costly loom-made garments and Indian piece goods, either on the borders only, or in stripes throughout, or in diapered figures. The gold-bordered loom embroideries are made chiefly at Sattara, and the gold or silver striped at Tanjore; the gold figured ''mashrus'' at Tanjore, Trichinopoly, and Hyderabad in the Deccau; and the highly ornamented gold-figured silks and gold and silver tissues principally at Ahmedabad, Benares, Murshedabad, and Trichinopoly." Among the Western communities the demand for gold and silver lace and embroideries arises chiefly in connexion with naval and military uniforms, court costumes, public and private liveries, ecclesiastical robes and draperies, theatrical dresses, and the badges and insignia of various orders. To a limited extent there is a trade in gold wire and lace to India and China. The metallic basis of the various fabrics is wire round and flattened, the wire being of three kinds — 1st, gold wire, which is invariably silver gilt wire; 2d, copper gilt wire, used for common liveries and theatrical purposes; and 3d, silver wire. These wires are drawn by the ordinary processes, and the flattening, when done, is accomplished by passing the wire between a pair of revolving rollers of fine polished steel. The various qualities of wire are prepared and used in precisely the same way as in India, — round wire, flat wire, thread made of flat gold wire twisted round orange-coloured silk or cotton, known in the trade as "orris," fine spirals and spangles, all being in use in the West as in the East. The lace is woven in the same manner as ribbons, and there are very numerous varieties in richness, pattern, and quality. Cloth of gold, and brocades rich in gold and silver, are woven for ecclesiastical vestments and draperies. The proportions of gold and silver in the gold thread for the lace trade varies, but in all cases the proportion of gold is exceedingly small. An ordinary gold lace wire is drawn from a bar containing 90 parts of silver and 7 of copper, coated with 3 parts of gold. On an average each ounce troy of a bar so plated is drawn into 1500 yards of wire; and therefore about 16 grains of gold cover a mile of wire. It is estimated that about 250,000 ounces of gold wire are made annually in Great Britain, of which about 20 per cent, is used for the headings of calico, muslin, &c., and the remainder is worked up in the gold lace trade.<ref>William Chandler Roberts-Austen and H. Bauerman [W.C.R. — H.B.]. "Gold and Silver Lace." In "Gold." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. 10 (X). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2010%20%28G-GOT%29%20193592738.23/page/753/mode/1up (accessed January 2023): 753, Col. 2c – 754, Cols. 1a–b – 2a–b.</ref></blockquote> ==== Honiton Lace ==== Kate Stradsin says,<blockquote>Honiton lace was the finest English equivalent of Brussels bobbin lace and was constructed in small ‘sprigs, in the cottages of lacemakers[.'] These sprigs were then joined together and bleached to form the large white flounces that were so sought after in the mid-nineteenth century.<ref>Strasdin, Kate. "Rediscovering Queen Alexandra’s Wardrobe: The Challenges and Rewards of Object-Based Research." ''The Court Historian'' 24.2 (2019): 181-196. Rpt http://repository.falmouth.ac.uk/3762/15/Rediscovering%20Queen%20Alexandra%27s%20Wardrobe.pdf: 13, and (for the little quotation) n. 37, which reads "Margaret Tomlinson, ''Three Generations in the Honiton Lace Trade: A Family History'', self-published, 1983."</ref></blockquote> [[File:Strook in Alençon naaldkant, 1750-1775.jpg|thumb|alt=A long piece of complex white lace with garlands, flowers and bows|Point d'Alençon lace, 1750-1775]] ==== Passementerie ==== ''Passementerie'' is the French term for trim on clothing or furniture. The 19th century (especially during the First and Second Empire) was a time of great "''exubérance''" in passementerie in French design, including the development and widespread use of the Jacquard loom.<ref>{{Cite journal|date=2023-06-10|title=Passementerie|url=https://fr.wikipedia.org/w/index.php?title=Passementerie&oldid=205068926|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Passementerie.</ref> ==== Point d'Alençon Lace ==== A lace made by hand using a number of complex steps and layers. The lacemakers build the point d'Alençon design on some kind of mesh and sometimes leave some of the mesh in as part of the lace and perhaps to provide structure. Elizabeth Lewandowski defines point d'Alençon lace and Alençon lace separately. Point lace is needlepoint lace,<ref name=":7">Lewandowski, Elizabeth J. ''The Complete Costume Dictionary''. Scarecrow Press, 2011.</ref>{{rp|233}} so Alençon point is "a two thread [needlepoint] lace."<ref name=":7" />{{rp|7}} Alençon lace has a "floral design on [a] fine net ground [and is] referred to as [the] queen of French handmade needlepoint laces. The original handmade Alençon was a fine needlepoint lace made of linen thread."<ref name=":7" />{{rp|7}} The sample of point d'Alençon lace (right), from 1750–1775, shows the linen mesh that the lace was constructed on.<ref>{{Cite web|url=http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689|title=MoMu - Open Fashion|website=openfashion.momu.be|access-date=2024-02-26}} ModeMuseum Antwerpen. http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689.</ref> The consistency in this sample suggests it may have been made by machine. == Fabric == === Brocatelle === Brocatelle is a kind of brocade, more simple than most brocades because it uses fewer warp and weft threads and fewer colors to form the design. The article in the French ''Wikipédia'' defines it like this:<blockquote>La '''brocatelle''' est un type de tissu datant du <abbr>xvi^e</abbr> siècle qui comporte deux chaînes et deux trames, au minimum. Il est composé pour que le dessin ressorte avec un relief prononcé, grâce à la chaîne sur un fond en sergé. Les brocatelles les plus anciennes sont toujours fabriquées avec une des trames en lin.<ref>{{Cite journal|date=2023-06-01|title=Brocatelle|url=https://fr.wikipedia.org/w/index.php?title=Brocatelle&oldid=204796410|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Brocatelle.</ref></blockquote>Which translates to this:<blockquote>Brocatelle is a type of fabric dating from the 16th century that has two warps and two wefts, at a minimum. It is composed so that the design stands out with a pronounced relief, thanks to the weft threads on a twill background. The oldest brocades were always made with one of the wefts being linen.</blockquote>The ''Oxford English Dictionary'' says, brocatelle is an "imitation of brocade, usually made of silk or wool, used for tapestry, upholstery, etc., now also for dresses. Both the nature and the use of the stuff have changed" between the late 17th century and 1888, the last time this definition was revised.<ref>"brocatelle, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/23550. Accessed 4 July 2023.</ref> === Broché === === Ciselé === === Crépe de Chine === The ''Oxford English Dictionary'' distinguishes the use of ''crêpe'' (using a circumflex rather than an acute accent over the first ''e'') from ''crape'' in textiles, saying ''crêpe'' is "often borrowed [from the French] as a term for all crapy fabrics other than ordinary black mourning crape,"<ref>"crêpe, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44242. Accessed 10 February 2023.</ref> with usage examples ranging from 1797 to the mid 20th century. Crêpe de chine, it says is "a white or other coloured crape made of raw silk." === Épinglé Velvet === Often spelled ''épingle'' rather than ''épinglé'', this term appears to have been used for a fabric made of wool, or at least wool along with linen or cotton, that was heavier and stiffer than silk velvet. It was associated with outer garments and men's clothing. Nowadays, épinglé velvet is an upholstery fabric in which the pile is cut into designs and patterns, and the portrait of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary, Duchess of Hamilton]] shows a mantle described as épinglé velvet that does seem to be a velvet with a woven pattern perhaps cut into the pile. === Lace === While lace also functioned sometimes as fabric — at the décolletage, for example, on the stomacher or as a veil — here we organize it as a [[Social Victorians/Terminology#Trim and Lace|part of the elaboration of clothing]]. === Liberty Fabrics === === Lisse === According to the ''Oxford English Dictionary'', the term ''lisse'' as a "kind of silk gauze" was used in the 19th-century UK and US.<ref>"lisse, n.1." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/108978. Accessed 4 July 2023.</ref> === Satin === The pre-1927 ''History of Feminine Fashion'', probably commissioned by Charles Frederick Worth's sons, describes Worth's "insistence on new fabrics, new trimmings, new materials of every description" at the beginning of his career in the mid 19th century:<blockquote>When Worth first entered the business of dressmaking, the only materials of the richer sort used for woman's dress were velvet, faille, and watered silk. Satin, for example, was never used. M. Worth desired to use satin very extensively in the gowns he designed, but he was not satisfied with what could be had at the time; he wanted something very much richer than was produced by the mills at Lyons. That his requirements entailed the reconstruction of mills mattered little — the mills were reconstructed under his directions, and the Lyons looms turned out a richer satin than ever, and the manufacturers prospered accordingly.<ref name=":9">[Worth, House of.] {{Cite book|url=http://archive.org/details/AHistoryOfFeminineFashion|title=A History Of Feminine Fashion (1800s to 1920s)}} Before 1927. [Likely commissioned by Worth. Link is to Archive.org; info from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Worth_Biarritz_salon.jpg.]</ref>{{rp|6 in printed, 26 in digital book}}</blockquote> === Selesia === According to the ''Oxford English Dictionary'', ''silesia'' is "A fine linen or cotton fabric originally manufactured in Silesia in what is now Germany (''Schlesien'').<ref>"Silesia, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/179664. Accessed 9 February 2023.</ref> It may have been used as a lining — for pockets, for example — in garments made of more luxurious or more expensive cloth. The word ''sleazy'' — "Of textile fabrics or materials: Thin or flimsy in texture; having little substance or body."<ref>"sleazy, adj." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/181563. Accessed 9 February 2023.</ref> — may be related. === Shot Fabric === According to the ''Oxford English Dictionary'', "Of a textile fabric: Woven with warp-threads of one colour and weft-threads of another, so that the fabric (usually silk) changes in tint when viewed from different points."<ref>“Shot, ''Adj.''”  ''Oxford English Dictionary'', Oxford UP,  July 2023, https://doi.org/10.1093/OED/2977164390.</ref> A shot fabric might also be made of silk and cotton fibers. === Tissue === A lightly woven fabric like gauze or chiffon. The light weave can make the fabric translucent and make pleating and gathering flatter and less bulky. Tissue can be woven to be shot, sheer, stiff or soft. Historically, the term in English was used for a "rich kind of cloth, often interwoven with gold or silver" or "various rich or fine fabrics of delicate or gauzy texture."<ref>“Tissue, N.” ''Oxford English Dictionary'', Oxford UP, March 2024, https://doi.org/10.1093/OED/5896731814.</ref> == Fan == The ''Encyclopaedia Britannica'' (9th edition) has an article on the fan. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>FAN (Latin, ''vannus''; French, ''éventail''), a light implement used for giving motion to the air. ''Ventilabrum'' and ''flabellum'' are names under which ecclesiastical fans are mentioned in old inventories. Fans for cooling the face have been in use in hot climates from remote ages. A bas-relief in the British Museum represents Sennacherib with female figures carrying feather fans. They were attributes of royalty along with horse-hair fly-flappers and umbrellas. Examples may be seen in plates of the Egyptian sculptures at Thebes and other places, and also in the ruins of Persepolis. In the museum of Boulak, near Cairo, a wooden fan handle showing holes for feathers is still preserved. It is from the tomb of Amen-hotep, of the 18th dynasty, 17th century <small>B</small>.<small>C</small>. In India fans were also attributes of men in authority, and sometimes sacred emblems. A heartshaped fan, with an ivory handle, of unknown age, and held in great veneration by the Hindus, was given to the prince of Wales. Large punkahs or screens, moved by a servant who does nothing else, are in common use by Europeans in India at this day. Fans were used in the early Middle Ages to keep flies from the sacred elements during the celebrations of the Christian mysteries. Sometimes they were round, with bells attached — of silver, or silver gilt. Notices of such fans in the ancient records of St Paul’s, London, Salisbury cathedral, and many other churches, exist still. For these purposes they are no longer used in the Western church, though they are retained in some Oriental rites. The large feather fans, however, are still carried in the state processions of the supreme pontiff in Rome, though not used during the celebration of the mass. The fan of Queen Theodolinda (7th century) is still preserved in the treasury of the cathedral of Monza. Fans made part of the bridal outfit, or ''mundus muliebris'', of ancient Roman ladies. Folding fans had their origin in Japan, and were imported thence to China. They were in the shape still used—a segment of a circle of paper pasted on a light radiating frame-work of bamboo, and variously decorated, some in colours, others of white paper on which verses or sentences are written. It is a compliment in China to invite a friend or distinguished guest to write some sentiment on your fan as a memento of any special occasion, and this practice has continued. A fan that has some celebrity in France was presented by the Chinese ambassador to the Comtesse de Clauzel at the coronation of Napoleon I. in 1804. When a site was given in 1635, on an artificial island, for the settlement of Portuguese merchants in Nippo in Japan, the space was laid out in the form of a fan as emblematic of an object agreeable for general use. Men and women of every rank both in China and Japan carry fans, even artisans using them with one hand while working with the other. In China they are often made of carved ivory, the sticks being plates very thin and sometimes carved on both sides, the intervals between the carved parts pierced with astonishing delicacy, and the plates held together by a ribbon. The Japanese make the two outer guards of the stick, which cover the others, occasionally of beaten iron, extremely thin and light, damascened with gold and other metals. Fans were used by Portuguese ladies in the 14th century, and were well known in England before the close of the reign of Richard II. In France the inventory of Charles V. at the end of the 14th century mentions a folding ivory fan. They were brought into general use in that country by Catherine de’ Medici, probably from Italy, then in advance of other countries in all matters of personal luxury. The court ladies of Henry VIII.’s reign in England were used to handling fans, A lady in the Dance of Death by Holbein holds a fan. Queen Elizabeth is painted with a round leather fan in her portrait at Gorhambury; and as many as twenty-seven are enumerated in her inventory (1606). Coryat, an English traveller, in 1608 describes them as common in Italy. They also became of general use from that time in Spain. In Italy, France, and Spain fans had special conventional uses, and various actions in handling them grew into a code of signals, by which ladies were supposed to convey hints or signals to admirers or to rivals in society. A paper in the ''Spectator'' humorously proposes to establish a regular drill for these purposes. The chief seat of the European manufacture of fans during the 17th century was Paris, where the sticks or frames, whether of wood or ivory, were made, and the decorations painted on mounts of very carefully prepared vellum (called latterly ''chicken skin'', but not correctly), — a material stronger and tougher than paper, which breaks at the folds. Paris makers exported fans unpainted to Madrid and other Spanish cities, where they were decorated by native artists. Many were exported complete; of old fans called Spanish a great number were in fact made in France. Louis XIV. issued edicts at various times to regulate the manufacture. Besides fans mounted with parchment, Dutch fans of ivory were imported into Paris, and decorated by the heraldic painters in the process called “Vernis Martin,” after a famous carriage painter and inventor of colourless lac varnish. Fans of this kind belonging to the Queen and to the late baroness de Rothschild were exhibited in 1870 at Kensington. A fan of the date of 1660, representing sacred subjects, is attributed to Philippe de Champagne, another to Peter Oliver in England in the / 17th century. Cano de Arevalo, a Spanish painter of the 17th century devoted himself to fan painting. Some harsh expressions of Queen Christina to the young ladies of the French court are said to have caused an increased ostentation in the splendour of their fans, which were set with jewels and mounted in gold. Rosalba Carriera was the name of a fan painter of celebrity in the 17th century. Lebrun and Romanelli were much employed during the same period. Klingstet, a Dutch artist, enjoyed a considerable reputation for his fans from the latter part of the 17th and the first thirty years of the 18th century. The revocation of the edict of Nantes drove many fan-makers out of France to Holland and England. The trade in England was well established under the Stuart sovereigns. Petitions were addressed by the fan-makers to Charles II. against the importation of fans from India, and a duty was levied upon such fans in consequence. This importation of Indian fans, according to Savary, extended also to France. During the reign of Louis XV. carved Indian and China fans displaced to some extent those formerly imported from Italy, which had been painted on swanskin parchment prepared with various perfumes. During the 18th century all the luxurious ornamentation of the day was bestowed on fans as far as they could display it. The sticks were made of mother-of-pearl or ivory, carved with extraordinary skill in France, Italy, England, and other countries. They were painted from designs of Boucher, Watteau, Lancret, and other "genre" painters, Hébert, Rau, Chevalier, Jean Boquet, Mad. Verité, are known as fan painters. These fashions were followed in most countries of Europe, with certain national differences. Taffeta and silk, as well as fine parchment, were used for the mounts. Little circles of glass were let into the stick to be looked through, and small telescopic glasses were sometimes contrived at the pivot of the stick. They were occasionally mounted with the finest point lace. An interesting fan (belonging to Madame de Thiac in France), the work of Le Flamand, was presented by the municipality of Dieppe to Marie Antoinette on the birth of her son the dauphin. From the time of the Revolution the old luxury expended on fans died out. Fine examples ceased to be exported to England and other countries. The painting on them represented scenes or personages connected with political events. At a later period fan mounts were often prints coloured by hand. The events of the day mark the date of many examples found in modern collections. Amongst the fanmakers of the present time the names of Alexandre, Duvelleroy, Fayet, Vanier, may be mentioned as well known in Paris. The sticks are chiefly made in the department of Oise, at Le Déluge, Crèvecœur, Méry, Ste Geneviève, and other villages, where whole families are engaged in preparing them; ivory sticks are carved at Dieppe. Water-colour painters of distinction often design and paint the mounts, the best designs being figure subjects. A great impulse has been given to the manufacture and painting of fans in England since the exhibition which took place at South Kensington in 1870. Other exhibitions have since been held, and competitive prizes offered, one of which was gained by the Princess Louise. Modern collections of fans take their date from the emigration of many noble families from France at the time of the Revolution. Such objects were given as souvenirs and occasionally sold by families in straitened circumstances. A large number of fans of all sorts, principally those of the 18th century, French, English, German, Italian Spanish, &c., have been lately bequeathed to the South Kensington Museum. Regarding the different parts of folding fans it may be well to state that the sticks are called in French ''brins'', the two outer guards ''panaches'', and the mount ''feuille''.<ref>J. H. Pollen [J.H.P.]. "Fan." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. '''10''' ('''X'''). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%209%20%28FAL-FYZ%29%20193323016.23/page/26/mode/2up (accessed January 2023): 27, Col. 1b – 28, Col. 1c.</ref></blockquote> == Fancy-dress Ball == Fancy-dress (or costume) balls were popular and frequent in the U.K. and France as well as the rest of Europe during the 19th century. The themes and styles of the fancy-dress balls influenced those that followed. At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the guests came dressed in costume from times before 1820, as instructed on '''the invitation''', but their clothing was much more about late-Victorian standards of beauty and fashion than the standards of whatever time period the portraits they were copying or basing their costumes on. ''The Queen'' published dress and fashion information and advice under the byline of Ardern Holt, who regularly answered questions from readers about fashion as well as about fancy dress. (More about Ardern Holt, which is almost certainly a pseudonym, can be found on the [[Social Victorians/People/Working in Publishing#Journalists|People Working in Publishing]] page.) Holt also ran wrote entire articles with suggestions for what might make an appealing fancy-dress costume as well as pointing readers away from costumes that had been worn too frequently. The suggestions for costumes are based on familiar types or portraits available to readers, similar to Holt's books on fancy dress, which ran through a number of editions in the 1880s and 1890s. Fancy-dress questions sometimes asked for details about costumes worn in theatrical or operatic productions, which Holt provides. In November 1897, Holt refers to the Duchess of Devonshire's 2 July ball: "Since the famous fancy ball, given at Devonshire House during this year, historical fancy dresses have assumed a prominence that they had not hitherto known."<ref>Holt, Ardern. "Fancy Dress a la Mode." The ''Queen'' 27 November 1897, Saturday: 94 [of 145 in BNA; print p. 1026], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971127/459/0094.</ref> Holt goes on to provide a number of ideas for costumes for historical fancy dress, as always with a strong leaning toward Victorian standards of beauty and style and away from any concern for historical accuracy. Ardern Holt published books on fancy dress as well as writing for the ''Queen'' and other periodicals, but not all of them were about fancy dress. # ''Gentlemen's Fancy Dress: How to Choose It''. Wyman & Sons, 1882. (''Google Books'' has this: https://books.google.com/books/about/Gentlemen_s_Fancy_Dress.html?id=ED8CAAAAQAAJ.) Later editions: 1898 (HathiTrust) # ''Fancy Dresses Described; Or, What to Wear at Fancy Balls''. Debenham & Freebody, 1882. Illustr., Lillian Young. (HathiTrust has this.) Later editions: 4th ed — 1884; 1887 (HathiTrust); 6th ed. — 1896 (HathiTrust) As Leonore Davidoff says, "Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration symbolised some status category for the female wearer."<ref name=":1" />{{rp|93}} [handled under Elaborations] === Historical Accuracy === Many of the costumes at the ball were based on portraits, especially when the guest was dressed as a historical figure. If possible, we have found the portraits likely to have been the originals, or we have found, if possible, portraits that show the subjects from the two time periods at similar ages. The way clothing was cut changed quite a bit between the 18th and 19th centuries. We think of Victorian clothing — particularly women's clothing, and particularly at the end of the century — as inflexible and restrictive, especially compared to 20th- and 21st-century customs permitting freedom of movement. The difference is generally evolutionary rather than absolute — that is, as time has passed since the 18th century, clothing has allowed an increasingly greater range of movement, especially for people who did not do manual labor. By the end of the 19th century, garments like women's bodices and men's coats were made fitted and smooth by attention to the grain of the fabric and by the use of darts (rather than techniques that assembled many small, individual pieces of fabric). * clothing construction and flat-pattern techniques * Generally, the further back in time we go, the more 2-dimensional the clothing itself was. ==== Women's Versions of Historical Accuracy at the Ball ==== As always with this ball, whatever historical accuracy might be present in a woman's costume is altered so that the wearer is still a fashionable Victorian lady. What makes the costumes look "Victorian" to our eyes is the line of the silhouette caused by the foundation undergarments as well as the many "elaborations"<ref name=":1" />{{rp|93}}, mostly in the decorations, trim and accessories. Also, the clothing hangs and drapes differently because the fabric was cut on grain and the shoulders were freed by the way the sleeves were set in. ==== Men's Versions of Historical Accuracy at the Ball ==== Because men were not wearing a Victorian foundation garment at the end of the century, the men's costumes at the ball are more historically accurate in some ways. * Trim * Mixing neck treatments * Hair * Breeches * Shoes and boots * Military uniforms, arms, gloves, boots == Feathers and Plumes == === Aigrette === Elizabeth Lewandowski defines ''aigrette'' as "France. Feather or plume from an egret or heron."<ref name=":7" />(5) Sometimes the newspapers use the term to refer to an accessory (like a fan or ornament on a hat) that includes such a feather or plume. The straight and tapered feathers in an aigrette are in a bundle. === Prince of Wales's Feathers or White Plumes === The feathers in an aigrette came from egrets and herons; Prince of Wales's feathers came from ostriches. A fuller discussion of Prince of Wales's feathers and the white ostrich plumes worn at court appears on [[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|Victorian Things]]. For much of the late 18th and 19th centuries, white ostrich plumes were central to fashion at court, and at a certain point in the late 18th century they became required for women being presented to the monarch and for their sponsors. Our purpose here is to understand why women were wearing plumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] as part of their costumes. First published in 1893, [[Social Victorians/People/Lady Colin Campbell|Lady Colin Campbell]]'s ''Manners and Rules of Good Society'' (1911 edition) says that<blockquote>It was compulsory for both Married and Unmarried Ladies to Wear Plumes. The married lady’s Court plume consisted of three white feathers. An unmarried lady’s of two white feathers. The three white feathers should be mounted as a Prince of Wales plume and worn towards the left hand side of the head. Colored feathers may not be worn. In deep mourning, white feathers must be worn, black feathers are inadmissible.<p> White veils or lace lappets must be worn with the feathers. The veils should not be longer than 45 inches.<ref>{{Cite web|url=https://www.edwardianpromenade.com/etiquette/the-court-presentation/|title=The Court Presentation|last=Holl|first=Evangeline|date=2007-12-07|website=Edwardian Promenade|language=en-US|access-date=2022-12-18}} https://www.edwardianpromenade.com/etiquette/the-court-presentation/.</ref></blockquote>[[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|This fashion was imported from France]] in the mid 1770s.<ref>"Abstract" for Blackwell, Caitlin. "'<nowiki/>''The Feather'd Fair in a Fright''': The Emblem of the Feather in Graphic Satire of 1776." ''Journal for Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. ''Wiley Online'' DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x (accessed November 2022).</ref> Separately, a secondary heraldic emblem of the Prince of Wales has been a specific arrangement of 3 ostrich feathers in a gold coronet<ref>{{Cite journal|date=2022-11-07|title=Prince of Wales's feathers|url=https://en.wikipedia.org/w/index.php?title=Prince_of_Wales%27s_feathers&oldid=1120556015|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Prince_of_Wales's_feathers.</ref> since King Edward III (1312–1377<ref>{{Cite journal|date=2022-12-14|title=Edward III of England|url=https://en.wikipedia.org/w/index.php?title=Edward_III_of_England&oldid=1127343221|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_III_of_England.</ref>). Some women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] wore white ostrich feathers in their hair, but most of them are not Prince of Wales's feathers. Most of the plumes in these portraits are arrangements of some kind of headdress to accompany the costume. A few, wearing what looks like the Princes of Wales's feathers, might be signaling that their character is royal or has royal ancestry. '''One of the women [which one?] was presented to the royals at this ball?''' Here is the list of women who are wearing white ostrich plumes in their portraits in the [[Social Victorians/1897 Fancy Dress Ball/Photographs|''Diamond Jubilee Fancy Dress Ball'' album of 286 photogravure portraits]]: # Kathleen Pelham-Clinton, the [[Social Victorians/People/Newcastle|Duchess of Newcastle]] # [[Social Victorians/People/Louisa Montagu Cavendish|Luise Cavendish]], the Duchess of Devonshire # Jesusa Murrieta del Campo Mello y Urritio (née Bellido), [[Social Victorians/People/Santurce|Marquisa de Santurce]] # Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]] # Princess (Laura Williamina Seymour) Victor of  [[Social Victorians/People/Gleichen#Laura%20Williamina%20Seymour%20of%20Hohenlohe-Langenburg|Hohenlohe Langenburg]] # Louisa Acheson, [[Social Victorians/People/Gosford|Lady Gosford]] # Alice Emily White Coke, [[Social Victorians/People/Leicester|Viscountess Coke]] # Lady Mary Stewart, Helen Mary Theresa [[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]] #[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], Duchess of [[Social Victorians/People/Marlborough|Marlborough]], dressed as the wife of the French Ambassador at the Court of Catherine of Russia (not white, but some color that reads dark in the black-and-white photograph) #Mrs. Mary [[Social Victorians/People/Chamberlain|Chamberlain]] (at 491), wearing white plumes, as Madame d'Epinay #Lady Clementine [[Social Victorians/People/Tweeddale|Hay]] (at 629), wearing white plumes, as St. Bris (''Les Huguenots'') #[[Social Victorians/People/Meysey-Thompson|Lady Meysey-Thompson]] (at 391), wearing white plumes, as Elizabeth, Queen of Bohemia #Mrs. [[Social Victorians/People/Grosvenor|Algernon (Catherine) Grosvenor]] (at 510), wearing white plumes, as Marie Louise #Lady [[Social Victorians/People/Ancaster|Evelyn Ewart]], at 401), wearing white plumes, as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson #[[Social Victorians/People/Lyttelton|Edith Sophy Balfour Lyttelton]] (at 580), wearing what might be white plumes on a large-brimmed white hat, after a picture by Romney #[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), wearing what might be white plumes, as Cydalise of the Comedie Italienne from the time of Louis XV #Lady [[Social Victorians/People/Ilchester|Muriel Fox Strangways]] (at 403), wearing what might be two smallish white plumes, as Lady Sarah Lennox, one of the bridesmaids of Queen Charlotte A.D. 1761 #Lady [[Social Victorians/People/Lucan|Violet Bingham]] (at 586), wearing perhaps one white plume in a headdress not related to the Prince of Wales's feathers #Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), wearing a headdress that includes some white plumes, as Lady Burleigh #[[Social Victorians/People/Dupplin|Agnes Blanche Marie Hay-Drummond]] (at 682), in a big headdress topped with white plumes, as Mademoiselle Andrée de Taverney A.D. 1775 #Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]] (at 336), wearing what looks like Prince of Wales's plumes #[[Social Victorians/People/Suffolk|Marguerite Hyde "Daisy" Leiter]] (at 684), wearing what looks like Prince of Wales's plumes #Lady [[Social Victorians/People/Spicer|Margaret Spicer]] (at 281), wearing one smallish white and one black plume, as Countess Zinotriff, Lady-in-Waiting to the Empress Catherine of Russia #Mrs. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 318), wearing what looks like Prince of Wales's plumes, as Elizabeth Cavendish, daughter of Bess of Hardwick #Nellie, [[Social Victorians/People/Kilmorey|Countess of Kilmorey]] (at 207), wearing three tall plumes, 2 white and one dark, as Comtesse du Barri #Daisy, [[Social Victorians/People/Warwick|Countess of Warwick]] (at 53), wearing at least 1 white plume, as Marie Antoinette More men than women were wearing plumes reminiscent of the Prince of Wales's feathers: * ==== Bibliography for Plumes and Prince of Wales's Feathers ==== * Blackwell, Caitlin. "'''The Feather'd Fair in a Fright'<nowiki/>'': The Emblem of the Feather in Graphic Satire of 1776." Journal for ''Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. Wiley Online DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x. * "Prince of Wales's feathers." ''Wikipedia'' https://en.wikipedia.org/wiki/Prince_of_Wales%27s_feathers (accessed November 2022). ['''Add women to this page'''] * Simpson, William. "On the Origin of the Prince of Wales' Feathers." ''Fraser's magazine'' 617 (1881): 637-649. Hathi Trust https://babel.hathitrust.org/cgi/pt?id=chi.79253140&view=1up&seq=643&q1=feathers (accessed December 2022). Deals mostly with use of feathers in other cultures and in antiquity; makes brief mention of feathers and plumes in signs and pub names that may not be associated with the Prince of Wales. No mention of the use of plumes in women's headdresses or court dress. == Honors == === The Bath === The Most Honourable Order of the Bath (GCB, Knight or Dame Grand Cross; KCB or DCB, Knight or Dame Commander; CB, Companion) === The Garter === The Most Noble Order of the Knights of the Garter (KG, Knight Companion; LG, Lady Companion) [[File:The Golden Fleece - collar exhibited at MET, NYC.jpg|thumb|The Golden Fleece collar and pendant for the 2019 "Last Knight" exhibition at the MET, NYC.|alt=Recent photograph of a gold necklace on a wide band, with a gold skin of a sheep hanging from it as a pendant]] === The Golden Fleece === To wear the golden fleece is to wear the insignia of the Order of the Golden Fleece, said to be "the most prestigious and historic order of chivalry in the world" because of its long history and strict limitations on membership.<ref name=":10">{{Cite journal|date=2020-09-25|title=Order of the Golden Fleece|url=https://en.wikipedia.org/w/index.php?title=Order_of_the_Golden_Fleece&oldid=980340875|journal=Wikipedia|language=en}}</ref> The monarchs of the U.K. were members of the originally Spanish order, as were others who could afford it, like the Duke of Wellington,<ref name=":12">Thompson, R[obert]. H[ugh]. "The Golden Fleece in Britain." Publication of the ''British Numismatic Society''. 2009 https://www.britnumsoc.org/publications/Digital%20BNJ/pdfs/2009_BNJ_79_8.pdf (accessed January 2023).</ref> the first Protestant to be admitted to the order.<ref name=":10" /> Founded in 1429/30 by Philip the Good, Duke of Burgundy, the order separated into two branches in 1714, one Spanish and the other Austrian, still led by the House of Habsburg.<ref name=":10" /> [[File:Prince Albert - Franz Xaver Winterhalter 1842.jpg|thumb|1842 Winterhalter portrait of Prince Albert wearing the insignia of the Order of the Golden Fleece, 1842|left|alt=1842 Portrait of Prince Albert by Winterhalter, wearing the insignia of the Golden Fleece]] The photograph (upper right) is of a Polish badge dating from the "turn of the XV and XVI centuries."<ref>{{Citation|title=Polski: Kolana orderowa orderu Złotego Runa, przełom XV i XVI wieku.|url=https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg|date=2019-11-10|accessdate=2023-01-10|last=Wulfstan}}. https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg.</ref> The collar to this Golden Fleece might be similar to the one the [[Social Victorians/People/Spencer Compton Cavendish#The Insignia of the Order of the Golden Fleece|Duke of Devonshire is wearing in the 1897 Lafayette portrait]]. The badges and collars that Knights of the Order actually wore vary quite a bit. The 1842 Franz Xaver Winterhalter portrait (left) of Prince Consort Albert, Victoria's husband and father of the Prince of Wales, shows him wearing the Golden Fleece on a red ribbon around his neck and the star of the Garter on the front of his coat.<ref>Winterhalter, Franz Xaver. ''Prince Albert''. {{Cite web|url=https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61|title=Explore the Royal Collection Online|website=www.rct.uk|access-date=2023-01-16}} https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61.</ref> === Royal Victorian Order === (GCVO, Knight or Dame Grand Cross; KCVO or DCVO, Knight or Dame Commander; CVO, Commander; LVO, Lieutenant; MVO, Member) === St. John === The Order of the Knights of St. John === Star of India === Most Exalted Order of the Star of India (GCSI, Knight Grand Commander; KCSI, Knight Commander; CSI, Companion) === Thistle === The Most Ancient and Most Noble Order of the Thistle == Jewelry and Stones == === Cabochon === This term describes both the treatment and shape of a precious or semiprecious stone. A cabochon treatment does not facet the stone but merely polishes it, removing "the rough parts" and the parts that are not the right stone.<ref>"cabochon, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/25778. Accessed 7 February 2023.</ref> A cabochon shape is often flat on one side and oval or round, forming a mound in the setting. === Jet === === ''Orfèvrerie'' === Sometimes misspelled in the newspapers as ''orvfèvrerie''. ''Orfèvrerie'' is the artistic work of a goldsmith, silversmith, or jeweler. === Turquoises === == Military == Several men from the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]] were dressed in military uniforms, some historical and some, possibly, not. === Baldric === According to the ''Oxford English Dictionary'', the primary sense of ''baldric'' is "A belt or girdle, usually of leather and richly ornamented, worn pendent from one shoulder across the breast and under the opposite arm, and used to support the wearer's sword, bugle, etc."<ref>"baldric, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/14849. Accessed 17 May 2023.</ref> This sense has been in existence since c. 1300. === Cuirass === According to the ''Oxford English Dictionary'', the primary sense of ''cuirass'' is "A piece of armour for the body (originally of leather); ''spec.'' a piece reaching down to the waist, and consisting of a breast-plate and a back-plate, buckled or otherwise fastened together ...."<ref>"cuirass, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/45604. Accessed 17 May 2023.</ref> [[File:Knötel IV, 04.jpg|thumb|alt=An Old drawing in color of British soldiers on horses brandishing swords in 1815.|1890 illustration of the Household Cavalry (Life Guard, left; Horse Guard, right) at the Battle of Waterloo, 1815]] === Household Cavalry === The Royal Household contains the Household Cavalry, a corps of British Army units assigned to the monarch. It is made up of 2 regiments, the Life Guards and what is now called The Blues and Royals, which were formed around the time of "the Restoration of the Monarchy in 1660."<ref name=":3">Joll, Christopher. "Tales of the Household Cavalry, No. 1. Roles." The Household Cavalry Museum, https://householdcavalry.co.uk/app/uploads/sites/2/2021/06/Household-Cavalry-Museum-video-series-large-print-text-Tales-episode-01.pdf.</ref>{{rp|1}} Regimental Historian Christopher Joll says, "the original Life Guards were formed as a mounted bodyguard for the exiled King Charles II, The Blues were raised as Cromwellian cavalry and The Royals were established to defend Tangier."<ref name=":3" />{{rp|1–2}} The 1st and 2nd Life Guards were formed from "the Troops of Horse and Horse Grenadier Guards ... in 1788."<ref name=":3" />{{rp|3}} The Life Guards were and are still official bodyguards of the queen or king, but through history they have been required to do quite a bit more than serve as bodyguards for the monarch. The Household Cavalry fought in the Battle of Waterloo on Sunday, 18 June 1815 as heavy cavalry.<ref name=":3" />{{rp|3}} Besides arresting the Cato Steet conspirators in 1820 "and guarding their subsequent execution," the Household Cavalry contributed to the "the expedition to rescue General Gordon, who was trapped in Khartoum by The Mahdi and his army of insurgents" in 1884.<ref name=":3" />{{rp|3}} In 1887 they "were involved ... in the suppression of rioters in Trafalgar Square on Bloody Sunday."<ref name=":3" />{{rp|3}} ==== Grenadier Guards ==== Three men — [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Lord Algernon Gordon-Lennox]], [[Social Victorians/People/Stanley#Edward George Villiers Stanley, Lord Stanley|Lord Stanley]], and [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|Hon. F. C. Stanley]] — attended the ball as officers of the Grenadier Guards, wearing "scarlet tunics, ... full blue breeches, scarlet hose and shoes, lappet wigs" as well as items associated with weapons and armor.<ref name=":14">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 34, Col. 2a}} Founded in England in 1656 as Foot Guards, this infantry regiment "was granted the 'Grenadier' designation by a Royal Proclamation" at the end of the Napoleonic Wars.<ref>{{Cite journal|date=2023-04-22|title=Grenadier Guards|url=https://en.wikipedia.org/w/index.php?title=Grenadier_Guards&oldid=1151238350|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Grenadier_Guards.</ref> They were not called Grenadier Guards, then, before about 1815. In 1660, the Stuart Restoration, they were called Lord Wentworth's Regiment, because they were under the command of Thomas Wentworth, 5th Baron Wentworth.<ref>{{Cite journal|date=2022-07-24|title=Lord Wentworth's Regiment|url=https://en.wikipedia.org/w/index.php?title=Lord_Wentworth%27s_Regiment&oldid=1100069077|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Wentworth%27s_Regiment.</ref> At the time of Lord Wentworth's Regiment, the style of the French cavalier had begun to influence wealthy British royalists. In the British military, a Cavalier was a wealthy follower of Charles I and Charles II — a commander, perhaps, or a field officer, but probably not a soldier.<ref>{{Cite journal|date=2023-04-22|title=Cavalier|url=https://en.wikipedia.org/w/index.php?title=Cavalier&oldid=1151166569|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier.</ref> The Guards were busy as infantry in the 17th century, engaging in a number of armed conflicts for Great Britain, but they also served the sovereign. According to the Guards Museum,<blockquote>In 1678 the Guards were ordered to form Grenadier Companies, these men were the strongest and tallest of the regiment, they carried axes, hatches and grenades, they were the shock troops of their day. Instead of wearing tri-corn hats they wore a mitre shaped cap.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/|title=Service to the Crown|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/.</ref></blockquote>The name comes from ''grenades'', then, and we are accustomed to seeing them in front of Buckingham Palace, with their tall mitre hats. The Guard fought in the American Revolution, and in the 19th century, the Grenadier Guards fought in the Crimean War, Sudan and the Boer War. They have roles as front-line troops and as ceremonial for the sovereign, which makes them elite:<blockquote>Queen Victoria decreed that she did not want to see a single chevron soldier within her Guards. Other then [sic] the two senior Warrant Officers of the British Army, the senior Warrant Officers of the Foot Guards wear a large Sovereigns personal coat of arms badge on their upper arm. No other regiments of the British Army are allowed to do so; all the others wear a small coat of arms of their lower arms. Up until 1871 all officers in the Foot Guards had the privilege of having double rankings. An Ensign was ranked as an Ensign and Lieutenant, a Lieutenant as Lieutenant and Captain and a Captain as Captain and Lieutenant Colonel. This was because at the time officers purchased their own ranks and it cost more to purchase a commission in the Foot Guards than any other regiments in the British Army. For example if it cost an officer in the Foot Guards £1,000 for his first rank, in the rest of the Army it would be £500 so if he transferred to another regiment he would loose [sic] £500, hence the higher rank, if he was an Ensign in the Guards and he transferred to a Line Regiment he went in at the higher rank of Lieutenant.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/|title=Formation and role of the Regiments|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/.</ref></blockquote> ==== Life Guards ==== [[Social Victorians/People/Shrewsbury#Reginald Talbot's Costume|General the Hon. Reginald Talbot]], a member of the 1st Life Guards, attended the Duchess of Devonshire's ball dressed in the uniform of his regiment during the Battle of Waterloo.<ref name=":14" />{{rp|p. 36, Col. 3b}} At the Battle of Waterloo the 1st Life Guards were part of the 1st Brigade — the Household Brigade — and were commanded by Major-General Lord Edward Somerset.<ref name=":4">{{Cite journal|date=2023-09-30|title=Battle of Waterloo|url=https://en.wikipedia.org/w/index.php?title=Battle_of_Waterloo&oldid=1177893566|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Battle_of_Waterloo.</ref> The 1st Life Guards were on "the extreme right" of a French countercharge and "kept their cohesion and consequently suffered significantly fewer casualties."<ref name=":4" /> == Photography == == Footnotes == {{reflist}} obfx9x5m1nl5zpn6i8lzr8c0kn5jtk6 2690304 2690288 2024-12-04T20:22:39Z Scogdill 1331941 /* Hoops */ 2690304 wikitext text/x-wiki Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. The definitions on this page are to provide a sense of what someone in the late 19th century might have meant by the term rather than a definition of what we might mean by it today. In the absence of a specialized glossary from the end of the 19th century in the U.K., we use the ''Oxford English Dictionary'' because the senses of a word are illustrated with examples that have dates so we can be sure that the senses we pick are appropriate for when they are used in the quotations we have. We also sometimes use the French ''Wikipédia'' to define a word because many technical terms of fashion were borrowings from the French. Also, often the French ''Wikipédia'' provides historical context for the uses of a word similar to the way the OED does. == Articles or Parts of Clothing: Non-gender-specific == === Mantle, Cloak, Cape === In 19th-century newspaper accounts, these terms are sometimes used without precision as synonyms. These are all outer garments. '''Mantle''' A mantle — often a long outer garment — might have elements like a train, sleeves, collars, revers, fur, and a cape. A late-19th-century writer making a distinction between a mantle and a cloak might use ''mantle'' if the garment is more voluminous. '''Cloak''' '''Cape''' === Peplum === According to the French ''Wiktionnaire'', a peplum is a "Short skirt or flared flounce layered at the waist of a jacket, blouse or dress" [translation by Google Translate].<ref>{{Cite journal|date=2021-07-02|title=péplum|url=https://fr.wiktionary.org/w/index.php?title=p%C3%A9plum&oldid=29547727|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/p%C3%A9plum.</ref> The ''Oxford English Dictionary'' has a fuller definition, although, it focuses on women's clothing because the sense is written for the present day:<blockquote>''Fashion''. ... a kind of overskirt resembling the ancient peplos (''obsolete''). Hence (now usually) in modern use: a short flared, gathered, or pleated strip of fabric attached at the waist of a woman's jacket, dress, or blouse to create a hanging frill or flounce.<ref name=":5">“peplum, n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1832614702>.</ref></blockquote>Men haven't worn peplums since the 18th century, except when wearing costumes based on historical portraits. The ''Daily News'' reported in 1896 that peplums had been revived as a fashion item for women.<ref name=":5" /> === Revers === According to the ''Oxford English Dictionary'', ''revers'' are the "edge[s] of a garment turned back to reveal the undersurface (often at the lapel or cuff) (chiefly in ''plural''); the material covering such an edge."<ref>"revers, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/164777. Accessed 17 April 2023.</ref> The term is French and was used this way in the 19th century (according to the ''Wiktionnaire'').<ref>{{Cite journal|date=2023-03-07|title=revers|url=https://fr.wiktionary.org/w/index.php?title=revers&oldid=31706560|journal=Wiktionnaire|language=fr}} https://fr.wiktionary.org/wiki/revers.</ref> == Articles or Parts of Clothing: Men's == [[Social Victorians/Terminology#Military|Men's military uniforms]] are discussed below. === À la Romaine === [[File:Johann Baptist Straub - Mars um 1772-1.jpg|thumb|left|alt=Old and damaged marble statue of a Roman god of war with flowing cloak, big helmet with a plume on top, and armor|Johann Baptist Straub's 1772 ''à la romaine'' ''Mars'']] A few people who attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball in 1897]] personated Roman gods or people. They were dressed not as Romans, however, but ''à la romaine'', which was a standardized style of depicting Roman figures that was used in paintings, sculpture and the theatre for historical dress from the 17th until the 20th century. The codification of the style was developed in France in the 17th century for theatre and ballet, when it became popular for masked balls. Women as well as men could be dressed ''à la romaine'', but much sculpture, portraiture and theatre offered opportunities for men to dress in Roman style — with armor and helmets — and so it was most common for men. In large part because of the codification of the style as well as the painting and sculpture, the style persisted and remained influential into the 20th century and can be found in museums and galleries and on monuments. For example, Johann Baptist Straub's 1772 statue of Mars (left), now in the Bayerisches Nationalmuseum, Munich, missing part of an arm, shows Mars ''à la romaine''. In London, an early 17th-century example of a figure of Mars ''à la romaine'', with a helmet, '''was''' "at the foot of the Buckingham tomb in Henry VII's Chapel at Westminster Abbey."<ref>Webb, Geoffrey. “Notes on Hubert Le Sueur-II.” ''The Burlington Magazine for Connoisseurs'' 52, no. 299 (1928): 81–89. http://www.jstor.org/stable/863535.</ref>{{rp|81, Col. 2c}} === Cavalier === [[File:Sir-Anthony-van-Dyck-Lord-John-Stuart-and-His-Brother-Lord-Bernard-Stuart.jpg|thumb|alt=Old painting of 2 men flamboyantly and stylishly dressed in colorful silk, with white lace, high-heeled boots and long hair|Van Dyck's c. 1638 painting of cavaliers Lord John Stuart and his brother Lord Bernard Stuart]] As a signifier in the form of clothing of a royalist political and social ideology begun in France in the early 17th century, the cavalier established France as the leader in fashion and taste. Adopted by [[Social Victorians/Terminology#Military|wealthy royalist British military officers]] during the time of the Restoration, the style signified a political and social position, both because of the loyalty to Charles I and II as well the wealth required to achieve the cavalier look. The style spread beyond the political, however, to become associated generally with dress as well as a style of poetry.<ref>{{Cite journal|date=2023-04-25|title=Cavalier poet|url=https://en.wikipedia.org/w/index.php?title=Cavalier_poet&oldid=1151690299|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier_poet.</ref> Van Dyck's 1638 painting of two brothers (right) emphasizes the cavalier style of dress. === Coats === ==== Doublet ==== * In the 19th-century newspaper accounts we have seen that use this word, doublet seems always to refer to a garment worn by a man, but historically women may have worn doublets. In fact, a doublet worn by Queen Elizabeth I exists and '''is somewhere'''. * Technically doublets were long sleeved, although we cannot be certain what this or that Victorian tailor would have done for a costume. For example, the [[Social Victorians/People/Spencer Compton Cavendish#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Duke of Devonshire's costume as Charles V]] shows long sleeves that may be part of the surcoat but should be the long sleeves of the doublet. ==== Pourpoint ==== A padded doublet worn under armor to protect the warrior from the metal chafing. A pourpoint could also be worn without the armor. ==== Surcoat ==== Sometimes just called ''coat''. [[File:Oscar Wilde by Sarony 1882 18.jpg|thumb|alt=Old photograph of a young man wearing a velvet jacket, knee breeches, silk hose and shiny pointed shoes with bows, seated on a sofa and leaning on his left hand and holding a book in his right| Oscar Wilde, 1882, by Napoleon Sarony]] === Hose, Stockings and Tights === Newspaper accounts from the late 19th century of men's clothing use the term ''hose'' for what we might call stockings or tights. In fact, the terminology is specific. ''Stockings'' is the more general term and could refer to hose or tights. With knee breeches men wore hose, which ended above the knee, and women wore hose under their dresses. The ''Oxford English Dictionary'' defines tights as "Tight-fitting breeches, worn by men in the 18th and early 19th centuries, and still forming part of court-dress."<ref>“Tights, N.” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2693287467.</ref> By 1897, the term was in use for women's stockings, which may have come up only to the knee. Tights were also worn by dancers and acrobats. This general sense of ''tights'' does not assume that they were knitted. ''Clocking'' is decorative embroidery on hose, usually, at the ankles on either the inside or the outside of the leg. It started at the ankle and went up the leg, sometimes as far as the knee. On women's hose, the clocking could be quite colorful and elaborate, while the clocking on men's hose was more inconspicuous. In many photographs men's hose are wrinkled, especially at the ankles and the knees, because they were shaped from woven fabric. Silk hose were knitted instead of woven, which gave them elasticity and reduced the wrinkling. The famous Sarony carte de visite photograph of Oscar Wilde (right) shows him in 1882 wearing knee breeches and silk hose, which are shiny and quite smoothly fitted although they show a few wrinkles at the ankles and knees. In the portraits of people in costume at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the men's hose are sometimes quite smooth, which means they were made of knitted silk and may have been smoothed for the portrait. In painted portraits the hose are almost always depicted as smooth, part of the artist's improvement of the appearance of the subject. === Shoes and Boots === == Articles or Parts of Clothing: Women's == === '''Chérusque''' === According to the French ''Wikipedia'', ''chérusque'' is a 19th-century term for the kind of standing collar like the ones worn by ladies in the Renaissance.<ref>{{Cite journal|date=2021-06-26|title=Collerette (costume)|url=https://fr.wikipedia.org/w/index.php?title=Collerette_(costume)&oldid=184136746|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Collerette_(costume)#Au+xixe+siècle+:+la+Chérusque.</ref> === Corsage === According to the ''Oxford English Dictionary'', the corsage is the "'body' of a woman's dress; a bodice."<ref>"corsage, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/42056. Accessed 7 February 2023.</ref> This sense is well documented in the ''OED'' for the mid and late 19th-century, used this way in fiction as well as in a publication like ''Godey's Lady's Book'', which would be expected to use appropriate terminology associated with fashion and dress making. The sense of "a bouquet worn on the bodice" is, according to the ''OED'', American. === Décolletage === === Girdle === === Mancheron === According to the ''Oxford English Dictionary'', a ''mancheron'' is a "historical" word for "A piece of trimming on the upper part of a sleeve on a woman's dress."<ref>"mancheron, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/113251. Accessed 17 April 2023.</ref> At the present, in French, a ''mancheron'' is a cap sleeve "cut directly on the bodice."<ref>{{Cite journal|date=2022-11-28|title=Manche (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manche_(v%C3%AAtement)&oldid=199054843|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Manche_(v%C3%AAtement).</ref> === Petticoat === According to the ''O.E.D.'', a petticoat is a <blockquote>skirt, as distinguished from a bodice, worn either externally or showing beneath a dress as part of the costume (often trimmed or ornamented); an outer skirt; a decorative underskirt. Frequently in ''plural'': a woman's or girl's upper skirts and underskirts collectively. Now ''archaic'' or ''historical''.<ref>“petticoat, n., sense 2.b”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1021034245></ref> </blockquote>This sense is, according to the ''O.E.D.'', "The usual sense between the 17th and 19th centuries." However, while petticoats belong in both outer- and undergarments — that is, meant to be seen or hidden, like underwear — they were always under another garment, for example, underneath an open overskirt. The primary sense seems to have shifted through the 19th century so that, by the end, petticoats were underwear and the term ''underskirt'' was used to describe what showed under an open overskirt. === Stomacher === According to the ''O.E.D.'', a stomacher is "An ornamental covering for the chest (often covered with jewels) worn by women under the lacing of the bodice,"<ref>“stomacher, n.¹, sense 3.a”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1169498955></ref> although by the end of the 19th century, the bodice did not often have visible laces. Some stomachers were so decorated that they were thought of as part of the jewelry. === Train === A train is The Length of the Train '''For the monarch [or a royal?]''' According to Debrett's,<blockquote>A peeress's coronation robe is a long-trained crimson velvet mantle, edged with miniver pure, with a miniver pure cape. The length of the train varies with the rank of the wearer: * Duchess: for rows of ermine; train to be six feet * Marchioness: three and a half rows of ermine; train to be three and three-quarters feet * Countess: three rows of ermine; train to be three and a half feet * Viscountess: two and a half rows of ermine; train to be three and a quarter feet * Baroness: two rows of ermine; train to be three feet<ref name=":2">{{Cite web|url=https://debretts.com/royal-family/dress-codes/|title=Dress Codes|website=debretts.com|language=en-US|access-date=2023-07-27}} https://debretts.com/royal-family/dress-codes/.</ref> </blockquote>The pattern on the coronet worn was also quite specific, similar but not exactly the same for peers and peeresses. Debrett's also distinguishes between coronets and tiaras, which were classified more like jewelry, which was regulated only in very general terms. Peeresses put on their coronets after the Queen or Queen Consort has been crowned. ['''peers?'''] === Foundation Garments === Unlike undergarments, Victorian women's foundation garments created the distinctive silhouette. Victorian undergarments included the chemise, the bloomers, the corset cover — articles that are not structural. The corset was an important element of the understructure of foundation garments — hoops, bustles, petticoats and so on — but it has never been the only important element. === Corset === [[File:Corset - MET 1972.209.49a, b.jpg|thumb|alt=Photograph of an old silk corset on a mannequin, showing the closure down the front, similar to a button, and channels in the fabric for the boning. It is wider at the top and bottom, creating smooth curves from the bust to the compressed waist to the hips, with a long point below the waist in front.|French 1890s corset, now in the Metropolitan Museum of Art, NYC]] The understructure of Victorian women's clothing is what makes the costumes worn by the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] so distinctly Victorian in appearance. An example of a corset that has the kind of structure often worn by fashionably dressed women in 1897 is the one at right. This corset exaggerated the shape of the women's bodies and made possible a bodice that looked and was fitted in the way that is so distinctive of the time — very controlled and smooth. And, as a structural element, this foundation garment carried the weight of all those layers and all that fabric and decoration on the gowns, trains and mantles. (The trains and mantles could be attached directly to the corset itself.) * This foundation emphasizes the waist and the bust in particular, in part because of the contrast between the very small waist and the rounded fullness of the bust and hips. * The idealized waist is defined by its small span and the sexualizing point at the center-bottom of the bodice, which directs the eye downwards. Interestingly, the pointed waistline worn by Elizabethan men has become level in the Victorian age. Highly fashionable Victorian women wearing the traditional style, however, had extremely pointed waists. * The busk (a kind of boning in the front of a corset that is less flexible than the rest) smoothed the bodice, flattened the abdomen and prevented the point on the bodice from curling up. * The sharp definition of the waist was caused by ** length of the corset (especially on the sides) ** the stiffness of the boning ** the layers of fabric ** the lacing (especially if the woman used tightlacing) ** the over-all shape, which was so much wider at the top and the bottom ** the contrast between the waist and the wider top and bottom * The late-19th-century corset was long, ending below the waist even on the sides and back. * The boning and the top edge of the late 19th-century fashion corset pushed up the bust, rounding (rather than flattening, as in earlier styles) the breasts, drawing attention to their exposed curves and creating cleavage. * The exaggerated bust was larger than the hips, whenever possible, an impression reinforced by the A-line of the skirt and the inverted Vs in the decorative trim near the waist and on the skirt. * This corset made the bodice very smooth with a very precise fit, that had no wrinkles, folds or loose drapery. The bodice was also trimmed or decorated, but the base was always a smooth bodice. More formal gowns would still have the fitted bodice and more elaborate trim made from lace, embroidery, appliqué, beading and possibly even jewels. The advantages and disadvantages of corseting and especially tight lacing were the subject of thousands of articles and opinions in the periodical press for a great part of the century, but the fetishistic and politicized tight lacing was practiced by very few women. And no single approach to corsetry was practiced by all women all the time. Most of the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 ball]] were not tightly laced, but the progressive style does not dominate either, even though all the costumes are technically historical dress. Part of what gives most of the costumes their distinctive 19th-century "look" is the more traditional corset beneath them. Even though this highly fashionable look was widely present in the historical costumes at the ball, some women's waists were obviously very small and others were hardly '''emphasized''' at all. Women's waists are never mentioned in the newspaper coverage of the ball — or, indeed, of any of the social events attended by the network at the ball — so it is only in photographs that we can see the effects of how they used their corsets. === Hoops === ''Hoops'' is a mid-19th-century term for a cage-like structure worn under a skirt to hold it away from the body. Striking for how long they lasted and '''the ways''' they evolved, hoops were the foundation undergarment for the bottom half of a woman's body, for a skirt and petticoat. Women wore this cage-like structure from the '''15th century''' through the late 19th century. The 16th-century Katherine of Aragon is credited with making it fashionable outside Spain The cage caused the silhouette of skirts to change shape over time and enabled the extreme distortions of 17th-and-18th-century panniers and the late 19th-century bustle. Early hoops circled the body symmetrically in a cone or drum shape, then were moved to the sides with panniers, then ballooned around the body like the top half of a sphere, and finally were pulled to the rear with a bustle. That is, like corsets, the distorted shapes of high fashion were made possible by hoops. High fashion demanded these shapes, which disguised women's bodies, '''especially below the waist, while corsets did their work above it'''. Besides the shape, the structure used to construct hoops evolved — from cane and wood to whalebone, then steel '''rods''' and wire. Add fabric structural stuff [[File:Pedro García de Benabarre St John Retable Detail.jpg|thumb|Pedro García de Benabarre St John Retable Detail]] [[File:Alonso Sánchez Coello 011.jpg|thumb|Alonso Sánchez Coello 011]] ==== 15th Century ==== Hoops first appeared in Spain in the 15th century and influenced European fashion for many years. Illustrations ==== 16th Century ==== [[File:Farthingale 2 (PSF).png|thumb|Farthingale 2 (PSF)]] In the 16th century, the garment we call ''hoops'' was called a farthingale. ''Vertugadin'' is a French term for ''farthingale'', a cage made of hoops supporting a skirt — "un élément essentiel de la mode Tudor en Angleterre [an essential element of Tudor fashion in England]."<ref name=":0">{{Cite journal|date=2022-03-12|title=Vertugadin|url=https://fr.wikipedia.org/w/index.php?title=Vertugadin&oldid=191825729|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Vertugadin.</ref> ''Farthingale'' is the term in English; in French, it's ''vertugadin'', and in Spanish ''vertugado''. Blanche Payne says,<blockquote>Katherine of Aragon is reputed to have introduced the Spanish farthingale (a hooped underskirt) into England early in the century. The result was to convert the columnar skirt of the fifteenth century into the cone shape of the sixteenth. ...<p> Spanish influence had introduced the hoop-supported skirt, smooth in contour, which was quite generally worn.<ref name=":11" /> (291)</blockquote>In fact, "The Spanish princess Catherine of Aragon brought the fashion to England for her marriage to Prince Arthur, eldest son of Henry VII in 1501 [La princesse espagnole Catherine d'Aragon amena la mode en Angleterre pour son mariage avec le prince Arthur, fils aîné d'Henri VII en 1501]."<ref name=":0" /> Catherine of Aragon, of course, married Henry VIII after Arthur's death. By the end of the 16th century the French and Spanish farthingales were not identical. The Spanish farthingale shaped the skirt into an A-line with a graduated series of hoops sewn to an undergarment. The French farthingale was a flattish "cartwheel" or platter of hoops worn below the waist and above the hips held the skirt out more or less horizontally. Once past the vertugadin, the skirt then fell straight to the floor, shaping the skirt into a kind of drum. The shoes show in the portraits of women wearing the French farthingale. <blockquote>"FARTHINGALE:  Renaissance (1450-1550 C.E. to Elizabethan (1550-1625 C.E.). Linen underskirt with wire supports which, when shaped, produced a variety of dome, bell, and oblong shapes."<ref name=":7" /> (105)</blockquote>[[File:Panniers 1.jpg|thumb|alt=Photograph of the wooden and fabric skeleton of an 18th-century women's foundation garment|Panniers 1]] ==== 17th Century ==== Generally, panniers were a kind of undergarment worn in the 17th and 18th centuries. Their design evolved during the century. Made of hoops of wood, they were "baskets" or cages worn on either side of the waist to broaden the skirts to the sides. Illustration ==== 18th Century ==== By the 18th century, the farthingale was called hoops, which were at this point were made of wood. Blanche Payne outlines the evolution of hoops, and thus the shape of the skirt, in the 18th century:<blockquote>SKIRT FASHIONS. Since skirts experienced the greatest alterations, a brief summary of the successive silhouettes should help to place individual costumes in their proper niches. Six basic forms appeared during the century, in the following order: # The bustle was a continuation of the 1690 mode. # The bell or dome shape resulted from the reintroduction of hoops; in England by 1710, in France by 1720. # The ellipse, the second phase of the hoop skirt, was achieved by broadening the support from side to side and compressing it from front to back. It had a long run of popularity, from 1740 to 1770, the extreme width being retained in court costumes. In France it persisted until the revolution, except that skirts were allowed to curve outward in [the] back again. English court costume followed this fashion well into the nineteenth century. # The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, which were used to form the three great ‘poufs’ known as the polonaise (see Fig. 452). These diversions appeared in the late sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period. # The return of the bustle in the 1780s. # The tubular form, drawn from classic art, in the 1790s.<ref name=":11" /> (411–413) </blockquote>While we think of the bustle as a 19th-century look, it can be found in the 18th century, as Payne says. Polonaise may need its own section. ==== 19th Century ==== [[File:Cutaway sketch of crinoline.gif|thumb|Cutaway sketch of crinoline]] [[File:Paris voulant englober la banlieue.JPG|thumb|Paris voulant englober la banlieue]] [[File:Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723.jpg|thumb|Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723]] In the 19th century, the hoops were made of wire and became lighter. By the 1860s, hoops made for huge round skirts. In Laura Ingalls Wilder's 1941 ''Little Town on the Prairie'', the '''16-year-old Laura wears hoops. Laura Ingalls Wilder and thus the character Laura were born in 1862, so this moment is set in 1883'''.<ref>Hill, Pamela Smith, ed. ''Pioneer Girl: The Annotated Autobiography''.</ref> She makes the comment that she wants to be in style, but that would be on the prairie and not necessarily the latest Parisian style.<blockquote>“Well,” Laura began; then she stopped and spun round and round, for the strong wind blowing against her always made the wires of her hoop skirt creep slowly upward under her skirts until they bunched around her knees. Then she must whirl around and around until the wires shook loose and spiraled down to the bottom of her skirts where they should be. “As she and Carrie hurried '''ob''' she began again. “I think it was silly, the way they dressed when Ma was a girl, don’t you? Drat this wind!” she exclaimed as the hoops began creeping upward again. “Quietly Carrie stood by while Laura whirled. “I’m glad I’m not old enough to have to wear hoops,” she said. “They’d make me dizzy.” “They are rather a nuisance,” Laura admitted. “But they are stylish, and when you’re my age you’ll want to be in style.”<ref>Wilder, Laura Ingalls. ''Little Town on the Prairie.'' Harper and Row, 1941. Pp. 272–273.</ref></blockquote>This very unusual description of the way the wind could make hoops creep and the solution. It must have been happening to other women wearing hoops at the time. == '''Traditional vs Progressive Style''' == === Progressive Style === The terms ''artistic dress'' and ''aesthetic dress'' are not synonymous and were in use at different times to refer to different groups of people in different contexts, but we recognize them as referring to a similar kind of personal style in clothing, a style we call progressive dress or the progressive style. Used in a very precise way, ''artistic dress'' is associated with the Pre-Raphaelite artists and the women in their circle beginning in the 1860s. Similarly, ''aesthetic dress'' is associated with the 1880s and 1890s and dress reform movements. In general, the progressive style is characterized by its resistance to the highly structured fashion of its day, especially corseting, aniline dyes and an extremely close fit. === Traditional Style === By the end of the century designs from the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]] (or Maison Worth) define what we think of as the traditional Victorian look, which was very stylish and expensive. Blanche Payne describes an example of the 1895 "high style" in a gown by Worth with "the idiosyncrasies of the [1890s] full blown":<blockquote>The dress is white silk with wine-red stripes. Sleeves, collars, bows, bag, hat, and hem border match the stripes. The sleeve has reached its maximum volume; the bosom full and emphasized with added lace; the waistline is elongated, pointed, and laced to the point of distress; the skirt is smooth over the hips, gradually swinging out to sweep the floor. This is the much vaunted hourglass figure.<ref name=":11">Payne, Blanche. ''History of Costume from the Ancient Egyptians to the Twentieth Century''. Harper & Row, 1965.</ref>{{rp|530}}</blockquote> The Victorian-looking gowns at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] are stylish in a way that recalls the designs of the House of Worth. The elements that make their look so Victorian are anachronisms on the costumes representing fashion of earlier eras. The women wearing these gowns preferred the standards of beauty from their own day to a more-or-less historically accurate look. The style competing at the very end of the century with the Worth look was not the historical, however, but a progressive style called at the time ''artistic'' or ''aesthetic''. William Powell Frith's 1883 painting ''A Private View at the Royal Academy, 1881'' (discussion below) pits this kind of traditional style against the progressive or artistic style. === The Styles === [[File:Frith A Private View.jpg|thumb|William Powell Frith, ''A Private View at the Royal Academy, 1881'']] We typically think of the late-Victorian silhouette as universal but, in the periods in which corsets dominated women's dress, not all women wore corsets and not all corsets were the same, as William Powell Frith's 1883 ''A Private View at the Royal Academy, 1881'' (right) illustrates. Frith is clear in his memoir that this painting — "recording for posterity the aesthetic craze as regards dress" — deliberately contrasts what he calls the "folly" of the Artistic Dress movement and the look of the traditional corseted waist.<ref>Frith, William Powell. ''My Autobiography and Reminiscences''. 1887.</ref> Frith considered the Artistic Movement and Artistic Dress "ephemeral," but its rejection of corsetry looks far more consequential to us in hindsight than it did in the 19th century. As Frith sees it, his painting critiques the "craze" associated with the women in this set of identifiable portraits who are not corseted, but his commitment to realism shows us a spectrum, a range, of conservatism and if not political then at least stylistic progressivism among the women. The progressives, oddly, are the women wearing artistic (that is, somewhat historical) dress, because they’re not corseted. It is a misreading to see the presentation of the women’s fashion as a simple opposition. Constance, Countess of Lonsdale — situated at the center of this painting with Frederick Leighton, president of the Royal Academy of Art — is the most conservatively dressed of the women depicted, with her narrow sleeves, tight waist and almost perfectly smooth bodice, which tells us that her corset has eyelets so that it can be laced precisely and tightly, and it has stays (or "bones") to prevent wrinkles or natural folds in the overclothing. Lillie Langtry, in the white dress, with her stylish narrow sleeves, does not have such a tightly bound waist or smooth bodice, suggesting she may not be corseted at all, as we know she sometimes was not.['''citation'''] Jenny Trip, a painter’s model, is the woman in the green dress in the aesthetic group being inspected by Anthony Trollope, who may be taking notes. She looks like she is not wearing a corset. Both Langtry and Trip are toward the middle of this spectrum: neither is dressed in the more extreme artistic dress of, say, the two figures between Trip and Trollope. A lot has been written about the late-Victorian attraction to historical dress, especially in the context of fancy-dress balls and the Gothic revival in social events as well as art and music. Part of the appeal has to have been the way those costumes could just be beautiful clothing beautifully made. Historical dress provided an opportunity for some elite women to wear less-structured but still beautiful and influential clothing. ['''Calvert'''<ref>Calvert, Robyne Erica. ''Fashioning the Artist: Artistic Dress in Victorian Britain 1848-1900''. Ph.D. thesis, University of Glasgow, 2012. <nowiki>https://theses.gla.ac.uk/3279/</nowiki></ref>] The standards for beauty, then, with historical dress were Victorian, with the added benefit of possibly less structure. So, at the Duchess of Devonshire's ball, "while some attendees tried to hew closely to historical precedent, many rendered their historical or mythological personage in the sartorial vocabulary they knew best. The [photographs of people in their costumes at the ball offer] a glimpse into how Victorians understood history, not a glimpse into the costume of an authentic historical past."<ref>Mitchell, Rebecca N. "The Victorian Fancy Dress Ball, 1870–1900." ''Fashion Theory'' 2017 (21: 3): 291–315. DOI: 10.1080/1362704X.2016.1172817.</ref> (294) * historical dress: beautiful clothing. * the range at the ball, from Minnie Paget to Gwladys * "In light of such efforts, the ball remains to this day one of the best documented outings of the period, and a quick glance at the album shows that ..." Women had more choices about their waists than the simple opposition between no corset and tightlacing can accommodate. The range of choices is illustrated in Frith's painting, with a woman locating herself on it at a particular moment for particular reasons. Much analysis of 19th-century corsetry focuses on its sexualizing effects — corsets dominated Victorian photographic pornography ['''citations'''] and at the same time, the absence of a corset was sexual because it suggested nudity.['''citations'''] A great deal of analysis of 19th-century corsetry, on the other hand, assumes that women wore corsets for the male gaze ['''citations'''] or that they tightened their waists to compete with other women.['''citations'''] But as we can see in Frith's painting, the sexualizing effect was not universal or sweeping, and these analyses do not account for the choices women had in which corset to wear or how tightly to lace it. Especially given the way that some photographic portraits were mechanically altered to make the waist appear smaller, the size of a woman's waist had to do with how she was presenting herself to the world. That is, the fact that women made choices about the size of or emphasis on their waists suggests that they had agency that needs to be taken into account. As they navigated the complex social world, women's fashion choices had meaning. Society or political hostesses had agency not only in their clothing but generally in that complex social world. They had roles managing social events of the upper classes, especially of the upper aristocracy and oligarchy, like the Duchess of Devonshire's ball. Their class and rank, then, were essential to their agency, including to some degree their freedom to choose what kind of corset to wear and how to wear it. Also, by the end of the century lots of different kinds of corsets were available for lots of different purposes. Special corsets existed for pregnancy, sports (like tennis, bicycling, horseback riding, golf, fencing, archery, stalking and hunting), theatre and dance and, of course, for these women corsets could be made to support the special dress worn over it. Women's choices in how they presented themselves to the world included more than just their foundation garments, of course. "Every cap, bow, streamer, ruffle, fringe, bustle, glove," that is, the trim and decorations on their garments, their jewelry and accessories — which Davidoff calls "elaborations"<ref name=":1">Davidoff, Leonore. ''The Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig 1973).</ref>{{rp|93}} — pointed to a host of status categories, like class, rank, wealth, age, marital status, engagement with the empire, how sexual they wanted to seem, political alignment and purpose at the social event. For example, when women were being presented to the monarch, they were expected to wear three ostrich plumes, often called the [[Social Victorians/Terminology#Prince of Wales's Feathers or White Plumes|Prince of Wales's feathers]]. Like all fashions, the corset, which was quite long-lasting in all its various forms, eventually went out of style. Of the many factors that might have influenced its demise, perhaps most important was the women's movement, in which women's rights, freedom, employment and access to their own money and children were less slogan-worthy but at least as essential as votes for women. The activities of the animal-rights movements drew attention not only to the profligate use of the bodies and feathers of birds but also to the looming extinction of the baleen whale, which made whale bone scarce and expensive. Perhaps the century's debates over corseting and especially tightlacing were relevant to some decisions not to be corseted. And, of course, perhaps no other reason is required than that the nature of fashion is to change. == Cinque Cento == According to the ''Oxford English Dictionary'', ''Cinque Cento'' is a shortening of ''mil cinque cento'', or 1500.<ref>"cinquecento, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/33143. Accessed 7 February 2023.</ref> The term, then would refer, perhaps informally, to the sixteenth century. == Crevé == ''Creve'', without the accent, is an old word in English (c. 1450) for burst or split.<ref>"creve, v." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44339. Accessed 8 February 2023.</ref> ['''With the acute accent, it looks like a past participle in French.'''] == Elastic == Elastic had been invented and was in use by the end of the 19th century. For the sense of "Elastic cord or string, usually woven with india-rubber,"<ref name=":6">“elastic, adj. & n.”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1199670313>.</ref> the ''Oxford English Dictionary'' has usage examples beginning in 1847. The example for 1886 is vivid: "The thorough-going prim man will always place a circle of elastic round his hair previous to putting on his college cap."<ref name=":6" /> == Elaborations == In her 1973 ''The Best Circles: Society, Etiquette and the Season'', Leonore Davidoff notes that women’s status was indicated by dress and especially ornament: “Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration,” she says, “symbolised some status category for the female wearer.”<ref name=":1" />{{rp|93}} Looking at these elaborations as meaningful rather than dismissing them as failed attempts at "historical accuracy" reveals a great deal about the individual women who wore or carried them — and about the society women and political hostesses in their roles as managers of the social world. In her review of ''The House of Worth: Portrait of an Archive'', Mary Frances Gormally says,<blockquote>In a socially regulated year, garments custom made with a Worth label provided women with total reassurance, whatever the season, time of day or occasion, setting them apart as members of the “Best Circles” dressed in luxurious, fashionable and always appropriate attire (Davidoff 1973). The woman with a Worth wardrobe was a woman of elegance, lineage, status, extreme wealth and faultless taste.<ref>Gormally, Mary Frances. Review essay of ''The House of Worth: Portrait of an Archive'', by Amy de la Haye and Valerie D. Mendes (V&A Publishing, 2014). ''Fashion Theory'' 2017 (21, 1): 109–126. DOI: 10.1080/1362704X.2016.1179400.</ref> (117)</blockquote> === Aglet, Aiglet === Historically, an aglet is a "point or metal piece that capped a string [or ribbon] used to attach two pieces of the garment together, i.e., sleeve and bodice."<ref name=":7" />{{rp|4}} Although they were decorative, they were not always visible on the outside of the clothing. They were often stuffed inside the layers at the waist (for example, attaching the bodice to a skirt or breeches). === Frou-frou === In French, ''frou-frou'' or, spelled as ''froufrou'', is the sound of the rustling of silk or sometimes of fabrics in general.<ref>{{Cite journal|date=2023-07-25|title=frou-frou|url=https://fr.wiktionary.org/w/index.php?title=frou-frou&oldid=32508509|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/frou-frou.</ref> The first use the French ''Wiktionnaire'' lists is Honoré Balzac, ''La Cousine Bette'', 1846.<ref>{{Cite journal|date=2023-06-03|title=froufrou|url=https://fr.wiktionary.org/w/index.php?title=froufrou&oldid=32330124|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/froufrou.</ref> ''Frou-frou'' is a term clothing historians use to describe decorative additions to an article of clothing; often the term has a slight negative connotation, suggesting that the additions are superficial. === Pouf, Puff, Poof === According to the French ''Wikipédia'', a pouf was, beginning in 1744, a "kind of women's hairstyle":<blockquote>The hairstyle in question, known as the “pouf”, had launched the reputation of the enterprising Rose Bertin, owner of the Grand Mogol, a very prominent fashion accessories boutique on Rue Saint-Honoré in Paris in 1774. Created in collaboration with the famous hairdresser, Monsieur Léonard, the pouf was built on a scaffolding of wire, fabric, gauze, horsehair, fake hair, and the client's own hair held up in an almost vertical position. — (Marie-Antoinette, ''Queen of Fashion'', translated from the American by Sylvie Lévy, in ''The Rules of the Game'', n° 40, 2009)</blockquote>''Puff'' and ''poof'' are used to describe clothing. === Shirring === ''Shirring'' is the gathering of fabric to make poufs or puffs. The 19th century is known for its use of this decorative technique. Even men's clothing had shirring: at the shoulder seam. === Sequins === Sequins, paillettes, spangles Sequins — or paillettes — are "small, scalelike glittering disks."<ref name=":7" />(216) The French ''Wiktionnaire'' defines ''paillette'' as "Lamelle de métal, brillante, mince, percée au milieu, ordinairement ronde, et qu’on applique sur une étoffe pour l’orner [A strip of metal, shiny, thin, pierced in the middle, usually round, and which is applied to a fabric in order to decorate it.]"<ref name=":8">{{Cite journal|date=2024-03-18|title=paillette|url=https://fr.wiktionary.org/w/index.php?title=paillette&oldid=33809572|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/paillette.</ref> According to the ''OED'', the use of ''sequin'' as a decorative device for clothing (as opposed to gold coins minted and used for international trade) goes back to the 1850s.<ref>“Sequin, N.” ''Oxford English Dictionary'', Oxford UP, September 2023, https://doi.org/10.1093/OED/4074851670.</ref> The first instance of ''spangle'' as "A small round thin piece of glittering metal (usually brass) with a hole in the centre to pass a thread through, used for the decoration of textile fabrics and other materials of various sorts" is from c. 1420.<ref>“Spangle, N. (1).” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/4727197141.</ref> The first use of ''paillette'' listed in the French ''Wiktionnaire'' is in Jules Verne in 1873 to describe colored spots on icy walls.<ref name=":8" /> Currently many distinguish between sequins (which are smaller) and paillettes (which are larger). Before the 20th century, sequins were metal discs or foil leaves, and so of course if they were silver or copper, they tarnished. It is not until well into the 20th century that plastics were invented and used for sequins. === Trim and Lace === ''A History of Feminine Fashion'', published sometime before 1927 and probably commissioned by [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|the Maison Worth]], describes Charles Frederick Worth's contributions to the development of embroidery and [[Social Victorians/Terminology#Passementerie|passementerie]] (trim) from about the middle of the 19th century:<blockquote>For it must be remembered that one of M. Worth's most important and lasting contributions to the prosperity of those who cater for women's needs, as well as to the variety and elegance of his clients' garments, was his insistence on new fabrics, new trimmings, new materials of every description. In his endeavours to restore in Paris the splendours of the days of La Pompadour, and of Marie Antoinette, he found himself confronted at the outset with a grave difficulty, which would have proved unsurmountable to a man of less energy, resource and initiative. The magnificent materials of those days were no longer to be had! The Revolution had destroyed the market for beautiful materials of this, type, and the Restoration and regime of Louis Philippe had left a dour aspect in the City of Light. ... On parallel lines [to his development of better [[Social Victorians/Terminology#Satin|satin]]], [Worth] stimulated also the manufacture of embroidery and ''passementerie''. It was he who first started the manufacture of laces copied from the designs of the real old laces. He was the / first dressmaker to use fur in the trimming of light materials — but he employed only the richer furs, such as sable and ermine, and had no use whatever for the inferior varieties of skins.<ref name=":9" />{{rp|6–7}}</blockquote> ==== Gold and Silver Fabric and Lace ==== The ''Encyclopaedia Britannica'' (9th edition) has an article on gold and silver fabric, threads and lace attached to the article on gold. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>GOLD AND SILVER LACE. Under this heading a general account may be given of the use of the precious metals in textiles of all descriptions into which they enter. That these metals were used largely in the sumptuous textiles of the earliest periods of civilization there is abundant testimony; and to this day, in the Oriental centres whence a knowledge and the use of fabrics inwoven, ornamented, and embroidered with gold and silver first spread, the passion for such brilliant and costly textiles is still most strongly and generally prevalent. The earliest mention of the use of gold in a woven fabric occurs in the description of the ephod made for Aaron (Exod. xxxix. 2, 3) — "And he made the ephod of gold, blue, and purple, and scarlet, and fine twined linen. And they did beat the gold into thin plates, and cut it into wires (strips), to work it in the blue, and in the purple, and in the scarlet, and in the fine linen, with cunning work." In both the ''Iliad'' and the ''Odyssey'' distinct allusion is frequently made to inwoven and embroidered golden textiles. Many circumstances point to the conclusion that the art of weaving and embroidering with gold and silver originated in India, where it is still principally prosecuted, and that from one great city to another the practice travelled westward, — Babylon, Tarsus, Baghdad, Damascus, the islands of Cyprus and Sicily, Con- / stantinople and Venice, all in the process of time becoming famous centres of these much prized manufactures. Alexander the Great found Indian kings and princes arrayed in robes of gold and purple; and the Persian monarch Darius, we are told, wore a war mantle of cloth of gold, on which were figured two golden hawks as if pecking at each other. There is reason, according to Josephus, to believe that the “royal apparel" worn by Herod on the day of his death (Acts xii. 21) was a tissue of silver. Agrippina, the wife of the emperor Claudius, had a robe woven entirely of gold, and from that period downwards royal personages and high ecclesiastical dignitaries used cloth and tissues of gold and silver for their state and ceremonial robes, as well as for costly hangings and decorations. In England, at different periods, various names were applied to cloths of gold, as ciclatoun, tartarium, naques or nac, baudekiu or baldachin, Cyprus damask, and twssewys or tissue. The thin flimsy paper known as tissue paper, is so called because it originally was placed between the folds of gold "tissue" to prevent the contiguous surfaces from fraying each other. At what time the drawing of gold wire for the preparation of these textiles was first practised is not accurately known. The art was probably introduced and applied in different localities at widely different dates, but down till mediaeval times the method graphically described in the Pentateuch continued to be practised with both gold and silver. Fabrics woven with gold and silver continue to be used on the largest scale to this day in India; and there the preparation of the varieties of wire, and the working of the various forms of lace, brocade, and embroidery, is at once an important and peculiar art. The basis of all modern fabrics of this kind is wire, the "gold wire" of the manufacturer being in all cases silver gilt wire, and silver wire being, of course, composed of pure silver. In India the wire is drawn by means of simple draw-plates, with rude and simple appliances, from rounded bars of silver, or gold-plated silver, as the case may be. The wire is flattened into the strip or ribbon-like form it generally assumes by passing it, fourteen or fifteen strands simultaneously, over a fine, smooth, round-topped anvil, and beating it as it passes with a heavy hammer having a slightly convex surface. From wire so flattened there is made in India soniri, a tissue or cloth of gold, the web or warp being composed entirely of golden strips, and ruperi, a similar tissue of silver. Gold lace is also made on a warp of thick yellow silk with a weft of flat wire, and in the case of ribbons the warp or web is composed of the metal. The flattened wires are twisted around orange (in the case of silver, white) coloured silk thread, so as completely to cover the thread and present the appearance of a continuous wire; and in this form it is chiefly employed for weaving into the rich brocades known as kincobs or kinkhábs. Wires flattened, or partially flattened, are also twisted into exceedingly fine spirals, and in this form they are the basis of numerous ornamental applications. Such spirals drawn out till they present a waved appearance, and in that state flattened, are much used for rich heavy embroideries termed karchobs. Spangles for embroideries, &c., are made from spirals of comparatively stout wire, by cutting them down ring by ring, laying each C-like ring on an anvil, and by a smart blow with a hammer flattening it out into a thin round disk with a slit extending from the centre to one edge. Fine spirals are also used for general embroidery purposes. The demand for various kinds of loom-woven and embroidered gold and silver work in India is immense; and the variety of textiles so ornamented is also very great. "Gold and silver," says Dr Birdwood in his ''Handbook to the British-Indian Section, Paris Exhibition'', 1878, "are worked into the decoration of all the more costly loom-made garments and Indian piece goods, either on the borders only, or in stripes throughout, or in diapered figures. The gold-bordered loom embroideries are made chiefly at Sattara, and the gold or silver striped at Tanjore; the gold figured ''mashrus'' at Tanjore, Trichinopoly, and Hyderabad in the Deccau; and the highly ornamented gold-figured silks and gold and silver tissues principally at Ahmedabad, Benares, Murshedabad, and Trichinopoly." Among the Western communities the demand for gold and silver lace and embroideries arises chiefly in connexion with naval and military uniforms, court costumes, public and private liveries, ecclesiastical robes and draperies, theatrical dresses, and the badges and insignia of various orders. To a limited extent there is a trade in gold wire and lace to India and China. The metallic basis of the various fabrics is wire round and flattened, the wire being of three kinds — 1st, gold wire, which is invariably silver gilt wire; 2d, copper gilt wire, used for common liveries and theatrical purposes; and 3d, silver wire. These wires are drawn by the ordinary processes, and the flattening, when done, is accomplished by passing the wire between a pair of revolving rollers of fine polished steel. The various qualities of wire are prepared and used in precisely the same way as in India, — round wire, flat wire, thread made of flat gold wire twisted round orange-coloured silk or cotton, known in the trade as "orris," fine spirals and spangles, all being in use in the West as in the East. The lace is woven in the same manner as ribbons, and there are very numerous varieties in richness, pattern, and quality. Cloth of gold, and brocades rich in gold and silver, are woven for ecclesiastical vestments and draperies. The proportions of gold and silver in the gold thread for the lace trade varies, but in all cases the proportion of gold is exceedingly small. An ordinary gold lace wire is drawn from a bar containing 90 parts of silver and 7 of copper, coated with 3 parts of gold. On an average each ounce troy of a bar so plated is drawn into 1500 yards of wire; and therefore about 16 grains of gold cover a mile of wire. It is estimated that about 250,000 ounces of gold wire are made annually in Great Britain, of which about 20 per cent, is used for the headings of calico, muslin, &c., and the remainder is worked up in the gold lace trade.<ref>William Chandler Roberts-Austen and H. Bauerman [W.C.R. — H.B.]. "Gold and Silver Lace." In "Gold." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. 10 (X). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2010%20%28G-GOT%29%20193592738.23/page/753/mode/1up (accessed January 2023): 753, Col. 2c – 754, Cols. 1a–b – 2a–b.</ref></blockquote> ==== Honiton Lace ==== Kate Stradsin says,<blockquote>Honiton lace was the finest English equivalent of Brussels bobbin lace and was constructed in small ‘sprigs, in the cottages of lacemakers[.'] These sprigs were then joined together and bleached to form the large white flounces that were so sought after in the mid-nineteenth century.<ref>Strasdin, Kate. "Rediscovering Queen Alexandra’s Wardrobe: The Challenges and Rewards of Object-Based Research." ''The Court Historian'' 24.2 (2019): 181-196. Rpt http://repository.falmouth.ac.uk/3762/15/Rediscovering%20Queen%20Alexandra%27s%20Wardrobe.pdf: 13, and (for the little quotation) n. 37, which reads "Margaret Tomlinson, ''Three Generations in the Honiton Lace Trade: A Family History'', self-published, 1983."</ref></blockquote> [[File:Strook in Alençon naaldkant, 1750-1775.jpg|thumb|alt=A long piece of complex white lace with garlands, flowers and bows|Point d'Alençon lace, 1750-1775]] ==== Passementerie ==== ''Passementerie'' is the French term for trim on clothing or furniture. The 19th century (especially during the First and Second Empire) was a time of great "''exubérance''" in passementerie in French design, including the development and widespread use of the Jacquard loom.<ref>{{Cite journal|date=2023-06-10|title=Passementerie|url=https://fr.wikipedia.org/w/index.php?title=Passementerie&oldid=205068926|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Passementerie.</ref> ==== Point d'Alençon Lace ==== A lace made by hand using a number of complex steps and layers. The lacemakers build the point d'Alençon design on some kind of mesh and sometimes leave some of the mesh in as part of the lace and perhaps to provide structure. Elizabeth Lewandowski defines point d'Alençon lace and Alençon lace separately. Point lace is needlepoint lace,<ref name=":7">Lewandowski, Elizabeth J. ''The Complete Costume Dictionary''. Scarecrow Press, 2011.</ref>{{rp|233}} so Alençon point is "a two thread [needlepoint] lace."<ref name=":7" />{{rp|7}} Alençon lace has a "floral design on [a] fine net ground [and is] referred to as [the] queen of French handmade needlepoint laces. The original handmade Alençon was a fine needlepoint lace made of linen thread."<ref name=":7" />{{rp|7}} The sample of point d'Alençon lace (right), from 1750–1775, shows the linen mesh that the lace was constructed on.<ref>{{Cite web|url=http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689|title=MoMu - Open Fashion|website=openfashion.momu.be|access-date=2024-02-26}} ModeMuseum Antwerpen. http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689.</ref> The consistency in this sample suggests it may have been made by machine. == Fabric == === Brocatelle === Brocatelle is a kind of brocade, more simple than most brocades because it uses fewer warp and weft threads and fewer colors to form the design. The article in the French ''Wikipédia'' defines it like this:<blockquote>La '''brocatelle''' est un type de tissu datant du <abbr>xvi^e</abbr> siècle qui comporte deux chaînes et deux trames, au minimum. Il est composé pour que le dessin ressorte avec un relief prononcé, grâce à la chaîne sur un fond en sergé. Les brocatelles les plus anciennes sont toujours fabriquées avec une des trames en lin.<ref>{{Cite journal|date=2023-06-01|title=Brocatelle|url=https://fr.wikipedia.org/w/index.php?title=Brocatelle&oldid=204796410|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Brocatelle.</ref></blockquote>Which translates to this:<blockquote>Brocatelle is a type of fabric dating from the 16th century that has two warps and two wefts, at a minimum. It is composed so that the design stands out with a pronounced relief, thanks to the weft threads on a twill background. The oldest brocades were always made with one of the wefts being linen.</blockquote>The ''Oxford English Dictionary'' says, brocatelle is an "imitation of brocade, usually made of silk or wool, used for tapestry, upholstery, etc., now also for dresses. Both the nature and the use of the stuff have changed" between the late 17th century and 1888, the last time this definition was revised.<ref>"brocatelle, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/23550. Accessed 4 July 2023.</ref> === Broché === === Ciselé === === Crépe de Chine === The ''Oxford English Dictionary'' distinguishes the use of ''crêpe'' (using a circumflex rather than an acute accent over the first ''e'') from ''crape'' in textiles, saying ''crêpe'' is "often borrowed [from the French] as a term for all crapy fabrics other than ordinary black mourning crape,"<ref>"crêpe, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44242. Accessed 10 February 2023.</ref> with usage examples ranging from 1797 to the mid 20th century. Crêpe de chine, it says is "a white or other coloured crape made of raw silk." === Épinglé Velvet === Often spelled ''épingle'' rather than ''épinglé'', this term appears to have been used for a fabric made of wool, or at least wool along with linen or cotton, that was heavier and stiffer than silk velvet. It was associated with outer garments and men's clothing. Nowadays, épinglé velvet is an upholstery fabric in which the pile is cut into designs and patterns, and the portrait of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary, Duchess of Hamilton]] shows a mantle described as épinglé velvet that does seem to be a velvet with a woven pattern perhaps cut into the pile. === Lace === While lace also functioned sometimes as fabric — at the décolletage, for example, on the stomacher or as a veil — here we organize it as a [[Social Victorians/Terminology#Trim and Lace|part of the elaboration of clothing]]. === Liberty Fabrics === === Lisse === According to the ''Oxford English Dictionary'', the term ''lisse'' as a "kind of silk gauze" was used in the 19th-century UK and US.<ref>"lisse, n.1." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/108978. Accessed 4 July 2023.</ref> === Satin === The pre-1927 ''History of Feminine Fashion'', probably commissioned by Charles Frederick Worth's sons, describes Worth's "insistence on new fabrics, new trimmings, new materials of every description" at the beginning of his career in the mid 19th century:<blockquote>When Worth first entered the business of dressmaking, the only materials of the richer sort used for woman's dress were velvet, faille, and watered silk. Satin, for example, was never used. M. Worth desired to use satin very extensively in the gowns he designed, but he was not satisfied with what could be had at the time; he wanted something very much richer than was produced by the mills at Lyons. That his requirements entailed the reconstruction of mills mattered little — the mills were reconstructed under his directions, and the Lyons looms turned out a richer satin than ever, and the manufacturers prospered accordingly.<ref name=":9">[Worth, House of.] {{Cite book|url=http://archive.org/details/AHistoryOfFeminineFashion|title=A History Of Feminine Fashion (1800s to 1920s)}} Before 1927. [Likely commissioned by Worth. Link is to Archive.org; info from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Worth_Biarritz_salon.jpg.]</ref>{{rp|6 in printed, 26 in digital book}}</blockquote> === Selesia === According to the ''Oxford English Dictionary'', ''silesia'' is "A fine linen or cotton fabric originally manufactured in Silesia in what is now Germany (''Schlesien'').<ref>"Silesia, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/179664. Accessed 9 February 2023.</ref> It may have been used as a lining — for pockets, for example — in garments made of more luxurious or more expensive cloth. The word ''sleazy'' — "Of textile fabrics or materials: Thin or flimsy in texture; having little substance or body."<ref>"sleazy, adj." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/181563. Accessed 9 February 2023.</ref> — may be related. === Shot Fabric === According to the ''Oxford English Dictionary'', "Of a textile fabric: Woven with warp-threads of one colour and weft-threads of another, so that the fabric (usually silk) changes in tint when viewed from different points."<ref>“Shot, ''Adj.''”  ''Oxford English Dictionary'', Oxford UP,  July 2023, https://doi.org/10.1093/OED/2977164390.</ref> A shot fabric might also be made of silk and cotton fibers. === Tissue === A lightly woven fabric like gauze or chiffon. The light weave can make the fabric translucent and make pleating and gathering flatter and less bulky. Tissue can be woven to be shot, sheer, stiff or soft. Historically, the term in English was used for a "rich kind of cloth, often interwoven with gold or silver" or "various rich or fine fabrics of delicate or gauzy texture."<ref>“Tissue, N.” ''Oxford English Dictionary'', Oxford UP, March 2024, https://doi.org/10.1093/OED/5896731814.</ref> == Fan == The ''Encyclopaedia Britannica'' (9th edition) has an article on the fan. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>FAN (Latin, ''vannus''; French, ''éventail''), a light implement used for giving motion to the air. ''Ventilabrum'' and ''flabellum'' are names under which ecclesiastical fans are mentioned in old inventories. Fans for cooling the face have been in use in hot climates from remote ages. A bas-relief in the British Museum represents Sennacherib with female figures carrying feather fans. They were attributes of royalty along with horse-hair fly-flappers and umbrellas. Examples may be seen in plates of the Egyptian sculptures at Thebes and other places, and also in the ruins of Persepolis. In the museum of Boulak, near Cairo, a wooden fan handle showing holes for feathers is still preserved. It is from the tomb of Amen-hotep, of the 18th dynasty, 17th century <small>B</small>.<small>C</small>. In India fans were also attributes of men in authority, and sometimes sacred emblems. A heartshaped fan, with an ivory handle, of unknown age, and held in great veneration by the Hindus, was given to the prince of Wales. Large punkahs or screens, moved by a servant who does nothing else, are in common use by Europeans in India at this day. Fans were used in the early Middle Ages to keep flies from the sacred elements during the celebrations of the Christian mysteries. Sometimes they were round, with bells attached — of silver, or silver gilt. Notices of such fans in the ancient records of St Paul’s, London, Salisbury cathedral, and many other churches, exist still. For these purposes they are no longer used in the Western church, though they are retained in some Oriental rites. The large feather fans, however, are still carried in the state processions of the supreme pontiff in Rome, though not used during the celebration of the mass. The fan of Queen Theodolinda (7th century) is still preserved in the treasury of the cathedral of Monza. Fans made part of the bridal outfit, or ''mundus muliebris'', of ancient Roman ladies. Folding fans had their origin in Japan, and were imported thence to China. They were in the shape still used—a segment of a circle of paper pasted on a light radiating frame-work of bamboo, and variously decorated, some in colours, others of white paper on which verses or sentences are written. It is a compliment in China to invite a friend or distinguished guest to write some sentiment on your fan as a memento of any special occasion, and this practice has continued. A fan that has some celebrity in France was presented by the Chinese ambassador to the Comtesse de Clauzel at the coronation of Napoleon I. in 1804. When a site was given in 1635, on an artificial island, for the settlement of Portuguese merchants in Nippo in Japan, the space was laid out in the form of a fan as emblematic of an object agreeable for general use. Men and women of every rank both in China and Japan carry fans, even artisans using them with one hand while working with the other. In China they are often made of carved ivory, the sticks being plates very thin and sometimes carved on both sides, the intervals between the carved parts pierced with astonishing delicacy, and the plates held together by a ribbon. The Japanese make the two outer guards of the stick, which cover the others, occasionally of beaten iron, extremely thin and light, damascened with gold and other metals. Fans were used by Portuguese ladies in the 14th century, and were well known in England before the close of the reign of Richard II. In France the inventory of Charles V. at the end of the 14th century mentions a folding ivory fan. They were brought into general use in that country by Catherine de’ Medici, probably from Italy, then in advance of other countries in all matters of personal luxury. The court ladies of Henry VIII.’s reign in England were used to handling fans, A lady in the Dance of Death by Holbein holds a fan. Queen Elizabeth is painted with a round leather fan in her portrait at Gorhambury; and as many as twenty-seven are enumerated in her inventory (1606). Coryat, an English traveller, in 1608 describes them as common in Italy. They also became of general use from that time in Spain. In Italy, France, and Spain fans had special conventional uses, and various actions in handling them grew into a code of signals, by which ladies were supposed to convey hints or signals to admirers or to rivals in society. A paper in the ''Spectator'' humorously proposes to establish a regular drill for these purposes. The chief seat of the European manufacture of fans during the 17th century was Paris, where the sticks or frames, whether of wood or ivory, were made, and the decorations painted on mounts of very carefully prepared vellum (called latterly ''chicken skin'', but not correctly), — a material stronger and tougher than paper, which breaks at the folds. Paris makers exported fans unpainted to Madrid and other Spanish cities, where they were decorated by native artists. Many were exported complete; of old fans called Spanish a great number were in fact made in France. Louis XIV. issued edicts at various times to regulate the manufacture. Besides fans mounted with parchment, Dutch fans of ivory were imported into Paris, and decorated by the heraldic painters in the process called “Vernis Martin,” after a famous carriage painter and inventor of colourless lac varnish. Fans of this kind belonging to the Queen and to the late baroness de Rothschild were exhibited in 1870 at Kensington. A fan of the date of 1660, representing sacred subjects, is attributed to Philippe de Champagne, another to Peter Oliver in England in the / 17th century. Cano de Arevalo, a Spanish painter of the 17th century devoted himself to fan painting. Some harsh expressions of Queen Christina to the young ladies of the French court are said to have caused an increased ostentation in the splendour of their fans, which were set with jewels and mounted in gold. Rosalba Carriera was the name of a fan painter of celebrity in the 17th century. Lebrun and Romanelli were much employed during the same period. Klingstet, a Dutch artist, enjoyed a considerable reputation for his fans from the latter part of the 17th and the first thirty years of the 18th century. The revocation of the edict of Nantes drove many fan-makers out of France to Holland and England. The trade in England was well established under the Stuart sovereigns. Petitions were addressed by the fan-makers to Charles II. against the importation of fans from India, and a duty was levied upon such fans in consequence. This importation of Indian fans, according to Savary, extended also to France. During the reign of Louis XV. carved Indian and China fans displaced to some extent those formerly imported from Italy, which had been painted on swanskin parchment prepared with various perfumes. During the 18th century all the luxurious ornamentation of the day was bestowed on fans as far as they could display it. The sticks were made of mother-of-pearl or ivory, carved with extraordinary skill in France, Italy, England, and other countries. They were painted from designs of Boucher, Watteau, Lancret, and other "genre" painters, Hébert, Rau, Chevalier, Jean Boquet, Mad. Verité, are known as fan painters. These fashions were followed in most countries of Europe, with certain national differences. Taffeta and silk, as well as fine parchment, were used for the mounts. Little circles of glass were let into the stick to be looked through, and small telescopic glasses were sometimes contrived at the pivot of the stick. They were occasionally mounted with the finest point lace. An interesting fan (belonging to Madame de Thiac in France), the work of Le Flamand, was presented by the municipality of Dieppe to Marie Antoinette on the birth of her son the dauphin. From the time of the Revolution the old luxury expended on fans died out. Fine examples ceased to be exported to England and other countries. The painting on them represented scenes or personages connected with political events. At a later period fan mounts were often prints coloured by hand. The events of the day mark the date of many examples found in modern collections. Amongst the fanmakers of the present time the names of Alexandre, Duvelleroy, Fayet, Vanier, may be mentioned as well known in Paris. The sticks are chiefly made in the department of Oise, at Le Déluge, Crèvecœur, Méry, Ste Geneviève, and other villages, where whole families are engaged in preparing them; ivory sticks are carved at Dieppe. Water-colour painters of distinction often design and paint the mounts, the best designs being figure subjects. A great impulse has been given to the manufacture and painting of fans in England since the exhibition which took place at South Kensington in 1870. Other exhibitions have since been held, and competitive prizes offered, one of which was gained by the Princess Louise. Modern collections of fans take their date from the emigration of many noble families from France at the time of the Revolution. Such objects were given as souvenirs and occasionally sold by families in straitened circumstances. A large number of fans of all sorts, principally those of the 18th century, French, English, German, Italian Spanish, &c., have been lately bequeathed to the South Kensington Museum. Regarding the different parts of folding fans it may be well to state that the sticks are called in French ''brins'', the two outer guards ''panaches'', and the mount ''feuille''.<ref>J. H. Pollen [J.H.P.]. "Fan." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. '''10''' ('''X'''). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%209%20%28FAL-FYZ%29%20193323016.23/page/26/mode/2up (accessed January 2023): 27, Col. 1b – 28, Col. 1c.</ref></blockquote> == Fancy-dress Ball == Fancy-dress (or costume) balls were popular and frequent in the U.K. and France as well as the rest of Europe during the 19th century. The themes and styles of the fancy-dress balls influenced those that followed. At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the guests came dressed in costume from times before 1820, as instructed on '''the invitation''', but their clothing was much more about late-Victorian standards of beauty and fashion than the standards of whatever time period the portraits they were copying or basing their costumes on. ''The Queen'' published dress and fashion information and advice under the byline of Ardern Holt, who regularly answered questions from readers about fashion as well as about fancy dress. (More about Ardern Holt, which is almost certainly a pseudonym, can be found on the [[Social Victorians/People/Working in Publishing#Journalists|People Working in Publishing]] page.) Holt also ran wrote entire articles with suggestions for what might make an appealing fancy-dress costume as well as pointing readers away from costumes that had been worn too frequently. The suggestions for costumes are based on familiar types or portraits available to readers, similar to Holt's books on fancy dress, which ran through a number of editions in the 1880s and 1890s. Fancy-dress questions sometimes asked for details about costumes worn in theatrical or operatic productions, which Holt provides. In November 1897, Holt refers to the Duchess of Devonshire's 2 July ball: "Since the famous fancy ball, given at Devonshire House during this year, historical fancy dresses have assumed a prominence that they had not hitherto known."<ref>Holt, Ardern. "Fancy Dress a la Mode." The ''Queen'' 27 November 1897, Saturday: 94 [of 145 in BNA; print p. 1026], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971127/459/0094.</ref> Holt goes on to provide a number of ideas for costumes for historical fancy dress, as always with a strong leaning toward Victorian standards of beauty and style and away from any concern for historical accuracy. Ardern Holt published books on fancy dress as well as writing for the ''Queen'' and other periodicals, but not all of them were about fancy dress. # ''Gentlemen's Fancy Dress: How to Choose It''. Wyman & Sons, 1882. (''Google Books'' has this: https://books.google.com/books/about/Gentlemen_s_Fancy_Dress.html?id=ED8CAAAAQAAJ.) Later editions: 1898 (HathiTrust) # ''Fancy Dresses Described; Or, What to Wear at Fancy Balls''. Debenham & Freebody, 1882. Illustr., Lillian Young. (HathiTrust has this.) Later editions: 4th ed — 1884; 1887 (HathiTrust); 6th ed. — 1896 (HathiTrust) As Leonore Davidoff says, "Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration symbolised some status category for the female wearer."<ref name=":1" />{{rp|93}} [handled under Elaborations] === Historical Accuracy === Many of the costumes at the ball were based on portraits, especially when the guest was dressed as a historical figure. If possible, we have found the portraits likely to have been the originals, or we have found, if possible, portraits that show the subjects from the two time periods at similar ages. The way clothing was cut changed quite a bit between the 18th and 19th centuries. We think of Victorian clothing — particularly women's clothing, and particularly at the end of the century — as inflexible and restrictive, especially compared to 20th- and 21st-century customs permitting freedom of movement. The difference is generally evolutionary rather than absolute — that is, as time has passed since the 18th century, clothing has allowed an increasingly greater range of movement, especially for people who did not do manual labor. By the end of the 19th century, garments like women's bodices and men's coats were made fitted and smooth by attention to the grain of the fabric and by the use of darts (rather than techniques that assembled many small, individual pieces of fabric). * clothing construction and flat-pattern techniques * Generally, the further back in time we go, the more 2-dimensional the clothing itself was. ==== Women's Versions of Historical Accuracy at the Ball ==== As always with this ball, whatever historical accuracy might be present in a woman's costume is altered so that the wearer is still a fashionable Victorian lady. What makes the costumes look "Victorian" to our eyes is the line of the silhouette caused by the foundation undergarments as well as the many "elaborations"<ref name=":1" />{{rp|93}}, mostly in the decorations, trim and accessories. Also, the clothing hangs and drapes differently because the fabric was cut on grain and the shoulders were freed by the way the sleeves were set in. ==== Men's Versions of Historical Accuracy at the Ball ==== Because men were not wearing a Victorian foundation garment at the end of the century, the men's costumes at the ball are more historically accurate in some ways. * Trim * Mixing neck treatments * Hair * Breeches * Shoes and boots * Military uniforms, arms, gloves, boots == Feathers and Plumes == === Aigrette === Elizabeth Lewandowski defines ''aigrette'' as "France. Feather or plume from an egret or heron."<ref name=":7" />(5) Sometimes the newspapers use the term to refer to an accessory (like a fan or ornament on a hat) that includes such a feather or plume. The straight and tapered feathers in an aigrette are in a bundle. === Prince of Wales's Feathers or White Plumes === The feathers in an aigrette came from egrets and herons; Prince of Wales's feathers came from ostriches. A fuller discussion of Prince of Wales's feathers and the white ostrich plumes worn at court appears on [[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|Victorian Things]]. For much of the late 18th and 19th centuries, white ostrich plumes were central to fashion at court, and at a certain point in the late 18th century they became required for women being presented to the monarch and for their sponsors. Our purpose here is to understand why women were wearing plumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] as part of their costumes. First published in 1893, [[Social Victorians/People/Lady Colin Campbell|Lady Colin Campbell]]'s ''Manners and Rules of Good Society'' (1911 edition) says that<blockquote>It was compulsory for both Married and Unmarried Ladies to Wear Plumes. The married lady’s Court plume consisted of three white feathers. An unmarried lady’s of two white feathers. The three white feathers should be mounted as a Prince of Wales plume and worn towards the left hand side of the head. Colored feathers may not be worn. In deep mourning, white feathers must be worn, black feathers are inadmissible.<p> White veils or lace lappets must be worn with the feathers. The veils should not be longer than 45 inches.<ref>{{Cite web|url=https://www.edwardianpromenade.com/etiquette/the-court-presentation/|title=The Court Presentation|last=Holl|first=Evangeline|date=2007-12-07|website=Edwardian Promenade|language=en-US|access-date=2022-12-18}} https://www.edwardianpromenade.com/etiquette/the-court-presentation/.</ref></blockquote>[[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|This fashion was imported from France]] in the mid 1770s.<ref>"Abstract" for Blackwell, Caitlin. "'<nowiki/>''The Feather'd Fair in a Fright''': The Emblem of the Feather in Graphic Satire of 1776." ''Journal for Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. ''Wiley Online'' DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x (accessed November 2022).</ref> Separately, a secondary heraldic emblem of the Prince of Wales has been a specific arrangement of 3 ostrich feathers in a gold coronet<ref>{{Cite journal|date=2022-11-07|title=Prince of Wales's feathers|url=https://en.wikipedia.org/w/index.php?title=Prince_of_Wales%27s_feathers&oldid=1120556015|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Prince_of_Wales's_feathers.</ref> since King Edward III (1312–1377<ref>{{Cite journal|date=2022-12-14|title=Edward III of England|url=https://en.wikipedia.org/w/index.php?title=Edward_III_of_England&oldid=1127343221|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_III_of_England.</ref>). Some women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] wore white ostrich feathers in their hair, but most of them are not Prince of Wales's feathers. Most of the plumes in these portraits are arrangements of some kind of headdress to accompany the costume. A few, wearing what looks like the Princes of Wales's feathers, might be signaling that their character is royal or has royal ancestry. '''One of the women [which one?] was presented to the royals at this ball?''' Here is the list of women who are wearing white ostrich plumes in their portraits in the [[Social Victorians/1897 Fancy Dress Ball/Photographs|''Diamond Jubilee Fancy Dress Ball'' album of 286 photogravure portraits]]: # Kathleen Pelham-Clinton, the [[Social Victorians/People/Newcastle|Duchess of Newcastle]] # [[Social Victorians/People/Louisa Montagu Cavendish|Luise Cavendish]], the Duchess of Devonshire # Jesusa Murrieta del Campo Mello y Urritio (née Bellido), [[Social Victorians/People/Santurce|Marquisa de Santurce]] # Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]] # Princess (Laura Williamina Seymour) Victor of  [[Social Victorians/People/Gleichen#Laura%20Williamina%20Seymour%20of%20Hohenlohe-Langenburg|Hohenlohe Langenburg]] # Louisa Acheson, [[Social Victorians/People/Gosford|Lady Gosford]] # Alice Emily White Coke, [[Social Victorians/People/Leicester|Viscountess Coke]] # Lady Mary Stewart, Helen Mary Theresa [[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]] #[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], Duchess of [[Social Victorians/People/Marlborough|Marlborough]], dressed as the wife of the French Ambassador at the Court of Catherine of Russia (not white, but some color that reads dark in the black-and-white photograph) #Mrs. Mary [[Social Victorians/People/Chamberlain|Chamberlain]] (at 491), wearing white plumes, as Madame d'Epinay #Lady Clementine [[Social Victorians/People/Tweeddale|Hay]] (at 629), wearing white plumes, as St. Bris (''Les Huguenots'') #[[Social Victorians/People/Meysey-Thompson|Lady Meysey-Thompson]] (at 391), wearing white plumes, as Elizabeth, Queen of Bohemia #Mrs. [[Social Victorians/People/Grosvenor|Algernon (Catherine) Grosvenor]] (at 510), wearing white plumes, as Marie Louise #Lady [[Social Victorians/People/Ancaster|Evelyn Ewart]], at 401), wearing white plumes, as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson #[[Social Victorians/People/Lyttelton|Edith Sophy Balfour Lyttelton]] (at 580), wearing what might be white plumes on a large-brimmed white hat, after a picture by Romney #[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), wearing what might be white plumes, as Cydalise of the Comedie Italienne from the time of Louis XV #Lady [[Social Victorians/People/Ilchester|Muriel Fox Strangways]] (at 403), wearing what might be two smallish white plumes, as Lady Sarah Lennox, one of the bridesmaids of Queen Charlotte A.D. 1761 #Lady [[Social Victorians/People/Lucan|Violet Bingham]] (at 586), wearing perhaps one white plume in a headdress not related to the Prince of Wales's feathers #Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), wearing a headdress that includes some white plumes, as Lady Burleigh #[[Social Victorians/People/Dupplin|Agnes Blanche Marie Hay-Drummond]] (at 682), in a big headdress topped with white plumes, as Mademoiselle Andrée de Taverney A.D. 1775 #Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]] (at 336), wearing what looks like Prince of Wales's plumes #[[Social Victorians/People/Suffolk|Marguerite Hyde "Daisy" Leiter]] (at 684), wearing what looks like Prince of Wales's plumes #Lady [[Social Victorians/People/Spicer|Margaret Spicer]] (at 281), wearing one smallish white and one black plume, as Countess Zinotriff, Lady-in-Waiting to the Empress Catherine of Russia #Mrs. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 318), wearing what looks like Prince of Wales's plumes, as Elizabeth Cavendish, daughter of Bess of Hardwick #Nellie, [[Social Victorians/People/Kilmorey|Countess of Kilmorey]] (at 207), wearing three tall plumes, 2 white and one dark, as Comtesse du Barri #Daisy, [[Social Victorians/People/Warwick|Countess of Warwick]] (at 53), wearing at least 1 white plume, as Marie Antoinette More men than women were wearing plumes reminiscent of the Prince of Wales's feathers: * ==== Bibliography for Plumes and Prince of Wales's Feathers ==== * Blackwell, Caitlin. "'''The Feather'd Fair in a Fright'<nowiki/>'': The Emblem of the Feather in Graphic Satire of 1776." Journal for ''Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. Wiley Online DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x. * "Prince of Wales's feathers." ''Wikipedia'' https://en.wikipedia.org/wiki/Prince_of_Wales%27s_feathers (accessed November 2022). ['''Add women to this page'''] * Simpson, William. "On the Origin of the Prince of Wales' Feathers." ''Fraser's magazine'' 617 (1881): 637-649. Hathi Trust https://babel.hathitrust.org/cgi/pt?id=chi.79253140&view=1up&seq=643&q1=feathers (accessed December 2022). Deals mostly with use of feathers in other cultures and in antiquity; makes brief mention of feathers and plumes in signs and pub names that may not be associated with the Prince of Wales. No mention of the use of plumes in women's headdresses or court dress. == Honors == === The Bath === The Most Honourable Order of the Bath (GCB, Knight or Dame Grand Cross; KCB or DCB, Knight or Dame Commander; CB, Companion) === The Garter === The Most Noble Order of the Knights of the Garter (KG, Knight Companion; LG, Lady Companion) [[File:The Golden Fleece - collar exhibited at MET, NYC.jpg|thumb|The Golden Fleece collar and pendant for the 2019 "Last Knight" exhibition at the MET, NYC.|alt=Recent photograph of a gold necklace on a wide band, with a gold skin of a sheep hanging from it as a pendant]] === The Golden Fleece === To wear the golden fleece is to wear the insignia of the Order of the Golden Fleece, said to be "the most prestigious and historic order of chivalry in the world" because of its long history and strict limitations on membership.<ref name=":10">{{Cite journal|date=2020-09-25|title=Order of the Golden Fleece|url=https://en.wikipedia.org/w/index.php?title=Order_of_the_Golden_Fleece&oldid=980340875|journal=Wikipedia|language=en}}</ref> The monarchs of the U.K. were members of the originally Spanish order, as were others who could afford it, like the Duke of Wellington,<ref name=":12">Thompson, R[obert]. H[ugh]. "The Golden Fleece in Britain." Publication of the ''British Numismatic Society''. 2009 https://www.britnumsoc.org/publications/Digital%20BNJ/pdfs/2009_BNJ_79_8.pdf (accessed January 2023).</ref> the first Protestant to be admitted to the order.<ref name=":10" /> Founded in 1429/30 by Philip the Good, Duke of Burgundy, the order separated into two branches in 1714, one Spanish and the other Austrian, still led by the House of Habsburg.<ref name=":10" /> [[File:Prince Albert - Franz Xaver Winterhalter 1842.jpg|thumb|1842 Winterhalter portrait of Prince Albert wearing the insignia of the Order of the Golden Fleece, 1842|left|alt=1842 Portrait of Prince Albert by Winterhalter, wearing the insignia of the Golden Fleece]] The photograph (upper right) is of a Polish badge dating from the "turn of the XV and XVI centuries."<ref>{{Citation|title=Polski: Kolana orderowa orderu Złotego Runa, przełom XV i XVI wieku.|url=https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg|date=2019-11-10|accessdate=2023-01-10|last=Wulfstan}}. https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg.</ref> The collar to this Golden Fleece might be similar to the one the [[Social Victorians/People/Spencer Compton Cavendish#The Insignia of the Order of the Golden Fleece|Duke of Devonshire is wearing in the 1897 Lafayette portrait]]. The badges and collars that Knights of the Order actually wore vary quite a bit. The 1842 Franz Xaver Winterhalter portrait (left) of Prince Consort Albert, Victoria's husband and father of the Prince of Wales, shows him wearing the Golden Fleece on a red ribbon around his neck and the star of the Garter on the front of his coat.<ref>Winterhalter, Franz Xaver. ''Prince Albert''. {{Cite web|url=https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61|title=Explore the Royal Collection Online|website=www.rct.uk|access-date=2023-01-16}} https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61.</ref> === Royal Victorian Order === (GCVO, Knight or Dame Grand Cross; KCVO or DCVO, Knight or Dame Commander; CVO, Commander; LVO, Lieutenant; MVO, Member) === St. John === The Order of the Knights of St. John === Star of India === Most Exalted Order of the Star of India (GCSI, Knight Grand Commander; KCSI, Knight Commander; CSI, Companion) === Thistle === The Most Ancient and Most Noble Order of the Thistle == Jewelry and Stones == === Cabochon === This term describes both the treatment and shape of a precious or semiprecious stone. A cabochon treatment does not facet the stone but merely polishes it, removing "the rough parts" and the parts that are not the right stone.<ref>"cabochon, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/25778. Accessed 7 February 2023.</ref> A cabochon shape is often flat on one side and oval or round, forming a mound in the setting. === Jet === === ''Orfèvrerie'' === Sometimes misspelled in the newspapers as ''orvfèvrerie''. ''Orfèvrerie'' is the artistic work of a goldsmith, silversmith, or jeweler. === Turquoises === == Military == Several men from the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]] were dressed in military uniforms, some historical and some, possibly, not. === Baldric === According to the ''Oxford English Dictionary'', the primary sense of ''baldric'' is "A belt or girdle, usually of leather and richly ornamented, worn pendent from one shoulder across the breast and under the opposite arm, and used to support the wearer's sword, bugle, etc."<ref>"baldric, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/14849. Accessed 17 May 2023.</ref> This sense has been in existence since c. 1300. === Cuirass === According to the ''Oxford English Dictionary'', the primary sense of ''cuirass'' is "A piece of armour for the body (originally of leather); ''spec.'' a piece reaching down to the waist, and consisting of a breast-plate and a back-plate, buckled or otherwise fastened together ...."<ref>"cuirass, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/45604. Accessed 17 May 2023.</ref> [[File:Knötel IV, 04.jpg|thumb|alt=An Old drawing in color of British soldiers on horses brandishing swords in 1815.|1890 illustration of the Household Cavalry (Life Guard, left; Horse Guard, right) at the Battle of Waterloo, 1815]] === Household Cavalry === The Royal Household contains the Household Cavalry, a corps of British Army units assigned to the monarch. It is made up of 2 regiments, the Life Guards and what is now called The Blues and Royals, which were formed around the time of "the Restoration of the Monarchy in 1660."<ref name=":3">Joll, Christopher. "Tales of the Household Cavalry, No. 1. Roles." The Household Cavalry Museum, https://householdcavalry.co.uk/app/uploads/sites/2/2021/06/Household-Cavalry-Museum-video-series-large-print-text-Tales-episode-01.pdf.</ref>{{rp|1}} Regimental Historian Christopher Joll says, "the original Life Guards were formed as a mounted bodyguard for the exiled King Charles II, The Blues were raised as Cromwellian cavalry and The Royals were established to defend Tangier."<ref name=":3" />{{rp|1–2}} The 1st and 2nd Life Guards were formed from "the Troops of Horse and Horse Grenadier Guards ... in 1788."<ref name=":3" />{{rp|3}} The Life Guards were and are still official bodyguards of the queen or king, but through history they have been required to do quite a bit more than serve as bodyguards for the monarch. The Household Cavalry fought in the Battle of Waterloo on Sunday, 18 June 1815 as heavy cavalry.<ref name=":3" />{{rp|3}} Besides arresting the Cato Steet conspirators in 1820 "and guarding their subsequent execution," the Household Cavalry contributed to the "the expedition to rescue General Gordon, who was trapped in Khartoum by The Mahdi and his army of insurgents" in 1884.<ref name=":3" />{{rp|3}} In 1887 they "were involved ... in the suppression of rioters in Trafalgar Square on Bloody Sunday."<ref name=":3" />{{rp|3}} ==== Grenadier Guards ==== Three men — [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Lord Algernon Gordon-Lennox]], [[Social Victorians/People/Stanley#Edward George Villiers Stanley, Lord Stanley|Lord Stanley]], and [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|Hon. F. C. Stanley]] — attended the ball as officers of the Grenadier Guards, wearing "scarlet tunics, ... full blue breeches, scarlet hose and shoes, lappet wigs" as well as items associated with weapons and armor.<ref name=":14">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 34, Col. 2a}} Founded in England in 1656 as Foot Guards, this infantry regiment "was granted the 'Grenadier' designation by a Royal Proclamation" at the end of the Napoleonic Wars.<ref>{{Cite journal|date=2023-04-22|title=Grenadier Guards|url=https://en.wikipedia.org/w/index.php?title=Grenadier_Guards&oldid=1151238350|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Grenadier_Guards.</ref> They were not called Grenadier Guards, then, before about 1815. In 1660, the Stuart Restoration, they were called Lord Wentworth's Regiment, because they were under the command of Thomas Wentworth, 5th Baron Wentworth.<ref>{{Cite journal|date=2022-07-24|title=Lord Wentworth's Regiment|url=https://en.wikipedia.org/w/index.php?title=Lord_Wentworth%27s_Regiment&oldid=1100069077|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Wentworth%27s_Regiment.</ref> At the time of Lord Wentworth's Regiment, the style of the French cavalier had begun to influence wealthy British royalists. In the British military, a Cavalier was a wealthy follower of Charles I and Charles II — a commander, perhaps, or a field officer, but probably not a soldier.<ref>{{Cite journal|date=2023-04-22|title=Cavalier|url=https://en.wikipedia.org/w/index.php?title=Cavalier&oldid=1151166569|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier.</ref> The Guards were busy as infantry in the 17th century, engaging in a number of armed conflicts for Great Britain, but they also served the sovereign. According to the Guards Museum,<blockquote>In 1678 the Guards were ordered to form Grenadier Companies, these men were the strongest and tallest of the regiment, they carried axes, hatches and grenades, they were the shock troops of their day. Instead of wearing tri-corn hats they wore a mitre shaped cap.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/|title=Service to the Crown|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/.</ref></blockquote>The name comes from ''grenades'', then, and we are accustomed to seeing them in front of Buckingham Palace, with their tall mitre hats. The Guard fought in the American Revolution, and in the 19th century, the Grenadier Guards fought in the Crimean War, Sudan and the Boer War. They have roles as front-line troops and as ceremonial for the sovereign, which makes them elite:<blockquote>Queen Victoria decreed that she did not want to see a single chevron soldier within her Guards. Other then [sic] the two senior Warrant Officers of the British Army, the senior Warrant Officers of the Foot Guards wear a large Sovereigns personal coat of arms badge on their upper arm. No other regiments of the British Army are allowed to do so; all the others wear a small coat of arms of their lower arms. Up until 1871 all officers in the Foot Guards had the privilege of having double rankings. An Ensign was ranked as an Ensign and Lieutenant, a Lieutenant as Lieutenant and Captain and a Captain as Captain and Lieutenant Colonel. This was because at the time officers purchased their own ranks and it cost more to purchase a commission in the Foot Guards than any other regiments in the British Army. For example if it cost an officer in the Foot Guards £1,000 for his first rank, in the rest of the Army it would be £500 so if he transferred to another regiment he would loose [sic] £500, hence the higher rank, if he was an Ensign in the Guards and he transferred to a Line Regiment he went in at the higher rank of Lieutenant.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/|title=Formation and role of the Regiments|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/.</ref></blockquote> ==== Life Guards ==== [[Social Victorians/People/Shrewsbury#Reginald Talbot's Costume|General the Hon. Reginald Talbot]], a member of the 1st Life Guards, attended the Duchess of Devonshire's ball dressed in the uniform of his regiment during the Battle of Waterloo.<ref name=":14" />{{rp|p. 36, Col. 3b}} At the Battle of Waterloo the 1st Life Guards were part of the 1st Brigade — the Household Brigade — and were commanded by Major-General Lord Edward Somerset.<ref name=":4">{{Cite journal|date=2023-09-30|title=Battle of Waterloo|url=https://en.wikipedia.org/w/index.php?title=Battle_of_Waterloo&oldid=1177893566|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Battle_of_Waterloo.</ref> The 1st Life Guards were on "the extreme right" of a French countercharge and "kept their cohesion and consequently suffered significantly fewer casualties."<ref name=":4" /> == Photography == == Footnotes == {{reflist}} 0eop5fasax0nln2f6580ya8x5okgfqm 2690305 2690304 2024-12-04T20:23:24Z Scogdill 1331941 /* Hoops */ 2690305 wikitext text/x-wiki Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. The definitions on this page are to provide a sense of what someone in the late 19th century might have meant by the term rather than a definition of what we might mean by it today. In the absence of a specialized glossary from the end of the 19th century in the U.K., we use the ''Oxford English Dictionary'' because the senses of a word are illustrated with examples that have dates so we can be sure that the senses we pick are appropriate for when they are used in the quotations we have. We also sometimes use the French ''Wikipédia'' to define a word because many technical terms of fashion were borrowings from the French. Also, often the French ''Wikipédia'' provides historical context for the uses of a word similar to the way the OED does. == Articles or Parts of Clothing: Non-gender-specific == === Mantle, Cloak, Cape === In 19th-century newspaper accounts, these terms are sometimes used without precision as synonyms. These are all outer garments. '''Mantle''' A mantle — often a long outer garment — might have elements like a train, sleeves, collars, revers, fur, and a cape. A late-19th-century writer making a distinction between a mantle and a cloak might use ''mantle'' if the garment is more voluminous. '''Cloak''' '''Cape''' === Peplum === According to the French ''Wiktionnaire'', a peplum is a "Short skirt or flared flounce layered at the waist of a jacket, blouse or dress" [translation by Google Translate].<ref>{{Cite journal|date=2021-07-02|title=péplum|url=https://fr.wiktionary.org/w/index.php?title=p%C3%A9plum&oldid=29547727|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/p%C3%A9plum.</ref> The ''Oxford English Dictionary'' has a fuller definition, although, it focuses on women's clothing because the sense is written for the present day:<blockquote>''Fashion''. ... a kind of overskirt resembling the ancient peplos (''obsolete''). Hence (now usually) in modern use: a short flared, gathered, or pleated strip of fabric attached at the waist of a woman's jacket, dress, or blouse to create a hanging frill or flounce.<ref name=":5">“peplum, n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1832614702>.</ref></blockquote>Men haven't worn peplums since the 18th century, except when wearing costumes based on historical portraits. The ''Daily News'' reported in 1896 that peplums had been revived as a fashion item for women.<ref name=":5" /> === Revers === According to the ''Oxford English Dictionary'', ''revers'' are the "edge[s] of a garment turned back to reveal the undersurface (often at the lapel or cuff) (chiefly in ''plural''); the material covering such an edge."<ref>"revers, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/164777. Accessed 17 April 2023.</ref> The term is French and was used this way in the 19th century (according to the ''Wiktionnaire'').<ref>{{Cite journal|date=2023-03-07|title=revers|url=https://fr.wiktionary.org/w/index.php?title=revers&oldid=31706560|journal=Wiktionnaire|language=fr}} https://fr.wiktionary.org/wiki/revers.</ref> == Articles or Parts of Clothing: Men's == [[Social Victorians/Terminology#Military|Men's military uniforms]] are discussed below. === À la Romaine === [[File:Johann Baptist Straub - Mars um 1772-1.jpg|thumb|left|alt=Old and damaged marble statue of a Roman god of war with flowing cloak, big helmet with a plume on top, and armor|Johann Baptist Straub's 1772 ''à la romaine'' ''Mars'']] A few people who attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball in 1897]] personated Roman gods or people. They were dressed not as Romans, however, but ''à la romaine'', which was a standardized style of depicting Roman figures that was used in paintings, sculpture and the theatre for historical dress from the 17th until the 20th century. The codification of the style was developed in France in the 17th century for theatre and ballet, when it became popular for masked balls. Women as well as men could be dressed ''à la romaine'', but much sculpture, portraiture and theatre offered opportunities for men to dress in Roman style — with armor and helmets — and so it was most common for men. In large part because of the codification of the style as well as the painting and sculpture, the style persisted and remained influential into the 20th century and can be found in museums and galleries and on monuments. For example, Johann Baptist Straub's 1772 statue of Mars (left), now in the Bayerisches Nationalmuseum, Munich, missing part of an arm, shows Mars ''à la romaine''. In London, an early 17th-century example of a figure of Mars ''à la romaine'', with a helmet, '''was''' "at the foot of the Buckingham tomb in Henry VII's Chapel at Westminster Abbey."<ref>Webb, Geoffrey. “Notes on Hubert Le Sueur-II.” ''The Burlington Magazine for Connoisseurs'' 52, no. 299 (1928): 81–89. http://www.jstor.org/stable/863535.</ref>{{rp|81, Col. 2c}} === Cavalier === [[File:Sir-Anthony-van-Dyck-Lord-John-Stuart-and-His-Brother-Lord-Bernard-Stuart.jpg|thumb|alt=Old painting of 2 men flamboyantly and stylishly dressed in colorful silk, with white lace, high-heeled boots and long hair|Van Dyck's c. 1638 painting of cavaliers Lord John Stuart and his brother Lord Bernard Stuart]] As a signifier in the form of clothing of a royalist political and social ideology begun in France in the early 17th century, the cavalier established France as the leader in fashion and taste. Adopted by [[Social Victorians/Terminology#Military|wealthy royalist British military officers]] during the time of the Restoration, the style signified a political and social position, both because of the loyalty to Charles I and II as well the wealth required to achieve the cavalier look. The style spread beyond the political, however, to become associated generally with dress as well as a style of poetry.<ref>{{Cite journal|date=2023-04-25|title=Cavalier poet|url=https://en.wikipedia.org/w/index.php?title=Cavalier_poet&oldid=1151690299|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier_poet.</ref> Van Dyck's 1638 painting of two brothers (right) emphasizes the cavalier style of dress. === Coats === ==== Doublet ==== * In the 19th-century newspaper accounts we have seen that use this word, doublet seems always to refer to a garment worn by a man, but historically women may have worn doublets. In fact, a doublet worn by Queen Elizabeth I exists and '''is somewhere'''. * Technically doublets were long sleeved, although we cannot be certain what this or that Victorian tailor would have done for a costume. For example, the [[Social Victorians/People/Spencer Compton Cavendish#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Duke of Devonshire's costume as Charles V]] shows long sleeves that may be part of the surcoat but should be the long sleeves of the doublet. ==== Pourpoint ==== A padded doublet worn under armor to protect the warrior from the metal chafing. A pourpoint could also be worn without the armor. ==== Surcoat ==== Sometimes just called ''coat''. [[File:Oscar Wilde by Sarony 1882 18.jpg|thumb|alt=Old photograph of a young man wearing a velvet jacket, knee breeches, silk hose and shiny pointed shoes with bows, seated on a sofa and leaning on his left hand and holding a book in his right| Oscar Wilde, 1882, by Napoleon Sarony]] === Hose, Stockings and Tights === Newspaper accounts from the late 19th century of men's clothing use the term ''hose'' for what we might call stockings or tights. In fact, the terminology is specific. ''Stockings'' is the more general term and could refer to hose or tights. With knee breeches men wore hose, which ended above the knee, and women wore hose under their dresses. The ''Oxford English Dictionary'' defines tights as "Tight-fitting breeches, worn by men in the 18th and early 19th centuries, and still forming part of court-dress."<ref>“Tights, N.” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2693287467.</ref> By 1897, the term was in use for women's stockings, which may have come up only to the knee. Tights were also worn by dancers and acrobats. This general sense of ''tights'' does not assume that they were knitted. ''Clocking'' is decorative embroidery on hose, usually, at the ankles on either the inside or the outside of the leg. It started at the ankle and went up the leg, sometimes as far as the knee. On women's hose, the clocking could be quite colorful and elaborate, while the clocking on men's hose was more inconspicuous. In many photographs men's hose are wrinkled, especially at the ankles and the knees, because they were shaped from woven fabric. Silk hose were knitted instead of woven, which gave them elasticity and reduced the wrinkling. The famous Sarony carte de visite photograph of Oscar Wilde (right) shows him in 1882 wearing knee breeches and silk hose, which are shiny and quite smoothly fitted although they show a few wrinkles at the ankles and knees. In the portraits of people in costume at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the men's hose are sometimes quite smooth, which means they were made of knitted silk and may have been smoothed for the portrait. In painted portraits the hose are almost always depicted as smooth, part of the artist's improvement of the appearance of the subject. === Shoes and Boots === == Articles or Parts of Clothing: Women's == === '''Chérusque''' === According to the French ''Wikipedia'', ''chérusque'' is a 19th-century term for the kind of standing collar like the ones worn by ladies in the Renaissance.<ref>{{Cite journal|date=2021-06-26|title=Collerette (costume)|url=https://fr.wikipedia.org/w/index.php?title=Collerette_(costume)&oldid=184136746|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Collerette_(costume)#Au+xixe+siècle+:+la+Chérusque.</ref> === Corsage === According to the ''Oxford English Dictionary'', the corsage is the "'body' of a woman's dress; a bodice."<ref>"corsage, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/42056. Accessed 7 February 2023.</ref> This sense is well documented in the ''OED'' for the mid and late 19th-century, used this way in fiction as well as in a publication like ''Godey's Lady's Book'', which would be expected to use appropriate terminology associated with fashion and dress making. The sense of "a bouquet worn on the bodice" is, according to the ''OED'', American. === Décolletage === === Girdle === === Mancheron === According to the ''Oxford English Dictionary'', a ''mancheron'' is a "historical" word for "A piece of trimming on the upper part of a sleeve on a woman's dress."<ref>"mancheron, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/113251. Accessed 17 April 2023.</ref> At the present, in French, a ''mancheron'' is a cap sleeve "cut directly on the bodice."<ref>{{Cite journal|date=2022-11-28|title=Manche (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manche_(v%C3%AAtement)&oldid=199054843|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Manche_(v%C3%AAtement).</ref> === Petticoat === According to the ''O.E.D.'', a petticoat is a <blockquote>skirt, as distinguished from a bodice, worn either externally or showing beneath a dress as part of the costume (often trimmed or ornamented); an outer skirt; a decorative underskirt. Frequently in ''plural'': a woman's or girl's upper skirts and underskirts collectively. Now ''archaic'' or ''historical''.<ref>“petticoat, n., sense 2.b”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1021034245></ref> </blockquote>This sense is, according to the ''O.E.D.'', "The usual sense between the 17th and 19th centuries." However, while petticoats belong in both outer- and undergarments — that is, meant to be seen or hidden, like underwear — they were always under another garment, for example, underneath an open overskirt. The primary sense seems to have shifted through the 19th century so that, by the end, petticoats were underwear and the term ''underskirt'' was used to describe what showed under an open overskirt. === Stomacher === According to the ''O.E.D.'', a stomacher is "An ornamental covering for the chest (often covered with jewels) worn by women under the lacing of the bodice,"<ref>“stomacher, n.¹, sense 3.a”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1169498955></ref> although by the end of the 19th century, the bodice did not often have visible laces. Some stomachers were so decorated that they were thought of as part of the jewelry. === Train === A train is The Length of the Train '''For the monarch [or a royal?]''' According to Debrett's,<blockquote>A peeress's coronation robe is a long-trained crimson velvet mantle, edged with miniver pure, with a miniver pure cape. The length of the train varies with the rank of the wearer: * Duchess: for rows of ermine; train to be six feet * Marchioness: three and a half rows of ermine; train to be three and three-quarters feet * Countess: three rows of ermine; train to be three and a half feet * Viscountess: two and a half rows of ermine; train to be three and a quarter feet * Baroness: two rows of ermine; train to be three feet<ref name=":2">{{Cite web|url=https://debretts.com/royal-family/dress-codes/|title=Dress Codes|website=debretts.com|language=en-US|access-date=2023-07-27}} https://debretts.com/royal-family/dress-codes/.</ref> </blockquote>The pattern on the coronet worn was also quite specific, similar but not exactly the same for peers and peeresses. Debrett's also distinguishes between coronets and tiaras, which were classified more like jewelry, which was regulated only in very general terms. Peeresses put on their coronets after the Queen or Queen Consort has been crowned. ['''peers?'''] === Foundation Garments === Unlike undergarments, Victorian women's foundation garments created the distinctive silhouette. Victorian undergarments included the chemise, the bloomers, the corset cover — articles that are not structural. The corset was an important element of the understructure of foundation garments — hoops, bustles, petticoats and so on — but it has never been the only important element. === Corset === [[File:Corset - MET 1972.209.49a, b.jpg|thumb|alt=Photograph of an old silk corset on a mannequin, showing the closure down the front, similar to a button, and channels in the fabric for the boning. It is wider at the top and bottom, creating smooth curves from the bust to the compressed waist to the hips, with a long point below the waist in front.|French 1890s corset, now in the Metropolitan Museum of Art, NYC]] The understructure of Victorian women's clothing is what makes the costumes worn by the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] so distinctly Victorian in appearance. An example of a corset that has the kind of structure often worn by fashionably dressed women in 1897 is the one at right. This corset exaggerated the shape of the women's bodies and made possible a bodice that looked and was fitted in the way that is so distinctive of the time — very controlled and smooth. And, as a structural element, this foundation garment carried the weight of all those layers and all that fabric and decoration on the gowns, trains and mantles. (The trains and mantles could be attached directly to the corset itself.) * This foundation emphasizes the waist and the bust in particular, in part because of the contrast between the very small waist and the rounded fullness of the bust and hips. * The idealized waist is defined by its small span and the sexualizing point at the center-bottom of the bodice, which directs the eye downwards. Interestingly, the pointed waistline worn by Elizabethan men has become level in the Victorian age. Highly fashionable Victorian women wearing the traditional style, however, had extremely pointed waists. * The busk (a kind of boning in the front of a corset that is less flexible than the rest) smoothed the bodice, flattened the abdomen and prevented the point on the bodice from curling up. * The sharp definition of the waist was caused by ** length of the corset (especially on the sides) ** the stiffness of the boning ** the layers of fabric ** the lacing (especially if the woman used tightlacing) ** the over-all shape, which was so much wider at the top and the bottom ** the contrast between the waist and the wider top and bottom * The late-19th-century corset was long, ending below the waist even on the sides and back. * The boning and the top edge of the late 19th-century fashion corset pushed up the bust, rounding (rather than flattening, as in earlier styles) the breasts, drawing attention to their exposed curves and creating cleavage. * The exaggerated bust was larger than the hips, whenever possible, an impression reinforced by the A-line of the skirt and the inverted Vs in the decorative trim near the waist and on the skirt. * This corset made the bodice very smooth with a very precise fit, that had no wrinkles, folds or loose drapery. The bodice was also trimmed or decorated, but the base was always a smooth bodice. More formal gowns would still have the fitted bodice and more elaborate trim made from lace, embroidery, appliqué, beading and possibly even jewels. The advantages and disadvantages of corseting and especially tight lacing were the subject of thousands of articles and opinions in the periodical press for a great part of the century, but the fetishistic and politicized tight lacing was practiced by very few women. And no single approach to corsetry was practiced by all women all the time. Most of the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 ball]] were not tightly laced, but the progressive style does not dominate either, even though all the costumes are technically historical dress. Part of what gives most of the costumes their distinctive 19th-century "look" is the more traditional corset beneath them. Even though this highly fashionable look was widely present in the historical costumes at the ball, some women's waists were obviously very small and others were hardly '''emphasized''' at all. Women's waists are never mentioned in the newspaper coverage of the ball — or, indeed, of any of the social events attended by the network at the ball — so it is only in photographs that we can see the effects of how they used their corsets. === Hoops === ''Hoops'' is a mid-19th-century term for a cage-like structure worn under a skirt to hold it away from the body. Striking for how long they lasted and '''the ways''' they evolved, hoops were the foundation undergarment for the bottom half of a woman's body, for a skirt and petticoat. Women wore this cage-like structure from the '''15th century''' through the late 19th century. The 16th-century Katherine of Aragon is credited with making it fashionable outside Spain The cage caused the silhouette of skirts to change shape over time and enabled the extreme distortions of 17th-and-18th-century panniers and the late 19th-century bustle. Early hoops circled the body symmetrically in a cone or drum shape, then were moved to the sides with panniers, then ballooned around the body like the top half of a sphere, and finally were pulled to the rear with a bustle. That is, like corsets, the distorted shapes of high fashion were made possible by hoops. High fashion demanded these shapes, which disguised women's bodies, '''especially below the waist, while corsets did their work above it'''. Besides the shape, the structure used to construct hoops evolved — from cane and wood to whalebone, then steel '''rods''' and wire. Add fabric structural stuff [[File:Pedro García de Benabarre St John Retable Detail.jpg|thumb|Pedro García de Benabarre St John Retable Detail]] [[File:Alonso Sánchez Coello 011.jpg|thumb|Alonso Sánchez Coello 011]] ==== 15th Century ==== Hoops first appeared in Spain in the 15th century and influenced European fashion for many years. Illustrations ==== 16th Century ==== [[File:Farthingale 2 (PSF).png|thumb|Farthingale 2 (PSF)]] In the 16th century, the garment we call ''hoops'' was called a farthingale. ''Vertugadin'' is a French term for ''farthingale'', a cage made of hoops supporting a skirt — "un élément essentiel de la mode Tudor en Angleterre [an essential element of Tudor fashion in England]."<ref name=":0">{{Cite journal|date=2022-03-12|title=Vertugadin|url=https://fr.wikipedia.org/w/index.php?title=Vertugadin&oldid=191825729|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Vertugadin.</ref> ''Farthingale'' is the term in English; in French, it's ''vertugadin'', and in Spanish ''vertugado''. Blanche Payne says,<blockquote>Katherine of Aragon is reputed to have introduced the Spanish farthingale (a hooped underskirt) into England early in the century. The result was to convert the columnar skirt of the fifteenth century into the cone shape of the sixteenth. ...<p> Spanish influence had introduced the hoop-supported skirt, smooth in contour, which was quite generally worn.<ref name=":11" /> (291)</blockquote>In fact, "The Spanish princess Catherine of Aragon brought the fashion to England for her marriage to Prince Arthur, eldest son of Henry VII in 1501 [La princesse espagnole Catherine d'Aragon amena la mode en Angleterre pour son mariage avec le prince Arthur, fils aîné d'Henri VII en 1501]."<ref name=":0" /> Catherine of Aragon, of course, married Henry VIII after Arthur's death. By the end of the 16th century the French and Spanish farthingales were not identical. The Spanish farthingale shaped the skirt into an A-line with a graduated series of hoops sewn to an undergarment. The French farthingale was a flattish "cartwheel" or platter of hoops worn below the waist and above the hips held the skirt out more or less horizontally. Once past the vertugadin, the skirt then fell straight to the floor, shaping the skirt into a kind of drum. The shoes show in the portraits of women wearing the French farthingale. <blockquote>"FARTHINGALE:  Renaissance (1450-1550 C.E. to Elizabethan (1550-1625 C.E.). Linen underskirt with wire supports which, when shaped, produced a variety of dome, bell, and oblong shapes."<ref name=":7" /> (105)</blockquote> ==== 17th Century ==== [[File:Panniers 1.jpg|thumb|left|alt=Photograph of the wooden and fabric skeleton of an 18th-century women's foundation garment|Panniers 1]] Generally, panniers were a kind of undergarment worn in the 17th and 18th centuries. Their design evolved during the century. Made of hoops of wood, they were "baskets" or cages worn on either side of the waist to broaden the skirts to the sides. Illustration ==== 18th Century ==== By the 18th century, the farthingale was called hoops, which were at this point were made of wood. Blanche Payne outlines the evolution of hoops, and thus the shape of the skirt, in the 18th century:<blockquote>SKIRT FASHIONS. Since skirts experienced the greatest alterations, a brief summary of the successive silhouettes should help to place individual costumes in their proper niches. Six basic forms appeared during the century, in the following order: # The bustle was a continuation of the 1690 mode. # The bell or dome shape resulted from the reintroduction of hoops; in England by 1710, in France by 1720. # The ellipse, the second phase of the hoop skirt, was achieved by broadening the support from side to side and compressing it from front to back. It had a long run of popularity, from 1740 to 1770, the extreme width being retained in court costumes. In France it persisted until the revolution, except that skirts were allowed to curve outward in [the] back again. English court costume followed this fashion well into the nineteenth century. # The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, which were used to form the three great ‘poufs’ known as the polonaise (see Fig. 452). These diversions appeared in the late sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period. # The return of the bustle in the 1780s. # The tubular form, drawn from classic art, in the 1790s.<ref name=":11" /> (411–413) </blockquote>While we think of the bustle as a 19th-century look, it can be found in the 18th century, as Payne says. Polonaise may need its own section. ==== 19th Century ==== [[File:Cutaway sketch of crinoline.gif|thumb|Cutaway sketch of crinoline]] [[File:Paris voulant englober la banlieue.JPG|thumb|Paris voulant englober la banlieue]] [[File:Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723.jpg|thumb|Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723]] In the 19th century, the hoops were made of wire and became lighter. By the 1860s, hoops made for huge round skirts. In Laura Ingalls Wilder's 1941 ''Little Town on the Prairie'', the '''16-year-old Laura wears hoops. Laura Ingalls Wilder and thus the character Laura were born in 1862, so this moment is set in 1883'''.<ref>Hill, Pamela Smith, ed. ''Pioneer Girl: The Annotated Autobiography''.</ref> She makes the comment that she wants to be in style, but that would be on the prairie and not necessarily the latest Parisian style.<blockquote>“Well,” Laura began; then she stopped and spun round and round, for the strong wind blowing against her always made the wires of her hoop skirt creep slowly upward under her skirts until they bunched around her knees. Then she must whirl around and around until the wires shook loose and spiraled down to the bottom of her skirts where they should be. “As she and Carrie hurried '''ob''' she began again. “I think it was silly, the way they dressed when Ma was a girl, don’t you? Drat this wind!” she exclaimed as the hoops began creeping upward again. “Quietly Carrie stood by while Laura whirled. “I’m glad I’m not old enough to have to wear hoops,” she said. “They’d make me dizzy.” “They are rather a nuisance,” Laura admitted. “But they are stylish, and when you’re my age you’ll want to be in style.”<ref>Wilder, Laura Ingalls. ''Little Town on the Prairie.'' Harper and Row, 1941. Pp. 272–273.</ref></blockquote>This very unusual description of the way the wind could make hoops creep and the solution. It must have been happening to other women wearing hoops at the time. == '''Traditional vs Progressive Style''' == === Progressive Style === The terms ''artistic dress'' and ''aesthetic dress'' are not synonymous and were in use at different times to refer to different groups of people in different contexts, but we recognize them as referring to a similar kind of personal style in clothing, a style we call progressive dress or the progressive style. Used in a very precise way, ''artistic dress'' is associated with the Pre-Raphaelite artists and the women in their circle beginning in the 1860s. Similarly, ''aesthetic dress'' is associated with the 1880s and 1890s and dress reform movements. In general, the progressive style is characterized by its resistance to the highly structured fashion of its day, especially corseting, aniline dyes and an extremely close fit. === Traditional Style === By the end of the century designs from the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]] (or Maison Worth) define what we think of as the traditional Victorian look, which was very stylish and expensive. Blanche Payne describes an example of the 1895 "high style" in a gown by Worth with "the idiosyncrasies of the [1890s] full blown":<blockquote>The dress is white silk with wine-red stripes. Sleeves, collars, bows, bag, hat, and hem border match the stripes. The sleeve has reached its maximum volume; the bosom full and emphasized with added lace; the waistline is elongated, pointed, and laced to the point of distress; the skirt is smooth over the hips, gradually swinging out to sweep the floor. This is the much vaunted hourglass figure.<ref name=":11">Payne, Blanche. ''History of Costume from the Ancient Egyptians to the Twentieth Century''. Harper & Row, 1965.</ref>{{rp|530}}</blockquote> The Victorian-looking gowns at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] are stylish in a way that recalls the designs of the House of Worth. The elements that make their look so Victorian are anachronisms on the costumes representing fashion of earlier eras. The women wearing these gowns preferred the standards of beauty from their own day to a more-or-less historically accurate look. The style competing at the very end of the century with the Worth look was not the historical, however, but a progressive style called at the time ''artistic'' or ''aesthetic''. William Powell Frith's 1883 painting ''A Private View at the Royal Academy, 1881'' (discussion below) pits this kind of traditional style against the progressive or artistic style. === The Styles === [[File:Frith A Private View.jpg|thumb|William Powell Frith, ''A Private View at the Royal Academy, 1881'']] We typically think of the late-Victorian silhouette as universal but, in the periods in which corsets dominated women's dress, not all women wore corsets and not all corsets were the same, as William Powell Frith's 1883 ''A Private View at the Royal Academy, 1881'' (right) illustrates. Frith is clear in his memoir that this painting — "recording for posterity the aesthetic craze as regards dress" — deliberately contrasts what he calls the "folly" of the Artistic Dress movement and the look of the traditional corseted waist.<ref>Frith, William Powell. ''My Autobiography and Reminiscences''. 1887.</ref> Frith considered the Artistic Movement and Artistic Dress "ephemeral," but its rejection of corsetry looks far more consequential to us in hindsight than it did in the 19th century. As Frith sees it, his painting critiques the "craze" associated with the women in this set of identifiable portraits who are not corseted, but his commitment to realism shows us a spectrum, a range, of conservatism and if not political then at least stylistic progressivism among the women. The progressives, oddly, are the women wearing artistic (that is, somewhat historical) dress, because they’re not corseted. It is a misreading to see the presentation of the women’s fashion as a simple opposition. Constance, Countess of Lonsdale — situated at the center of this painting with Frederick Leighton, president of the Royal Academy of Art — is the most conservatively dressed of the women depicted, with her narrow sleeves, tight waist and almost perfectly smooth bodice, which tells us that her corset has eyelets so that it can be laced precisely and tightly, and it has stays (or "bones") to prevent wrinkles or natural folds in the overclothing. Lillie Langtry, in the white dress, with her stylish narrow sleeves, does not have such a tightly bound waist or smooth bodice, suggesting she may not be corseted at all, as we know she sometimes was not.['''citation'''] Jenny Trip, a painter’s model, is the woman in the green dress in the aesthetic group being inspected by Anthony Trollope, who may be taking notes. She looks like she is not wearing a corset. Both Langtry and Trip are toward the middle of this spectrum: neither is dressed in the more extreme artistic dress of, say, the two figures between Trip and Trollope. A lot has been written about the late-Victorian attraction to historical dress, especially in the context of fancy-dress balls and the Gothic revival in social events as well as art and music. Part of the appeal has to have been the way those costumes could just be beautiful clothing beautifully made. Historical dress provided an opportunity for some elite women to wear less-structured but still beautiful and influential clothing. ['''Calvert'''<ref>Calvert, Robyne Erica. ''Fashioning the Artist: Artistic Dress in Victorian Britain 1848-1900''. Ph.D. thesis, University of Glasgow, 2012. <nowiki>https://theses.gla.ac.uk/3279/</nowiki></ref>] The standards for beauty, then, with historical dress were Victorian, with the added benefit of possibly less structure. So, at the Duchess of Devonshire's ball, "while some attendees tried to hew closely to historical precedent, many rendered their historical or mythological personage in the sartorial vocabulary they knew best. The [photographs of people in their costumes at the ball offer] a glimpse into how Victorians understood history, not a glimpse into the costume of an authentic historical past."<ref>Mitchell, Rebecca N. "The Victorian Fancy Dress Ball, 1870–1900." ''Fashion Theory'' 2017 (21: 3): 291–315. DOI: 10.1080/1362704X.2016.1172817.</ref> (294) * historical dress: beautiful clothing. * the range at the ball, from Minnie Paget to Gwladys * "In light of such efforts, the ball remains to this day one of the best documented outings of the period, and a quick glance at the album shows that ..." Women had more choices about their waists than the simple opposition between no corset and tightlacing can accommodate. The range of choices is illustrated in Frith's painting, with a woman locating herself on it at a particular moment for particular reasons. Much analysis of 19th-century corsetry focuses on its sexualizing effects — corsets dominated Victorian photographic pornography ['''citations'''] and at the same time, the absence of a corset was sexual because it suggested nudity.['''citations'''] A great deal of analysis of 19th-century corsetry, on the other hand, assumes that women wore corsets for the male gaze ['''citations'''] or that they tightened their waists to compete with other women.['''citations'''] But as we can see in Frith's painting, the sexualizing effect was not universal or sweeping, and these analyses do not account for the choices women had in which corset to wear or how tightly to lace it. Especially given the way that some photographic portraits were mechanically altered to make the waist appear smaller, the size of a woman's waist had to do with how she was presenting herself to the world. That is, the fact that women made choices about the size of or emphasis on their waists suggests that they had agency that needs to be taken into account. As they navigated the complex social world, women's fashion choices had meaning. Society or political hostesses had agency not only in their clothing but generally in that complex social world. They had roles managing social events of the upper classes, especially of the upper aristocracy and oligarchy, like the Duchess of Devonshire's ball. Their class and rank, then, were essential to their agency, including to some degree their freedom to choose what kind of corset to wear and how to wear it. Also, by the end of the century lots of different kinds of corsets were available for lots of different purposes. Special corsets existed for pregnancy, sports (like tennis, bicycling, horseback riding, golf, fencing, archery, stalking and hunting), theatre and dance and, of course, for these women corsets could be made to support the special dress worn over it. Women's choices in how they presented themselves to the world included more than just their foundation garments, of course. "Every cap, bow, streamer, ruffle, fringe, bustle, glove," that is, the trim and decorations on their garments, their jewelry and accessories — which Davidoff calls "elaborations"<ref name=":1">Davidoff, Leonore. ''The Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig 1973).</ref>{{rp|93}} — pointed to a host of status categories, like class, rank, wealth, age, marital status, engagement with the empire, how sexual they wanted to seem, political alignment and purpose at the social event. For example, when women were being presented to the monarch, they were expected to wear three ostrich plumes, often called the [[Social Victorians/Terminology#Prince of Wales's Feathers or White Plumes|Prince of Wales's feathers]]. Like all fashions, the corset, which was quite long-lasting in all its various forms, eventually went out of style. Of the many factors that might have influenced its demise, perhaps most important was the women's movement, in which women's rights, freedom, employment and access to their own money and children were less slogan-worthy but at least as essential as votes for women. The activities of the animal-rights movements drew attention not only to the profligate use of the bodies and feathers of birds but also to the looming extinction of the baleen whale, which made whale bone scarce and expensive. Perhaps the century's debates over corseting and especially tightlacing were relevant to some decisions not to be corseted. And, of course, perhaps no other reason is required than that the nature of fashion is to change. == Cinque Cento == According to the ''Oxford English Dictionary'', ''Cinque Cento'' is a shortening of ''mil cinque cento'', or 1500.<ref>"cinquecento, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/33143. Accessed 7 February 2023.</ref> The term, then would refer, perhaps informally, to the sixteenth century. == Crevé == ''Creve'', without the accent, is an old word in English (c. 1450) for burst or split.<ref>"creve, v." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44339. Accessed 8 February 2023.</ref> ['''With the acute accent, it looks like a past participle in French.'''] == Elastic == Elastic had been invented and was in use by the end of the 19th century. For the sense of "Elastic cord or string, usually woven with india-rubber,"<ref name=":6">“elastic, adj. & n.”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1199670313>.</ref> the ''Oxford English Dictionary'' has usage examples beginning in 1847. The example for 1886 is vivid: "The thorough-going prim man will always place a circle of elastic round his hair previous to putting on his college cap."<ref name=":6" /> == Elaborations == In her 1973 ''The Best Circles: Society, Etiquette and the Season'', Leonore Davidoff notes that women’s status was indicated by dress and especially ornament: “Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration,” she says, “symbolised some status category for the female wearer.”<ref name=":1" />{{rp|93}} Looking at these elaborations as meaningful rather than dismissing them as failed attempts at "historical accuracy" reveals a great deal about the individual women who wore or carried them — and about the society women and political hostesses in their roles as managers of the social world. In her review of ''The House of Worth: Portrait of an Archive'', Mary Frances Gormally says,<blockquote>In a socially regulated year, garments custom made with a Worth label provided women with total reassurance, whatever the season, time of day or occasion, setting them apart as members of the “Best Circles” dressed in luxurious, fashionable and always appropriate attire (Davidoff 1973). The woman with a Worth wardrobe was a woman of elegance, lineage, status, extreme wealth and faultless taste.<ref>Gormally, Mary Frances. Review essay of ''The House of Worth: Portrait of an Archive'', by Amy de la Haye and Valerie D. Mendes (V&A Publishing, 2014). ''Fashion Theory'' 2017 (21, 1): 109–126. DOI: 10.1080/1362704X.2016.1179400.</ref> (117)</blockquote> === Aglet, Aiglet === Historically, an aglet is a "point or metal piece that capped a string [or ribbon] used to attach two pieces of the garment together, i.e., sleeve and bodice."<ref name=":7" />{{rp|4}} Although they were decorative, they were not always visible on the outside of the clothing. They were often stuffed inside the layers at the waist (for example, attaching the bodice to a skirt or breeches). === Frou-frou === In French, ''frou-frou'' or, spelled as ''froufrou'', is the sound of the rustling of silk or sometimes of fabrics in general.<ref>{{Cite journal|date=2023-07-25|title=frou-frou|url=https://fr.wiktionary.org/w/index.php?title=frou-frou&oldid=32508509|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/frou-frou.</ref> The first use the French ''Wiktionnaire'' lists is Honoré Balzac, ''La Cousine Bette'', 1846.<ref>{{Cite journal|date=2023-06-03|title=froufrou|url=https://fr.wiktionary.org/w/index.php?title=froufrou&oldid=32330124|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/froufrou.</ref> ''Frou-frou'' is a term clothing historians use to describe decorative additions to an article of clothing; often the term has a slight negative connotation, suggesting that the additions are superficial. === Pouf, Puff, Poof === According to the French ''Wikipédia'', a pouf was, beginning in 1744, a "kind of women's hairstyle":<blockquote>The hairstyle in question, known as the “pouf”, had launched the reputation of the enterprising Rose Bertin, owner of the Grand Mogol, a very prominent fashion accessories boutique on Rue Saint-Honoré in Paris in 1774. Created in collaboration with the famous hairdresser, Monsieur Léonard, the pouf was built on a scaffolding of wire, fabric, gauze, horsehair, fake hair, and the client's own hair held up in an almost vertical position. — (Marie-Antoinette, ''Queen of Fashion'', translated from the American by Sylvie Lévy, in ''The Rules of the Game'', n° 40, 2009)</blockquote>''Puff'' and ''poof'' are used to describe clothing. === Shirring === ''Shirring'' is the gathering of fabric to make poufs or puffs. The 19th century is known for its use of this decorative technique. Even men's clothing had shirring: at the shoulder seam. === Sequins === Sequins, paillettes, spangles Sequins — or paillettes — are "small, scalelike glittering disks."<ref name=":7" />(216) The French ''Wiktionnaire'' defines ''paillette'' as "Lamelle de métal, brillante, mince, percée au milieu, ordinairement ronde, et qu’on applique sur une étoffe pour l’orner [A strip of metal, shiny, thin, pierced in the middle, usually round, and which is applied to a fabric in order to decorate it.]"<ref name=":8">{{Cite journal|date=2024-03-18|title=paillette|url=https://fr.wiktionary.org/w/index.php?title=paillette&oldid=33809572|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/paillette.</ref> According to the ''OED'', the use of ''sequin'' as a decorative device for clothing (as opposed to gold coins minted and used for international trade) goes back to the 1850s.<ref>“Sequin, N.” ''Oxford English Dictionary'', Oxford UP, September 2023, https://doi.org/10.1093/OED/4074851670.</ref> The first instance of ''spangle'' as "A small round thin piece of glittering metal (usually brass) with a hole in the centre to pass a thread through, used for the decoration of textile fabrics and other materials of various sorts" is from c. 1420.<ref>“Spangle, N. (1).” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/4727197141.</ref> The first use of ''paillette'' listed in the French ''Wiktionnaire'' is in Jules Verne in 1873 to describe colored spots on icy walls.<ref name=":8" /> Currently many distinguish between sequins (which are smaller) and paillettes (which are larger). Before the 20th century, sequins were metal discs or foil leaves, and so of course if they were silver or copper, they tarnished. It is not until well into the 20th century that plastics were invented and used for sequins. === Trim and Lace === ''A History of Feminine Fashion'', published sometime before 1927 and probably commissioned by [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|the Maison Worth]], describes Charles Frederick Worth's contributions to the development of embroidery and [[Social Victorians/Terminology#Passementerie|passementerie]] (trim) from about the middle of the 19th century:<blockquote>For it must be remembered that one of M. Worth's most important and lasting contributions to the prosperity of those who cater for women's needs, as well as to the variety and elegance of his clients' garments, was his insistence on new fabrics, new trimmings, new materials of every description. In his endeavours to restore in Paris the splendours of the days of La Pompadour, and of Marie Antoinette, he found himself confronted at the outset with a grave difficulty, which would have proved unsurmountable to a man of less energy, resource and initiative. The magnificent materials of those days were no longer to be had! The Revolution had destroyed the market for beautiful materials of this, type, and the Restoration and regime of Louis Philippe had left a dour aspect in the City of Light. ... On parallel lines [to his development of better [[Social Victorians/Terminology#Satin|satin]]], [Worth] stimulated also the manufacture of embroidery and ''passementerie''. It was he who first started the manufacture of laces copied from the designs of the real old laces. He was the / first dressmaker to use fur in the trimming of light materials — but he employed only the richer furs, such as sable and ermine, and had no use whatever for the inferior varieties of skins.<ref name=":9" />{{rp|6–7}}</blockquote> ==== Gold and Silver Fabric and Lace ==== The ''Encyclopaedia Britannica'' (9th edition) has an article on gold and silver fabric, threads and lace attached to the article on gold. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>GOLD AND SILVER LACE. Under this heading a general account may be given of the use of the precious metals in textiles of all descriptions into which they enter. That these metals were used largely in the sumptuous textiles of the earliest periods of civilization there is abundant testimony; and to this day, in the Oriental centres whence a knowledge and the use of fabrics inwoven, ornamented, and embroidered with gold and silver first spread, the passion for such brilliant and costly textiles is still most strongly and generally prevalent. The earliest mention of the use of gold in a woven fabric occurs in the description of the ephod made for Aaron (Exod. xxxix. 2, 3) — "And he made the ephod of gold, blue, and purple, and scarlet, and fine twined linen. And they did beat the gold into thin plates, and cut it into wires (strips), to work it in the blue, and in the purple, and in the scarlet, and in the fine linen, with cunning work." In both the ''Iliad'' and the ''Odyssey'' distinct allusion is frequently made to inwoven and embroidered golden textiles. Many circumstances point to the conclusion that the art of weaving and embroidering with gold and silver originated in India, where it is still principally prosecuted, and that from one great city to another the practice travelled westward, — Babylon, Tarsus, Baghdad, Damascus, the islands of Cyprus and Sicily, Con- / stantinople and Venice, all in the process of time becoming famous centres of these much prized manufactures. Alexander the Great found Indian kings and princes arrayed in robes of gold and purple; and the Persian monarch Darius, we are told, wore a war mantle of cloth of gold, on which were figured two golden hawks as if pecking at each other. There is reason, according to Josephus, to believe that the “royal apparel" worn by Herod on the day of his death (Acts xii. 21) was a tissue of silver. Agrippina, the wife of the emperor Claudius, had a robe woven entirely of gold, and from that period downwards royal personages and high ecclesiastical dignitaries used cloth and tissues of gold and silver for their state and ceremonial robes, as well as for costly hangings and decorations. In England, at different periods, various names were applied to cloths of gold, as ciclatoun, tartarium, naques or nac, baudekiu or baldachin, Cyprus damask, and twssewys or tissue. The thin flimsy paper known as tissue paper, is so called because it originally was placed between the folds of gold "tissue" to prevent the contiguous surfaces from fraying each other. At what time the drawing of gold wire for the preparation of these textiles was first practised is not accurately known. The art was probably introduced and applied in different localities at widely different dates, but down till mediaeval times the method graphically described in the Pentateuch continued to be practised with both gold and silver. Fabrics woven with gold and silver continue to be used on the largest scale to this day in India; and there the preparation of the varieties of wire, and the working of the various forms of lace, brocade, and embroidery, is at once an important and peculiar art. The basis of all modern fabrics of this kind is wire, the "gold wire" of the manufacturer being in all cases silver gilt wire, and silver wire being, of course, composed of pure silver. In India the wire is drawn by means of simple draw-plates, with rude and simple appliances, from rounded bars of silver, or gold-plated silver, as the case may be. The wire is flattened into the strip or ribbon-like form it generally assumes by passing it, fourteen or fifteen strands simultaneously, over a fine, smooth, round-topped anvil, and beating it as it passes with a heavy hammer having a slightly convex surface. From wire so flattened there is made in India soniri, a tissue or cloth of gold, the web or warp being composed entirely of golden strips, and ruperi, a similar tissue of silver. Gold lace is also made on a warp of thick yellow silk with a weft of flat wire, and in the case of ribbons the warp or web is composed of the metal. The flattened wires are twisted around orange (in the case of silver, white) coloured silk thread, so as completely to cover the thread and present the appearance of a continuous wire; and in this form it is chiefly employed for weaving into the rich brocades known as kincobs or kinkhábs. Wires flattened, or partially flattened, are also twisted into exceedingly fine spirals, and in this form they are the basis of numerous ornamental applications. Such spirals drawn out till they present a waved appearance, and in that state flattened, are much used for rich heavy embroideries termed karchobs. Spangles for embroideries, &c., are made from spirals of comparatively stout wire, by cutting them down ring by ring, laying each C-like ring on an anvil, and by a smart blow with a hammer flattening it out into a thin round disk with a slit extending from the centre to one edge. Fine spirals are also used for general embroidery purposes. The demand for various kinds of loom-woven and embroidered gold and silver work in India is immense; and the variety of textiles so ornamented is also very great. "Gold and silver," says Dr Birdwood in his ''Handbook to the British-Indian Section, Paris Exhibition'', 1878, "are worked into the decoration of all the more costly loom-made garments and Indian piece goods, either on the borders only, or in stripes throughout, or in diapered figures. The gold-bordered loom embroideries are made chiefly at Sattara, and the gold or silver striped at Tanjore; the gold figured ''mashrus'' at Tanjore, Trichinopoly, and Hyderabad in the Deccau; and the highly ornamented gold-figured silks and gold and silver tissues principally at Ahmedabad, Benares, Murshedabad, and Trichinopoly." Among the Western communities the demand for gold and silver lace and embroideries arises chiefly in connexion with naval and military uniforms, court costumes, public and private liveries, ecclesiastical robes and draperies, theatrical dresses, and the badges and insignia of various orders. To a limited extent there is a trade in gold wire and lace to India and China. The metallic basis of the various fabrics is wire round and flattened, the wire being of three kinds — 1st, gold wire, which is invariably silver gilt wire; 2d, copper gilt wire, used for common liveries and theatrical purposes; and 3d, silver wire. These wires are drawn by the ordinary processes, and the flattening, when done, is accomplished by passing the wire between a pair of revolving rollers of fine polished steel. The various qualities of wire are prepared and used in precisely the same way as in India, — round wire, flat wire, thread made of flat gold wire twisted round orange-coloured silk or cotton, known in the trade as "orris," fine spirals and spangles, all being in use in the West as in the East. The lace is woven in the same manner as ribbons, and there are very numerous varieties in richness, pattern, and quality. Cloth of gold, and brocades rich in gold and silver, are woven for ecclesiastical vestments and draperies. The proportions of gold and silver in the gold thread for the lace trade varies, but in all cases the proportion of gold is exceedingly small. An ordinary gold lace wire is drawn from a bar containing 90 parts of silver and 7 of copper, coated with 3 parts of gold. On an average each ounce troy of a bar so plated is drawn into 1500 yards of wire; and therefore about 16 grains of gold cover a mile of wire. It is estimated that about 250,000 ounces of gold wire are made annually in Great Britain, of which about 20 per cent, is used for the headings of calico, muslin, &c., and the remainder is worked up in the gold lace trade.<ref>William Chandler Roberts-Austen and H. Bauerman [W.C.R. — H.B.]. "Gold and Silver Lace." In "Gold." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. 10 (X). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2010%20%28G-GOT%29%20193592738.23/page/753/mode/1up (accessed January 2023): 753, Col. 2c – 754, Cols. 1a–b – 2a–b.</ref></blockquote> ==== Honiton Lace ==== Kate Stradsin says,<blockquote>Honiton lace was the finest English equivalent of Brussels bobbin lace and was constructed in small ‘sprigs, in the cottages of lacemakers[.'] These sprigs were then joined together and bleached to form the large white flounces that were so sought after in the mid-nineteenth century.<ref>Strasdin, Kate. "Rediscovering Queen Alexandra’s Wardrobe: The Challenges and Rewards of Object-Based Research." ''The Court Historian'' 24.2 (2019): 181-196. Rpt http://repository.falmouth.ac.uk/3762/15/Rediscovering%20Queen%20Alexandra%27s%20Wardrobe.pdf: 13, and (for the little quotation) n. 37, which reads "Margaret Tomlinson, ''Three Generations in the Honiton Lace Trade: A Family History'', self-published, 1983."</ref></blockquote> [[File:Strook in Alençon naaldkant, 1750-1775.jpg|thumb|alt=A long piece of complex white lace with garlands, flowers and bows|Point d'Alençon lace, 1750-1775]] ==== Passementerie ==== ''Passementerie'' is the French term for trim on clothing or furniture. The 19th century (especially during the First and Second Empire) was a time of great "''exubérance''" in passementerie in French design, including the development and widespread use of the Jacquard loom.<ref>{{Cite journal|date=2023-06-10|title=Passementerie|url=https://fr.wikipedia.org/w/index.php?title=Passementerie&oldid=205068926|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Passementerie.</ref> ==== Point d'Alençon Lace ==== A lace made by hand using a number of complex steps and layers. The lacemakers build the point d'Alençon design on some kind of mesh and sometimes leave some of the mesh in as part of the lace and perhaps to provide structure. Elizabeth Lewandowski defines point d'Alençon lace and Alençon lace separately. Point lace is needlepoint lace,<ref name=":7">Lewandowski, Elizabeth J. ''The Complete Costume Dictionary''. Scarecrow Press, 2011.</ref>{{rp|233}} so Alençon point is "a two thread [needlepoint] lace."<ref name=":7" />{{rp|7}} Alençon lace has a "floral design on [a] fine net ground [and is] referred to as [the] queen of French handmade needlepoint laces. The original handmade Alençon was a fine needlepoint lace made of linen thread."<ref name=":7" />{{rp|7}} The sample of point d'Alençon lace (right), from 1750–1775, shows the linen mesh that the lace was constructed on.<ref>{{Cite web|url=http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689|title=MoMu - Open Fashion|website=openfashion.momu.be|access-date=2024-02-26}} ModeMuseum Antwerpen. http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689.</ref> The consistency in this sample suggests it may have been made by machine. == Fabric == === Brocatelle === Brocatelle is a kind of brocade, more simple than most brocades because it uses fewer warp and weft threads and fewer colors to form the design. The article in the French ''Wikipédia'' defines it like this:<blockquote>La '''brocatelle''' est un type de tissu datant du <abbr>xvi^e</abbr> siècle qui comporte deux chaînes et deux trames, au minimum. Il est composé pour que le dessin ressorte avec un relief prononcé, grâce à la chaîne sur un fond en sergé. Les brocatelles les plus anciennes sont toujours fabriquées avec une des trames en lin.<ref>{{Cite journal|date=2023-06-01|title=Brocatelle|url=https://fr.wikipedia.org/w/index.php?title=Brocatelle&oldid=204796410|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Brocatelle.</ref></blockquote>Which translates to this:<blockquote>Brocatelle is a type of fabric dating from the 16th century that has two warps and two wefts, at a minimum. It is composed so that the design stands out with a pronounced relief, thanks to the weft threads on a twill background. The oldest brocades were always made with one of the wefts being linen.</blockquote>The ''Oxford English Dictionary'' says, brocatelle is an "imitation of brocade, usually made of silk or wool, used for tapestry, upholstery, etc., now also for dresses. Both the nature and the use of the stuff have changed" between the late 17th century and 1888, the last time this definition was revised.<ref>"brocatelle, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/23550. Accessed 4 July 2023.</ref> === Broché === === Ciselé === === Crépe de Chine === The ''Oxford English Dictionary'' distinguishes the use of ''crêpe'' (using a circumflex rather than an acute accent over the first ''e'') from ''crape'' in textiles, saying ''crêpe'' is "often borrowed [from the French] as a term for all crapy fabrics other than ordinary black mourning crape,"<ref>"crêpe, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44242. Accessed 10 February 2023.</ref> with usage examples ranging from 1797 to the mid 20th century. Crêpe de chine, it says is "a white or other coloured crape made of raw silk." === Épinglé Velvet === Often spelled ''épingle'' rather than ''épinglé'', this term appears to have been used for a fabric made of wool, or at least wool along with linen or cotton, that was heavier and stiffer than silk velvet. It was associated with outer garments and men's clothing. Nowadays, épinglé velvet is an upholstery fabric in which the pile is cut into designs and patterns, and the portrait of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary, Duchess of Hamilton]] shows a mantle described as épinglé velvet that does seem to be a velvet with a woven pattern perhaps cut into the pile. === Lace === While lace also functioned sometimes as fabric — at the décolletage, for example, on the stomacher or as a veil — here we organize it as a [[Social Victorians/Terminology#Trim and Lace|part of the elaboration of clothing]]. === Liberty Fabrics === === Lisse === According to the ''Oxford English Dictionary'', the term ''lisse'' as a "kind of silk gauze" was used in the 19th-century UK and US.<ref>"lisse, n.1." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/108978. Accessed 4 July 2023.</ref> === Satin === The pre-1927 ''History of Feminine Fashion'', probably commissioned by Charles Frederick Worth's sons, describes Worth's "insistence on new fabrics, new trimmings, new materials of every description" at the beginning of his career in the mid 19th century:<blockquote>When Worth first entered the business of dressmaking, the only materials of the richer sort used for woman's dress were velvet, faille, and watered silk. Satin, for example, was never used. M. Worth desired to use satin very extensively in the gowns he designed, but he was not satisfied with what could be had at the time; he wanted something very much richer than was produced by the mills at Lyons. That his requirements entailed the reconstruction of mills mattered little — the mills were reconstructed under his directions, and the Lyons looms turned out a richer satin than ever, and the manufacturers prospered accordingly.<ref name=":9">[Worth, House of.] {{Cite book|url=http://archive.org/details/AHistoryOfFeminineFashion|title=A History Of Feminine Fashion (1800s to 1920s)}} Before 1927. [Likely commissioned by Worth. Link is to Archive.org; info from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Worth_Biarritz_salon.jpg.]</ref>{{rp|6 in printed, 26 in digital book}}</blockquote> === Selesia === According to the ''Oxford English Dictionary'', ''silesia'' is "A fine linen or cotton fabric originally manufactured in Silesia in what is now Germany (''Schlesien'').<ref>"Silesia, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/179664. Accessed 9 February 2023.</ref> It may have been used as a lining — for pockets, for example — in garments made of more luxurious or more expensive cloth. The word ''sleazy'' — "Of textile fabrics or materials: Thin or flimsy in texture; having little substance or body."<ref>"sleazy, adj." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/181563. Accessed 9 February 2023.</ref> — may be related. === Shot Fabric === According to the ''Oxford English Dictionary'', "Of a textile fabric: Woven with warp-threads of one colour and weft-threads of another, so that the fabric (usually silk) changes in tint when viewed from different points."<ref>“Shot, ''Adj.''”  ''Oxford English Dictionary'', Oxford UP,  July 2023, https://doi.org/10.1093/OED/2977164390.</ref> A shot fabric might also be made of silk and cotton fibers. === Tissue === A lightly woven fabric like gauze or chiffon. The light weave can make the fabric translucent and make pleating and gathering flatter and less bulky. Tissue can be woven to be shot, sheer, stiff or soft. Historically, the term in English was used for a "rich kind of cloth, often interwoven with gold or silver" or "various rich or fine fabrics of delicate or gauzy texture."<ref>“Tissue, N.” ''Oxford English Dictionary'', Oxford UP, March 2024, https://doi.org/10.1093/OED/5896731814.</ref> == Fan == The ''Encyclopaedia Britannica'' (9th edition) has an article on the fan. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>FAN (Latin, ''vannus''; French, ''éventail''), a light implement used for giving motion to the air. ''Ventilabrum'' and ''flabellum'' are names under which ecclesiastical fans are mentioned in old inventories. Fans for cooling the face have been in use in hot climates from remote ages. A bas-relief in the British Museum represents Sennacherib with female figures carrying feather fans. They were attributes of royalty along with horse-hair fly-flappers and umbrellas. Examples may be seen in plates of the Egyptian sculptures at Thebes and other places, and also in the ruins of Persepolis. In the museum of Boulak, near Cairo, a wooden fan handle showing holes for feathers is still preserved. It is from the tomb of Amen-hotep, of the 18th dynasty, 17th century <small>B</small>.<small>C</small>. In India fans were also attributes of men in authority, and sometimes sacred emblems. A heartshaped fan, with an ivory handle, of unknown age, and held in great veneration by the Hindus, was given to the prince of Wales. Large punkahs or screens, moved by a servant who does nothing else, are in common use by Europeans in India at this day. Fans were used in the early Middle Ages to keep flies from the sacred elements during the celebrations of the Christian mysteries. Sometimes they were round, with bells attached — of silver, or silver gilt. Notices of such fans in the ancient records of St Paul’s, London, Salisbury cathedral, and many other churches, exist still. For these purposes they are no longer used in the Western church, though they are retained in some Oriental rites. The large feather fans, however, are still carried in the state processions of the supreme pontiff in Rome, though not used during the celebration of the mass. The fan of Queen Theodolinda (7th century) is still preserved in the treasury of the cathedral of Monza. Fans made part of the bridal outfit, or ''mundus muliebris'', of ancient Roman ladies. Folding fans had their origin in Japan, and were imported thence to China. They were in the shape still used—a segment of a circle of paper pasted on a light radiating frame-work of bamboo, and variously decorated, some in colours, others of white paper on which verses or sentences are written. It is a compliment in China to invite a friend or distinguished guest to write some sentiment on your fan as a memento of any special occasion, and this practice has continued. A fan that has some celebrity in France was presented by the Chinese ambassador to the Comtesse de Clauzel at the coronation of Napoleon I. in 1804. When a site was given in 1635, on an artificial island, for the settlement of Portuguese merchants in Nippo in Japan, the space was laid out in the form of a fan as emblematic of an object agreeable for general use. Men and women of every rank both in China and Japan carry fans, even artisans using them with one hand while working with the other. In China they are often made of carved ivory, the sticks being plates very thin and sometimes carved on both sides, the intervals between the carved parts pierced with astonishing delicacy, and the plates held together by a ribbon. The Japanese make the two outer guards of the stick, which cover the others, occasionally of beaten iron, extremely thin and light, damascened with gold and other metals. Fans were used by Portuguese ladies in the 14th century, and were well known in England before the close of the reign of Richard II. In France the inventory of Charles V. at the end of the 14th century mentions a folding ivory fan. They were brought into general use in that country by Catherine de’ Medici, probably from Italy, then in advance of other countries in all matters of personal luxury. The court ladies of Henry VIII.’s reign in England were used to handling fans, A lady in the Dance of Death by Holbein holds a fan. Queen Elizabeth is painted with a round leather fan in her portrait at Gorhambury; and as many as twenty-seven are enumerated in her inventory (1606). Coryat, an English traveller, in 1608 describes them as common in Italy. They also became of general use from that time in Spain. In Italy, France, and Spain fans had special conventional uses, and various actions in handling them grew into a code of signals, by which ladies were supposed to convey hints or signals to admirers or to rivals in society. A paper in the ''Spectator'' humorously proposes to establish a regular drill for these purposes. The chief seat of the European manufacture of fans during the 17th century was Paris, where the sticks or frames, whether of wood or ivory, were made, and the decorations painted on mounts of very carefully prepared vellum (called latterly ''chicken skin'', but not correctly), — a material stronger and tougher than paper, which breaks at the folds. Paris makers exported fans unpainted to Madrid and other Spanish cities, where they were decorated by native artists. Many were exported complete; of old fans called Spanish a great number were in fact made in France. Louis XIV. issued edicts at various times to regulate the manufacture. Besides fans mounted with parchment, Dutch fans of ivory were imported into Paris, and decorated by the heraldic painters in the process called “Vernis Martin,” after a famous carriage painter and inventor of colourless lac varnish. Fans of this kind belonging to the Queen and to the late baroness de Rothschild were exhibited in 1870 at Kensington. A fan of the date of 1660, representing sacred subjects, is attributed to Philippe de Champagne, another to Peter Oliver in England in the / 17th century. Cano de Arevalo, a Spanish painter of the 17th century devoted himself to fan painting. Some harsh expressions of Queen Christina to the young ladies of the French court are said to have caused an increased ostentation in the splendour of their fans, which were set with jewels and mounted in gold. Rosalba Carriera was the name of a fan painter of celebrity in the 17th century. Lebrun and Romanelli were much employed during the same period. Klingstet, a Dutch artist, enjoyed a considerable reputation for his fans from the latter part of the 17th and the first thirty years of the 18th century. The revocation of the edict of Nantes drove many fan-makers out of France to Holland and England. The trade in England was well established under the Stuart sovereigns. Petitions were addressed by the fan-makers to Charles II. against the importation of fans from India, and a duty was levied upon such fans in consequence. This importation of Indian fans, according to Savary, extended also to France. During the reign of Louis XV. carved Indian and China fans displaced to some extent those formerly imported from Italy, which had been painted on swanskin parchment prepared with various perfumes. During the 18th century all the luxurious ornamentation of the day was bestowed on fans as far as they could display it. The sticks were made of mother-of-pearl or ivory, carved with extraordinary skill in France, Italy, England, and other countries. They were painted from designs of Boucher, Watteau, Lancret, and other "genre" painters, Hébert, Rau, Chevalier, Jean Boquet, Mad. Verité, are known as fan painters. These fashions were followed in most countries of Europe, with certain national differences. Taffeta and silk, as well as fine parchment, were used for the mounts. Little circles of glass were let into the stick to be looked through, and small telescopic glasses were sometimes contrived at the pivot of the stick. They were occasionally mounted with the finest point lace. An interesting fan (belonging to Madame de Thiac in France), the work of Le Flamand, was presented by the municipality of Dieppe to Marie Antoinette on the birth of her son the dauphin. From the time of the Revolution the old luxury expended on fans died out. Fine examples ceased to be exported to England and other countries. The painting on them represented scenes or personages connected with political events. At a later period fan mounts were often prints coloured by hand. The events of the day mark the date of many examples found in modern collections. Amongst the fanmakers of the present time the names of Alexandre, Duvelleroy, Fayet, Vanier, may be mentioned as well known in Paris. The sticks are chiefly made in the department of Oise, at Le Déluge, Crèvecœur, Méry, Ste Geneviève, and other villages, where whole families are engaged in preparing them; ivory sticks are carved at Dieppe. Water-colour painters of distinction often design and paint the mounts, the best designs being figure subjects. A great impulse has been given to the manufacture and painting of fans in England since the exhibition which took place at South Kensington in 1870. Other exhibitions have since been held, and competitive prizes offered, one of which was gained by the Princess Louise. Modern collections of fans take their date from the emigration of many noble families from France at the time of the Revolution. Such objects were given as souvenirs and occasionally sold by families in straitened circumstances. A large number of fans of all sorts, principally those of the 18th century, French, English, German, Italian Spanish, &c., have been lately bequeathed to the South Kensington Museum. Regarding the different parts of folding fans it may be well to state that the sticks are called in French ''brins'', the two outer guards ''panaches'', and the mount ''feuille''.<ref>J. H. Pollen [J.H.P.]. "Fan." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. '''10''' ('''X'''). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%209%20%28FAL-FYZ%29%20193323016.23/page/26/mode/2up (accessed January 2023): 27, Col. 1b – 28, Col. 1c.</ref></blockquote> == Fancy-dress Ball == Fancy-dress (or costume) balls were popular and frequent in the U.K. and France as well as the rest of Europe during the 19th century. The themes and styles of the fancy-dress balls influenced those that followed. At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the guests came dressed in costume from times before 1820, as instructed on '''the invitation''', but their clothing was much more about late-Victorian standards of beauty and fashion than the standards of whatever time period the portraits they were copying or basing their costumes on. ''The Queen'' published dress and fashion information and advice under the byline of Ardern Holt, who regularly answered questions from readers about fashion as well as about fancy dress. (More about Ardern Holt, which is almost certainly a pseudonym, can be found on the [[Social Victorians/People/Working in Publishing#Journalists|People Working in Publishing]] page.) Holt also ran wrote entire articles with suggestions for what might make an appealing fancy-dress costume as well as pointing readers away from costumes that had been worn too frequently. The suggestions for costumes are based on familiar types or portraits available to readers, similar to Holt's books on fancy dress, which ran through a number of editions in the 1880s and 1890s. Fancy-dress questions sometimes asked for details about costumes worn in theatrical or operatic productions, which Holt provides. In November 1897, Holt refers to the Duchess of Devonshire's 2 July ball: "Since the famous fancy ball, given at Devonshire House during this year, historical fancy dresses have assumed a prominence that they had not hitherto known."<ref>Holt, Ardern. "Fancy Dress a la Mode." The ''Queen'' 27 November 1897, Saturday: 94 [of 145 in BNA; print p. 1026], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971127/459/0094.</ref> Holt goes on to provide a number of ideas for costumes for historical fancy dress, as always with a strong leaning toward Victorian standards of beauty and style and away from any concern for historical accuracy. Ardern Holt published books on fancy dress as well as writing for the ''Queen'' and other periodicals, but not all of them were about fancy dress. # ''Gentlemen's Fancy Dress: How to Choose It''. Wyman & Sons, 1882. (''Google Books'' has this: https://books.google.com/books/about/Gentlemen_s_Fancy_Dress.html?id=ED8CAAAAQAAJ.) Later editions: 1898 (HathiTrust) # ''Fancy Dresses Described; Or, What to Wear at Fancy Balls''. Debenham & Freebody, 1882. Illustr., Lillian Young. (HathiTrust has this.) Later editions: 4th ed — 1884; 1887 (HathiTrust); 6th ed. — 1896 (HathiTrust) As Leonore Davidoff says, "Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration symbolised some status category for the female wearer."<ref name=":1" />{{rp|93}} [handled under Elaborations] === Historical Accuracy === Many of the costumes at the ball were based on portraits, especially when the guest was dressed as a historical figure. If possible, we have found the portraits likely to have been the originals, or we have found, if possible, portraits that show the subjects from the two time periods at similar ages. The way clothing was cut changed quite a bit between the 18th and 19th centuries. We think of Victorian clothing — particularly women's clothing, and particularly at the end of the century — as inflexible and restrictive, especially compared to 20th- and 21st-century customs permitting freedom of movement. The difference is generally evolutionary rather than absolute — that is, as time has passed since the 18th century, clothing has allowed an increasingly greater range of movement, especially for people who did not do manual labor. By the end of the 19th century, garments like women's bodices and men's coats were made fitted and smooth by attention to the grain of the fabric and by the use of darts (rather than techniques that assembled many small, individual pieces of fabric). * clothing construction and flat-pattern techniques * Generally, the further back in time we go, the more 2-dimensional the clothing itself was. ==== Women's Versions of Historical Accuracy at the Ball ==== As always with this ball, whatever historical accuracy might be present in a woman's costume is altered so that the wearer is still a fashionable Victorian lady. What makes the costumes look "Victorian" to our eyes is the line of the silhouette caused by the foundation undergarments as well as the many "elaborations"<ref name=":1" />{{rp|93}}, mostly in the decorations, trim and accessories. Also, the clothing hangs and drapes differently because the fabric was cut on grain and the shoulders were freed by the way the sleeves were set in. ==== Men's Versions of Historical Accuracy at the Ball ==== Because men were not wearing a Victorian foundation garment at the end of the century, the men's costumes at the ball are more historically accurate in some ways. * Trim * Mixing neck treatments * Hair * Breeches * Shoes and boots * Military uniforms, arms, gloves, boots == Feathers and Plumes == === Aigrette === Elizabeth Lewandowski defines ''aigrette'' as "France. Feather or plume from an egret or heron."<ref name=":7" />(5) Sometimes the newspapers use the term to refer to an accessory (like a fan or ornament on a hat) that includes such a feather or plume. The straight and tapered feathers in an aigrette are in a bundle. === Prince of Wales's Feathers or White Plumes === The feathers in an aigrette came from egrets and herons; Prince of Wales's feathers came from ostriches. A fuller discussion of Prince of Wales's feathers and the white ostrich plumes worn at court appears on [[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|Victorian Things]]. For much of the late 18th and 19th centuries, white ostrich plumes were central to fashion at court, and at a certain point in the late 18th century they became required for women being presented to the monarch and for their sponsors. Our purpose here is to understand why women were wearing plumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] as part of their costumes. First published in 1893, [[Social Victorians/People/Lady Colin Campbell|Lady Colin Campbell]]'s ''Manners and Rules of Good Society'' (1911 edition) says that<blockquote>It was compulsory for both Married and Unmarried Ladies to Wear Plumes. The married lady’s Court plume consisted of three white feathers. An unmarried lady’s of two white feathers. The three white feathers should be mounted as a Prince of Wales plume and worn towards the left hand side of the head. Colored feathers may not be worn. In deep mourning, white feathers must be worn, black feathers are inadmissible.<p> White veils or lace lappets must be worn with the feathers. The veils should not be longer than 45 inches.<ref>{{Cite web|url=https://www.edwardianpromenade.com/etiquette/the-court-presentation/|title=The Court Presentation|last=Holl|first=Evangeline|date=2007-12-07|website=Edwardian Promenade|language=en-US|access-date=2022-12-18}} https://www.edwardianpromenade.com/etiquette/the-court-presentation/.</ref></blockquote>[[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|This fashion was imported from France]] in the mid 1770s.<ref>"Abstract" for Blackwell, Caitlin. "'<nowiki/>''The Feather'd Fair in a Fright''': The Emblem of the Feather in Graphic Satire of 1776." ''Journal for Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. ''Wiley Online'' DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x (accessed November 2022).</ref> Separately, a secondary heraldic emblem of the Prince of Wales has been a specific arrangement of 3 ostrich feathers in a gold coronet<ref>{{Cite journal|date=2022-11-07|title=Prince of Wales's feathers|url=https://en.wikipedia.org/w/index.php?title=Prince_of_Wales%27s_feathers&oldid=1120556015|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Prince_of_Wales's_feathers.</ref> since King Edward III (1312–1377<ref>{{Cite journal|date=2022-12-14|title=Edward III of England|url=https://en.wikipedia.org/w/index.php?title=Edward_III_of_England&oldid=1127343221|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_III_of_England.</ref>). Some women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] wore white ostrich feathers in their hair, but most of them are not Prince of Wales's feathers. Most of the plumes in these portraits are arrangements of some kind of headdress to accompany the costume. A few, wearing what looks like the Princes of Wales's feathers, might be signaling that their character is royal or has royal ancestry. '''One of the women [which one?] was presented to the royals at this ball?''' Here is the list of women who are wearing white ostrich plumes in their portraits in the [[Social Victorians/1897 Fancy Dress Ball/Photographs|''Diamond Jubilee Fancy Dress Ball'' album of 286 photogravure portraits]]: # Kathleen Pelham-Clinton, the [[Social Victorians/People/Newcastle|Duchess of Newcastle]] # [[Social Victorians/People/Louisa Montagu Cavendish|Luise Cavendish]], the Duchess of Devonshire # Jesusa Murrieta del Campo Mello y Urritio (née Bellido), [[Social Victorians/People/Santurce|Marquisa de Santurce]] # Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]] # Princess (Laura Williamina Seymour) Victor of  [[Social Victorians/People/Gleichen#Laura%20Williamina%20Seymour%20of%20Hohenlohe-Langenburg|Hohenlohe Langenburg]] # Louisa Acheson, [[Social Victorians/People/Gosford|Lady Gosford]] # Alice Emily White Coke, [[Social Victorians/People/Leicester|Viscountess Coke]] # Lady Mary Stewart, Helen Mary Theresa [[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]] #[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], Duchess of [[Social Victorians/People/Marlborough|Marlborough]], dressed as the wife of the French Ambassador at the Court of Catherine of Russia (not white, but some color that reads dark in the black-and-white photograph) #Mrs. Mary [[Social Victorians/People/Chamberlain|Chamberlain]] (at 491), wearing white plumes, as Madame d'Epinay #Lady Clementine [[Social Victorians/People/Tweeddale|Hay]] (at 629), wearing white plumes, as St. Bris (''Les Huguenots'') #[[Social Victorians/People/Meysey-Thompson|Lady Meysey-Thompson]] (at 391), wearing white plumes, as Elizabeth, Queen of Bohemia #Mrs. [[Social Victorians/People/Grosvenor|Algernon (Catherine) Grosvenor]] (at 510), wearing white plumes, as Marie Louise #Lady [[Social Victorians/People/Ancaster|Evelyn Ewart]], at 401), wearing white plumes, as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson #[[Social Victorians/People/Lyttelton|Edith Sophy Balfour Lyttelton]] (at 580), wearing what might be white plumes on a large-brimmed white hat, after a picture by Romney #[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), wearing what might be white plumes, as Cydalise of the Comedie Italienne from the time of Louis XV #Lady [[Social Victorians/People/Ilchester|Muriel Fox Strangways]] (at 403), wearing what might be two smallish white plumes, as Lady Sarah Lennox, one of the bridesmaids of Queen Charlotte A.D. 1761 #Lady [[Social Victorians/People/Lucan|Violet Bingham]] (at 586), wearing perhaps one white plume in a headdress not related to the Prince of Wales's feathers #Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), wearing a headdress that includes some white plumes, as Lady Burleigh #[[Social Victorians/People/Dupplin|Agnes Blanche Marie Hay-Drummond]] (at 682), in a big headdress topped with white plumes, as Mademoiselle Andrée de Taverney A.D. 1775 #Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]] (at 336), wearing what looks like Prince of Wales's plumes #[[Social Victorians/People/Suffolk|Marguerite Hyde "Daisy" Leiter]] (at 684), wearing what looks like Prince of Wales's plumes #Lady [[Social Victorians/People/Spicer|Margaret Spicer]] (at 281), wearing one smallish white and one black plume, as Countess Zinotriff, Lady-in-Waiting to the Empress Catherine of Russia #Mrs. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 318), wearing what looks like Prince of Wales's plumes, as Elizabeth Cavendish, daughter of Bess of Hardwick #Nellie, [[Social Victorians/People/Kilmorey|Countess of Kilmorey]] (at 207), wearing three tall plumes, 2 white and one dark, as Comtesse du Barri #Daisy, [[Social Victorians/People/Warwick|Countess of Warwick]] (at 53), wearing at least 1 white plume, as Marie Antoinette More men than women were wearing plumes reminiscent of the Prince of Wales's feathers: * ==== Bibliography for Plumes and Prince of Wales's Feathers ==== * Blackwell, Caitlin. "'''The Feather'd Fair in a Fright'<nowiki/>'': The Emblem of the Feather in Graphic Satire of 1776." Journal for ''Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. Wiley Online DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x. * "Prince of Wales's feathers." ''Wikipedia'' https://en.wikipedia.org/wiki/Prince_of_Wales%27s_feathers (accessed November 2022). ['''Add women to this page'''] * Simpson, William. "On the Origin of the Prince of Wales' Feathers." ''Fraser's magazine'' 617 (1881): 637-649. Hathi Trust https://babel.hathitrust.org/cgi/pt?id=chi.79253140&view=1up&seq=643&q1=feathers (accessed December 2022). Deals mostly with use of feathers in other cultures and in antiquity; makes brief mention of feathers and plumes in signs and pub names that may not be associated with the Prince of Wales. No mention of the use of plumes in women's headdresses or court dress. == Honors == === The Bath === The Most Honourable Order of the Bath (GCB, Knight or Dame Grand Cross; KCB or DCB, Knight or Dame Commander; CB, Companion) === The Garter === The Most Noble Order of the Knights of the Garter (KG, Knight Companion; LG, Lady Companion) [[File:The Golden Fleece - collar exhibited at MET, NYC.jpg|thumb|The Golden Fleece collar and pendant for the 2019 "Last Knight" exhibition at the MET, NYC.|alt=Recent photograph of a gold necklace on a wide band, with a gold skin of a sheep hanging from it as a pendant]] === The Golden Fleece === To wear the golden fleece is to wear the insignia of the Order of the Golden Fleece, said to be "the most prestigious and historic order of chivalry in the world" because of its long history and strict limitations on membership.<ref name=":10">{{Cite journal|date=2020-09-25|title=Order of the Golden Fleece|url=https://en.wikipedia.org/w/index.php?title=Order_of_the_Golden_Fleece&oldid=980340875|journal=Wikipedia|language=en}}</ref> The monarchs of the U.K. were members of the originally Spanish order, as were others who could afford it, like the Duke of Wellington,<ref name=":12">Thompson, R[obert]. H[ugh]. "The Golden Fleece in Britain." Publication of the ''British Numismatic Society''. 2009 https://www.britnumsoc.org/publications/Digital%20BNJ/pdfs/2009_BNJ_79_8.pdf (accessed January 2023).</ref> the first Protestant to be admitted to the order.<ref name=":10" /> Founded in 1429/30 by Philip the Good, Duke of Burgundy, the order separated into two branches in 1714, one Spanish and the other Austrian, still led by the House of Habsburg.<ref name=":10" /> [[File:Prince Albert - Franz Xaver Winterhalter 1842.jpg|thumb|1842 Winterhalter portrait of Prince Albert wearing the insignia of the Order of the Golden Fleece, 1842|left|alt=1842 Portrait of Prince Albert by Winterhalter, wearing the insignia of the Golden Fleece]] The photograph (upper right) is of a Polish badge dating from the "turn of the XV and XVI centuries."<ref>{{Citation|title=Polski: Kolana orderowa orderu Złotego Runa, przełom XV i XVI wieku.|url=https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg|date=2019-11-10|accessdate=2023-01-10|last=Wulfstan}}. https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg.</ref> The collar to this Golden Fleece might be similar to the one the [[Social Victorians/People/Spencer Compton Cavendish#The Insignia of the Order of the Golden Fleece|Duke of Devonshire is wearing in the 1897 Lafayette portrait]]. The badges and collars that Knights of the Order actually wore vary quite a bit. The 1842 Franz Xaver Winterhalter portrait (left) of Prince Consort Albert, Victoria's husband and father of the Prince of Wales, shows him wearing the Golden Fleece on a red ribbon around his neck and the star of the Garter on the front of his coat.<ref>Winterhalter, Franz Xaver. ''Prince Albert''. {{Cite web|url=https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61|title=Explore the Royal Collection Online|website=www.rct.uk|access-date=2023-01-16}} https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61.</ref> === Royal Victorian Order === (GCVO, Knight or Dame Grand Cross; KCVO or DCVO, Knight or Dame Commander; CVO, Commander; LVO, Lieutenant; MVO, Member) === St. John === The Order of the Knights of St. John === Star of India === Most Exalted Order of the Star of India (GCSI, Knight Grand Commander; KCSI, Knight Commander; CSI, Companion) === Thistle === The Most Ancient and Most Noble Order of the Thistle == Jewelry and Stones == === Cabochon === This term describes both the treatment and shape of a precious or semiprecious stone. A cabochon treatment does not facet the stone but merely polishes it, removing "the rough parts" and the parts that are not the right stone.<ref>"cabochon, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/25778. Accessed 7 February 2023.</ref> A cabochon shape is often flat on one side and oval or round, forming a mound in the setting. === Jet === === ''Orfèvrerie'' === Sometimes misspelled in the newspapers as ''orvfèvrerie''. ''Orfèvrerie'' is the artistic work of a goldsmith, silversmith, or jeweler. === Turquoises === == Military == Several men from the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]] were dressed in military uniforms, some historical and some, possibly, not. === Baldric === According to the ''Oxford English Dictionary'', the primary sense of ''baldric'' is "A belt or girdle, usually of leather and richly ornamented, worn pendent from one shoulder across the breast and under the opposite arm, and used to support the wearer's sword, bugle, etc."<ref>"baldric, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/14849. Accessed 17 May 2023.</ref> This sense has been in existence since c. 1300. === Cuirass === According to the ''Oxford English Dictionary'', the primary sense of ''cuirass'' is "A piece of armour for the body (originally of leather); ''spec.'' a piece reaching down to the waist, and consisting of a breast-plate and a back-plate, buckled or otherwise fastened together ...."<ref>"cuirass, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/45604. Accessed 17 May 2023.</ref> [[File:Knötel IV, 04.jpg|thumb|alt=An Old drawing in color of British soldiers on horses brandishing swords in 1815.|1890 illustration of the Household Cavalry (Life Guard, left; Horse Guard, right) at the Battle of Waterloo, 1815]] === Household Cavalry === The Royal Household contains the Household Cavalry, a corps of British Army units assigned to the monarch. It is made up of 2 regiments, the Life Guards and what is now called The Blues and Royals, which were formed around the time of "the Restoration of the Monarchy in 1660."<ref name=":3">Joll, Christopher. "Tales of the Household Cavalry, No. 1. Roles." The Household Cavalry Museum, https://householdcavalry.co.uk/app/uploads/sites/2/2021/06/Household-Cavalry-Museum-video-series-large-print-text-Tales-episode-01.pdf.</ref>{{rp|1}} Regimental Historian Christopher Joll says, "the original Life Guards were formed as a mounted bodyguard for the exiled King Charles II, The Blues were raised as Cromwellian cavalry and The Royals were established to defend Tangier."<ref name=":3" />{{rp|1–2}} The 1st and 2nd Life Guards were formed from "the Troops of Horse and Horse Grenadier Guards ... in 1788."<ref name=":3" />{{rp|3}} The Life Guards were and are still official bodyguards of the queen or king, but through history they have been required to do quite a bit more than serve as bodyguards for the monarch. The Household Cavalry fought in the Battle of Waterloo on Sunday, 18 June 1815 as heavy cavalry.<ref name=":3" />{{rp|3}} Besides arresting the Cato Steet conspirators in 1820 "and guarding their subsequent execution," the Household Cavalry contributed to the "the expedition to rescue General Gordon, who was trapped in Khartoum by The Mahdi and his army of insurgents" in 1884.<ref name=":3" />{{rp|3}} In 1887 they "were involved ... in the suppression of rioters in Trafalgar Square on Bloody Sunday."<ref name=":3" />{{rp|3}} ==== Grenadier Guards ==== Three men — [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Lord Algernon Gordon-Lennox]], [[Social Victorians/People/Stanley#Edward George Villiers Stanley, Lord Stanley|Lord Stanley]], and [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|Hon. F. C. Stanley]] — attended the ball as officers of the Grenadier Guards, wearing "scarlet tunics, ... full blue breeches, scarlet hose and shoes, lappet wigs" as well as items associated with weapons and armor.<ref name=":14">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 34, Col. 2a}} Founded in England in 1656 as Foot Guards, this infantry regiment "was granted the 'Grenadier' designation by a Royal Proclamation" at the end of the Napoleonic Wars.<ref>{{Cite journal|date=2023-04-22|title=Grenadier Guards|url=https://en.wikipedia.org/w/index.php?title=Grenadier_Guards&oldid=1151238350|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Grenadier_Guards.</ref> They were not called Grenadier Guards, then, before about 1815. In 1660, the Stuart Restoration, they were called Lord Wentworth's Regiment, because they were under the command of Thomas Wentworth, 5th Baron Wentworth.<ref>{{Cite journal|date=2022-07-24|title=Lord Wentworth's Regiment|url=https://en.wikipedia.org/w/index.php?title=Lord_Wentworth%27s_Regiment&oldid=1100069077|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Wentworth%27s_Regiment.</ref> At the time of Lord Wentworth's Regiment, the style of the French cavalier had begun to influence wealthy British royalists. In the British military, a Cavalier was a wealthy follower of Charles I and Charles II — a commander, perhaps, or a field officer, but probably not a soldier.<ref>{{Cite journal|date=2023-04-22|title=Cavalier|url=https://en.wikipedia.org/w/index.php?title=Cavalier&oldid=1151166569|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier.</ref> The Guards were busy as infantry in the 17th century, engaging in a number of armed conflicts for Great Britain, but they also served the sovereign. According to the Guards Museum,<blockquote>In 1678 the Guards were ordered to form Grenadier Companies, these men were the strongest and tallest of the regiment, they carried axes, hatches and grenades, they were the shock troops of their day. Instead of wearing tri-corn hats they wore a mitre shaped cap.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/|title=Service to the Crown|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/.</ref></blockquote>The name comes from ''grenades'', then, and we are accustomed to seeing them in front of Buckingham Palace, with their tall mitre hats. The Guard fought in the American Revolution, and in the 19th century, the Grenadier Guards fought in the Crimean War, Sudan and the Boer War. They have roles as front-line troops and as ceremonial for the sovereign, which makes them elite:<blockquote>Queen Victoria decreed that she did not want to see a single chevron soldier within her Guards. Other then [sic] the two senior Warrant Officers of the British Army, the senior Warrant Officers of the Foot Guards wear a large Sovereigns personal coat of arms badge on their upper arm. No other regiments of the British Army are allowed to do so; all the others wear a small coat of arms of their lower arms. Up until 1871 all officers in the Foot Guards had the privilege of having double rankings. An Ensign was ranked as an Ensign and Lieutenant, a Lieutenant as Lieutenant and Captain and a Captain as Captain and Lieutenant Colonel. This was because at the time officers purchased their own ranks and it cost more to purchase a commission in the Foot Guards than any other regiments in the British Army. For example if it cost an officer in the Foot Guards £1,000 for his first rank, in the rest of the Army it would be £500 so if he transferred to another regiment he would loose [sic] £500, hence the higher rank, if he was an Ensign in the Guards and he transferred to a Line Regiment he went in at the higher rank of Lieutenant.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/|title=Formation and role of the Regiments|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/.</ref></blockquote> ==== Life Guards ==== [[Social Victorians/People/Shrewsbury#Reginald Talbot's Costume|General the Hon. Reginald Talbot]], a member of the 1st Life Guards, attended the Duchess of Devonshire's ball dressed in the uniform of his regiment during the Battle of Waterloo.<ref name=":14" />{{rp|p. 36, Col. 3b}} At the Battle of Waterloo the 1st Life Guards were part of the 1st Brigade — the Household Brigade — and were commanded by Major-General Lord Edward Somerset.<ref name=":4">{{Cite journal|date=2023-09-30|title=Battle of Waterloo|url=https://en.wikipedia.org/w/index.php?title=Battle_of_Waterloo&oldid=1177893566|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Battle_of_Waterloo.</ref> The 1st Life Guards were on "the extreme right" of a French countercharge and "kept their cohesion and consequently suffered significantly fewer casualties."<ref name=":4" /> == Photography == == Footnotes == {{reflist}} tjnfpz1rud8i3j3lhawced5zqnpnggj 2690306 2690305 2024-12-04T20:28:53Z Scogdill 1331941 /* Hoops */ 2690306 wikitext text/x-wiki Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. The definitions on this page are to provide a sense of what someone in the late 19th century might have meant by the term rather than a definition of what we might mean by it today. In the absence of a specialized glossary from the end of the 19th century in the U.K., we use the ''Oxford English Dictionary'' because the senses of a word are illustrated with examples that have dates so we can be sure that the senses we pick are appropriate for when they are used in the quotations we have. We also sometimes use the French ''Wikipédia'' to define a word because many technical terms of fashion were borrowings from the French. Also, often the French ''Wikipédia'' provides historical context for the uses of a word similar to the way the OED does. == Articles or Parts of Clothing: Non-gender-specific == === Mantle, Cloak, Cape === In 19th-century newspaper accounts, these terms are sometimes used without precision as synonyms. These are all outer garments. '''Mantle''' A mantle — often a long outer garment — might have elements like a train, sleeves, collars, revers, fur, and a cape. A late-19th-century writer making a distinction between a mantle and a cloak might use ''mantle'' if the garment is more voluminous. '''Cloak''' '''Cape''' === Peplum === According to the French ''Wiktionnaire'', a peplum is a "Short skirt or flared flounce layered at the waist of a jacket, blouse or dress" [translation by Google Translate].<ref>{{Cite journal|date=2021-07-02|title=péplum|url=https://fr.wiktionary.org/w/index.php?title=p%C3%A9plum&oldid=29547727|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/p%C3%A9plum.</ref> The ''Oxford English Dictionary'' has a fuller definition, although, it focuses on women's clothing because the sense is written for the present day:<blockquote>''Fashion''. ... a kind of overskirt resembling the ancient peplos (''obsolete''). Hence (now usually) in modern use: a short flared, gathered, or pleated strip of fabric attached at the waist of a woman's jacket, dress, or blouse to create a hanging frill or flounce.<ref name=":5">“peplum, n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1832614702>.</ref></blockquote>Men haven't worn peplums since the 18th century, except when wearing costumes based on historical portraits. The ''Daily News'' reported in 1896 that peplums had been revived as a fashion item for women.<ref name=":5" /> === Revers === According to the ''Oxford English Dictionary'', ''revers'' are the "edge[s] of a garment turned back to reveal the undersurface (often at the lapel or cuff) (chiefly in ''plural''); the material covering such an edge."<ref>"revers, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/164777. Accessed 17 April 2023.</ref> The term is French and was used this way in the 19th century (according to the ''Wiktionnaire'').<ref>{{Cite journal|date=2023-03-07|title=revers|url=https://fr.wiktionary.org/w/index.php?title=revers&oldid=31706560|journal=Wiktionnaire|language=fr}} https://fr.wiktionary.org/wiki/revers.</ref> == Articles or Parts of Clothing: Men's == [[Social Victorians/Terminology#Military|Men's military uniforms]] are discussed below. === À la Romaine === [[File:Johann Baptist Straub - Mars um 1772-1.jpg|thumb|left|alt=Old and damaged marble statue of a Roman god of war with flowing cloak, big helmet with a plume on top, and armor|Johann Baptist Straub's 1772 ''à la romaine'' ''Mars'']] A few people who attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball in 1897]] personated Roman gods or people. They were dressed not as Romans, however, but ''à la romaine'', which was a standardized style of depicting Roman figures that was used in paintings, sculpture and the theatre for historical dress from the 17th until the 20th century. The codification of the style was developed in France in the 17th century for theatre and ballet, when it became popular for masked balls. Women as well as men could be dressed ''à la romaine'', but much sculpture, portraiture and theatre offered opportunities for men to dress in Roman style — with armor and helmets — and so it was most common for men. In large part because of the codification of the style as well as the painting and sculpture, the style persisted and remained influential into the 20th century and can be found in museums and galleries and on monuments. For example, Johann Baptist Straub's 1772 statue of Mars (left), now in the Bayerisches Nationalmuseum, Munich, missing part of an arm, shows Mars ''à la romaine''. In London, an early 17th-century example of a figure of Mars ''à la romaine'', with a helmet, '''was''' "at the foot of the Buckingham tomb in Henry VII's Chapel at Westminster Abbey."<ref>Webb, Geoffrey. “Notes on Hubert Le Sueur-II.” ''The Burlington Magazine for Connoisseurs'' 52, no. 299 (1928): 81–89. http://www.jstor.org/stable/863535.</ref>{{rp|81, Col. 2c}} === Cavalier === [[File:Sir-Anthony-van-Dyck-Lord-John-Stuart-and-His-Brother-Lord-Bernard-Stuart.jpg|thumb|alt=Old painting of 2 men flamboyantly and stylishly dressed in colorful silk, with white lace, high-heeled boots and long hair|Van Dyck's c. 1638 painting of cavaliers Lord John Stuart and his brother Lord Bernard Stuart]] As a signifier in the form of clothing of a royalist political and social ideology begun in France in the early 17th century, the cavalier established France as the leader in fashion and taste. Adopted by [[Social Victorians/Terminology#Military|wealthy royalist British military officers]] during the time of the Restoration, the style signified a political and social position, both because of the loyalty to Charles I and II as well the wealth required to achieve the cavalier look. The style spread beyond the political, however, to become associated generally with dress as well as a style of poetry.<ref>{{Cite journal|date=2023-04-25|title=Cavalier poet|url=https://en.wikipedia.org/w/index.php?title=Cavalier_poet&oldid=1151690299|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier_poet.</ref> Van Dyck's 1638 painting of two brothers (right) emphasizes the cavalier style of dress. === Coats === ==== Doublet ==== * In the 19th-century newspaper accounts we have seen that use this word, doublet seems always to refer to a garment worn by a man, but historically women may have worn doublets. In fact, a doublet worn by Queen Elizabeth I exists and '''is somewhere'''. * Technically doublets were long sleeved, although we cannot be certain what this or that Victorian tailor would have done for a costume. For example, the [[Social Victorians/People/Spencer Compton Cavendish#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Duke of Devonshire's costume as Charles V]] shows long sleeves that may be part of the surcoat but should be the long sleeves of the doublet. ==== Pourpoint ==== A padded doublet worn under armor to protect the warrior from the metal chafing. A pourpoint could also be worn without the armor. ==== Surcoat ==== Sometimes just called ''coat''. [[File:Oscar Wilde by Sarony 1882 18.jpg|thumb|alt=Old photograph of a young man wearing a velvet jacket, knee breeches, silk hose and shiny pointed shoes with bows, seated on a sofa and leaning on his left hand and holding a book in his right| Oscar Wilde, 1882, by Napoleon Sarony]] === Hose, Stockings and Tights === Newspaper accounts from the late 19th century of men's clothing use the term ''hose'' for what we might call stockings or tights. In fact, the terminology is specific. ''Stockings'' is the more general term and could refer to hose or tights. With knee breeches men wore hose, which ended above the knee, and women wore hose under their dresses. The ''Oxford English Dictionary'' defines tights as "Tight-fitting breeches, worn by men in the 18th and early 19th centuries, and still forming part of court-dress."<ref>“Tights, N.” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2693287467.</ref> By 1897, the term was in use for women's stockings, which may have come up only to the knee. Tights were also worn by dancers and acrobats. This general sense of ''tights'' does not assume that they were knitted. ''Clocking'' is decorative embroidery on hose, usually, at the ankles on either the inside or the outside of the leg. It started at the ankle and went up the leg, sometimes as far as the knee. On women's hose, the clocking could be quite colorful and elaborate, while the clocking on men's hose was more inconspicuous. In many photographs men's hose are wrinkled, especially at the ankles and the knees, because they were shaped from woven fabric. Silk hose were knitted instead of woven, which gave them elasticity and reduced the wrinkling. The famous Sarony carte de visite photograph of Oscar Wilde (right) shows him in 1882 wearing knee breeches and silk hose, which are shiny and quite smoothly fitted although they show a few wrinkles at the ankles and knees. In the portraits of people in costume at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the men's hose are sometimes quite smooth, which means they were made of knitted silk and may have been smoothed for the portrait. In painted portraits the hose are almost always depicted as smooth, part of the artist's improvement of the appearance of the subject. === Shoes and Boots === == Articles or Parts of Clothing: Women's == === '''Chérusque''' === According to the French ''Wikipedia'', ''chérusque'' is a 19th-century term for the kind of standing collar like the ones worn by ladies in the Renaissance.<ref>{{Cite journal|date=2021-06-26|title=Collerette (costume)|url=https://fr.wikipedia.org/w/index.php?title=Collerette_(costume)&oldid=184136746|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Collerette_(costume)#Au+xixe+siècle+:+la+Chérusque.</ref> === Corsage === According to the ''Oxford English Dictionary'', the corsage is the "'body' of a woman's dress; a bodice."<ref>"corsage, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/42056. Accessed 7 February 2023.</ref> This sense is well documented in the ''OED'' for the mid and late 19th-century, used this way in fiction as well as in a publication like ''Godey's Lady's Book'', which would be expected to use appropriate terminology associated with fashion and dress making. The sense of "a bouquet worn on the bodice" is, according to the ''OED'', American. === Décolletage === === Girdle === === Mancheron === According to the ''Oxford English Dictionary'', a ''mancheron'' is a "historical" word for "A piece of trimming on the upper part of a sleeve on a woman's dress."<ref>"mancheron, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/113251. Accessed 17 April 2023.</ref> At the present, in French, a ''mancheron'' is a cap sleeve "cut directly on the bodice."<ref>{{Cite journal|date=2022-11-28|title=Manche (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manche_(v%C3%AAtement)&oldid=199054843|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Manche_(v%C3%AAtement).</ref> === Petticoat === According to the ''O.E.D.'', a petticoat is a <blockquote>skirt, as distinguished from a bodice, worn either externally or showing beneath a dress as part of the costume (often trimmed or ornamented); an outer skirt; a decorative underskirt. Frequently in ''plural'': a woman's or girl's upper skirts and underskirts collectively. Now ''archaic'' or ''historical''.<ref>“petticoat, n., sense 2.b”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1021034245></ref> </blockquote>This sense is, according to the ''O.E.D.'', "The usual sense between the 17th and 19th centuries." However, while petticoats belong in both outer- and undergarments — that is, meant to be seen or hidden, like underwear — they were always under another garment, for example, underneath an open overskirt. The primary sense seems to have shifted through the 19th century so that, by the end, petticoats were underwear and the term ''underskirt'' was used to describe what showed under an open overskirt. === Stomacher === According to the ''O.E.D.'', a stomacher is "An ornamental covering for the chest (often covered with jewels) worn by women under the lacing of the bodice,"<ref>“stomacher, n.¹, sense 3.a”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1169498955></ref> although by the end of the 19th century, the bodice did not often have visible laces. Some stomachers were so decorated that they were thought of as part of the jewelry. === Train === A train is The Length of the Train '''For the monarch [or a royal?]''' According to Debrett's,<blockquote>A peeress's coronation robe is a long-trained crimson velvet mantle, edged with miniver pure, with a miniver pure cape. The length of the train varies with the rank of the wearer: * Duchess: for rows of ermine; train to be six feet * Marchioness: three and a half rows of ermine; train to be three and three-quarters feet * Countess: three rows of ermine; train to be three and a half feet * Viscountess: two and a half rows of ermine; train to be three and a quarter feet * Baroness: two rows of ermine; train to be three feet<ref name=":2">{{Cite web|url=https://debretts.com/royal-family/dress-codes/|title=Dress Codes|website=debretts.com|language=en-US|access-date=2023-07-27}} https://debretts.com/royal-family/dress-codes/.</ref> </blockquote>The pattern on the coronet worn was also quite specific, similar but not exactly the same for peers and peeresses. Debrett's also distinguishes between coronets and tiaras, which were classified more like jewelry, which was regulated only in very general terms. Peeresses put on their coronets after the Queen or Queen Consort has been crowned. ['''peers?'''] === Foundation Garments === Unlike undergarments, Victorian women's foundation garments created the distinctive silhouette. Victorian undergarments included the chemise, the bloomers, the corset cover — articles that are not structural. The corset was an important element of the understructure of foundation garments — hoops, bustles, petticoats and so on — but it has never been the only important element. === Corset === [[File:Corset - MET 1972.209.49a, b.jpg|thumb|alt=Photograph of an old silk corset on a mannequin, showing the closure down the front, similar to a button, and channels in the fabric for the boning. It is wider at the top and bottom, creating smooth curves from the bust to the compressed waist to the hips, with a long point below the waist in front.|French 1890s corset, now in the Metropolitan Museum of Art, NYC]] The understructure of Victorian women's clothing is what makes the costumes worn by the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] so distinctly Victorian in appearance. An example of a corset that has the kind of structure often worn by fashionably dressed women in 1897 is the one at right. This corset exaggerated the shape of the women's bodies and made possible a bodice that looked and was fitted in the way that is so distinctive of the time — very controlled and smooth. And, as a structural element, this foundation garment carried the weight of all those layers and all that fabric and decoration on the gowns, trains and mantles. (The trains and mantles could be attached directly to the corset itself.) * This foundation emphasizes the waist and the bust in particular, in part because of the contrast between the very small waist and the rounded fullness of the bust and hips. * The idealized waist is defined by its small span and the sexualizing point at the center-bottom of the bodice, which directs the eye downwards. Interestingly, the pointed waistline worn by Elizabethan men has become level in the Victorian age. Highly fashionable Victorian women wearing the traditional style, however, had extremely pointed waists. * The busk (a kind of boning in the front of a corset that is less flexible than the rest) smoothed the bodice, flattened the abdomen and prevented the point on the bodice from curling up. * The sharp definition of the waist was caused by ** length of the corset (especially on the sides) ** the stiffness of the boning ** the layers of fabric ** the lacing (especially if the woman used tightlacing) ** the over-all shape, which was so much wider at the top and the bottom ** the contrast between the waist and the wider top and bottom * The late-19th-century corset was long, ending below the waist even on the sides and back. * The boning and the top edge of the late 19th-century fashion corset pushed up the bust, rounding (rather than flattening, as in earlier styles) the breasts, drawing attention to their exposed curves and creating cleavage. * The exaggerated bust was larger than the hips, whenever possible, an impression reinforced by the A-line of the skirt and the inverted Vs in the decorative trim near the waist and on the skirt. * This corset made the bodice very smooth with a very precise fit, that had no wrinkles, folds or loose drapery. The bodice was also trimmed or decorated, but the base was always a smooth bodice. More formal gowns would still have the fitted bodice and more elaborate trim made from lace, embroidery, appliqué, beading and possibly even jewels. The advantages and disadvantages of corseting and especially tight lacing were the subject of thousands of articles and opinions in the periodical press for a great part of the century, but the fetishistic and politicized tight lacing was practiced by very few women. And no single approach to corsetry was practiced by all women all the time. Most of the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 ball]] were not tightly laced, but the progressive style does not dominate either, even though all the costumes are technically historical dress. Part of what gives most of the costumes their distinctive 19th-century "look" is the more traditional corset beneath them. Even though this highly fashionable look was widely present in the historical costumes at the ball, some women's waists were obviously very small and others were hardly '''emphasized''' at all. Women's waists are never mentioned in the newspaper coverage of the ball — or, indeed, of any of the social events attended by the network at the ball — so it is only in photographs that we can see the effects of how they used their corsets. === Hoops === ''Hoops'' is a mid-19th-century term for a cage-like structure worn under a skirt to hold it away from the body. Striking for how long they lasted and '''the ways''' they evolved, hoops were the foundation undergarment for the bottom half of a woman's body, for a skirt and petticoat. Women wore this cage-like structure from the '''15th century''' through the late 19th century. The 16th-century Katherine of Aragon is credited with making it fashionable outside Spain The cage caused the silhouette of skirts to change shape over time and enabled the extreme distortions of 17th-and-18th-century panniers and the late 19th-century bustle. Early hoops circled the body symmetrically in a cone or drum shape, then were moved to the sides with panniers, then ballooned around the body like the top half of a sphere, and finally were pulled to the rear with a bustle. That is, like corsets, the distorted shapes of high fashion were made possible by hoops. High fashion demanded these shapes, which disguised women's bodies, '''especially below the waist, while corsets did their work above it'''. Besides the shape, the structure used to construct hoops evolved — from cane and wood to whalebone, then steel '''rods''' and wire. Add fabric structural stuff [[File:Pedro García de Benabarre St John Retable Detail.jpg|thumb|Pedro García de Benabarre St John Retable Detail]] [[File:Alonso Sánchez Coello 011.jpg|thumb|Alonso Sánchez Coello 011]] ==== 15th Century ==== Hoops first appeared in Spain in the 15th century and influenced European fashion for many years. Illustrations ==== 16th Century ==== [[File:Farthingale 2 (PSF).png|thumb|Farthingale 2 (PSF)]] In the 16th century, the garment we call ''hoops'' was called a farthingale. ''Vertugadin'' is a French term for ''farthingale'', a cage made of hoops supporting a skirt — "un élément essentiel de la mode Tudor en Angleterre [an essential element of Tudor fashion in England]."<ref name=":0">{{Cite journal|date=2022-03-12|title=Vertugadin|url=https://fr.wikipedia.org/w/index.php?title=Vertugadin&oldid=191825729|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Vertugadin.</ref> ''Farthingale'' is the term in English; in French, it's ''vertugadin'', and in Spanish ''vertugado''. Blanche Payne says,<blockquote>Katherine of Aragon is reputed to have introduced the Spanish farthingale (a hooped underskirt) into England early in the century. The result was to convert the columnar skirt of the fifteenth century into the cone shape of the sixteenth. ...<p> Spanish influence had introduced the hoop-supported skirt, smooth in contour, which was quite generally worn.<ref name=":11" /> (291)</blockquote>In fact, "The Spanish princess Catherine of Aragon brought the fashion to England for her marriage to Prince Arthur, eldest son of Henry VII in 1501 [La princesse espagnole Catherine d'Aragon amena la mode en Angleterre pour son mariage avec le prince Arthur, fils aîné d'Henri VII en 1501]."<ref name=":0" /> Catherine of Aragon, of course, married Henry VIII after Arthur's death. By the end of the 16th century the French and Spanish farthingales were not identical. The Spanish farthingale shaped the skirt into an A-line with a graduated series of hoops sewn to an undergarment. The French farthingale was a flattish "cartwheel" or platter of hoops worn below the waist and above the hips held the skirt out more or less horizontally. Once past the vertugadin, the skirt then fell straight to the floor, shaping the skirt into a kind of drum. The shoes show in the portraits of women wearing the French farthingale. <blockquote>"FARTHINGALE:  Renaissance (1450-1550 C.E. to Elizabethan (1550-1625 C.E.). Linen underskirt with wire supports which, when shaped, produced a variety of dome, bell, and oblong shapes."<ref name=":7" /> (105)</blockquote> ==== 17th Century ==== [[File:Panniers 1.jpg|thumb|alt=Photograph of the wooden and fabric skeleton of an 18th-century women's foundation garment|Panniers 1]] Generally, panniers were a kind of undergarment worn in the 17th and 18th centuries. Their design evolved during the century. Made of hoops of wood, they were "baskets" or cages worn on either side of the waist to broaden the skirts to the sides. Illustration ==== 18th Century ==== By the 18th century, the farthingale was called hoops, which were at this point were made of wood. Blanche Payne outlines the evolution of hoops, and thus the shape of the skirt, in the 18th century:<blockquote>SKIRT FASHIONS. Since skirts experienced the greatest alterations, a brief summary of the successive silhouettes should help to place individual costumes in their proper niches. Six basic forms appeared during the century, in the following order: # The bustle was a continuation of the 1690 mode. # The bell or dome shape resulted from the reintroduction of hoops; in England by 1710, in France by 1720. # The ellipse, the second phase of the hoop skirt, was achieved by broadening the support from side to side and compressing it from front to back. It had a long run of popularity, from 1740 to 1770, the extreme width being retained in court costumes. In France it persisted until the revolution, except that skirts were allowed to curve outward in [the] back again. English court costume followed this fashion well into the nineteenth century. # The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, which were used to form the three great ‘poufs’ known as the polonaise (see Fig. 452). These diversions appeared in the late sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period. # The return of the bustle in the 1780s. # The tubular form, drawn from classic art, in the 1790s.<ref name=":11" /> (411–413) </blockquote>While we think of the bustle as a 19th-century look, it can be found in the 18th century, as Payne says. Polonaise may need its own section. ==== 19th Century ==== [[File:Cutaway sketch of crinoline.gif|thumb|Cutaway sketch of crinoline]] [[File:Paris voulant englober la banlieue.JPG|thumb|Paris voulant englober la banlieue]] [[File:Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723.jpg|thumb|Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723]] In the 19th century, the hoops were made of wire and became lighter. By the 1860s, hoops made for huge round skirts. In Laura Ingalls Wilder's 1941 ''Little Town on the Prairie'', the '''16-year-old Laura wears hoops. Laura Ingalls Wilder and thus the character Laura were born in 1862, so this moment is set in 1883'''.<ref>Hill, Pamela Smith, ed. ''Pioneer Girl: The Annotated Autobiography''.</ref> She makes the comment that she wants to be in style, but that would be on the prairie and not necessarily the latest Parisian style.<blockquote>“Well,” Laura began; then she stopped and spun round and round, for the strong wind blowing against her always made the wires of her hoop skirt creep slowly upward under her skirts until they bunched around her knees. Then she must whirl around and around until the wires shook loose and spiraled down to the bottom of her skirts where they should be. “As she and Carrie hurried '''ob''' she began again. “I think it was silly, the way they dressed when Ma was a girl, don’t you? Drat this wind!” she exclaimed as the hoops began creeping upward again. “Quietly Carrie stood by while Laura whirled. “I’m glad I’m not old enough to have to wear hoops,” she said. “They’d make me dizzy.” “They are rather a nuisance,” Laura admitted. “But they are stylish, and when you’re my age you’ll want to be in style.”<ref>Wilder, Laura Ingalls. ''Little Town on the Prairie.'' Harper and Row, 1941. Pp. 272–273.</ref></blockquote>This very unusual description of the way the wind could make hoops creep and the solution. It must have been happening to other women wearing hoops at the time. == '''Traditional vs Progressive Style''' == === Progressive Style === The terms ''artistic dress'' and ''aesthetic dress'' are not synonymous and were in use at different times to refer to different groups of people in different contexts, but we recognize them as referring to a similar kind of personal style in clothing, a style we call progressive dress or the progressive style. Used in a very precise way, ''artistic dress'' is associated with the Pre-Raphaelite artists and the women in their circle beginning in the 1860s. Similarly, ''aesthetic dress'' is associated with the 1880s and 1890s and dress reform movements. In general, the progressive style is characterized by its resistance to the highly structured fashion of its day, especially corseting, aniline dyes and an extremely close fit. === Traditional Style === By the end of the century designs from the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]] (or Maison Worth) define what we think of as the traditional Victorian look, which was very stylish and expensive. Blanche Payne describes an example of the 1895 "high style" in a gown by Worth with "the idiosyncrasies of the [1890s] full blown":<blockquote>The dress is white silk with wine-red stripes. Sleeves, collars, bows, bag, hat, and hem border match the stripes. The sleeve has reached its maximum volume; the bosom full and emphasized with added lace; the waistline is elongated, pointed, and laced to the point of distress; the skirt is smooth over the hips, gradually swinging out to sweep the floor. This is the much vaunted hourglass figure.<ref name=":11">Payne, Blanche. ''History of Costume from the Ancient Egyptians to the Twentieth Century''. Harper & Row, 1965.</ref>{{rp|530}}</blockquote> The Victorian-looking gowns at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] are stylish in a way that recalls the designs of the House of Worth. The elements that make their look so Victorian are anachronisms on the costumes representing fashion of earlier eras. The women wearing these gowns preferred the standards of beauty from their own day to a more-or-less historically accurate look. The style competing at the very end of the century with the Worth look was not the historical, however, but a progressive style called at the time ''artistic'' or ''aesthetic''. William Powell Frith's 1883 painting ''A Private View at the Royal Academy, 1881'' (discussion below) pits this kind of traditional style against the progressive or artistic style. === The Styles === [[File:Frith A Private View.jpg|thumb|William Powell Frith, ''A Private View at the Royal Academy, 1881'']] We typically think of the late-Victorian silhouette as universal but, in the periods in which corsets dominated women's dress, not all women wore corsets and not all corsets were the same, as William Powell Frith's 1883 ''A Private View at the Royal Academy, 1881'' (right) illustrates. Frith is clear in his memoir that this painting — "recording for posterity the aesthetic craze as regards dress" — deliberately contrasts what he calls the "folly" of the Artistic Dress movement and the look of the traditional corseted waist.<ref>Frith, William Powell. ''My Autobiography and Reminiscences''. 1887.</ref> Frith considered the Artistic Movement and Artistic Dress "ephemeral," but its rejection of corsetry looks far more consequential to us in hindsight than it did in the 19th century. As Frith sees it, his painting critiques the "craze" associated with the women in this set of identifiable portraits who are not corseted, but his commitment to realism shows us a spectrum, a range, of conservatism and if not political then at least stylistic progressivism among the women. The progressives, oddly, are the women wearing artistic (that is, somewhat historical) dress, because they’re not corseted. It is a misreading to see the presentation of the women’s fashion as a simple opposition. Constance, Countess of Lonsdale — situated at the center of this painting with Frederick Leighton, president of the Royal Academy of Art — is the most conservatively dressed of the women depicted, with her narrow sleeves, tight waist and almost perfectly smooth bodice, which tells us that her corset has eyelets so that it can be laced precisely and tightly, and it has stays (or "bones") to prevent wrinkles or natural folds in the overclothing. Lillie Langtry, in the white dress, with her stylish narrow sleeves, does not have such a tightly bound waist or smooth bodice, suggesting she may not be corseted at all, as we know she sometimes was not.['''citation'''] Jenny Trip, a painter’s model, is the woman in the green dress in the aesthetic group being inspected by Anthony Trollope, who may be taking notes. She looks like she is not wearing a corset. Both Langtry and Trip are toward the middle of this spectrum: neither is dressed in the more extreme artistic dress of, say, the two figures between Trip and Trollope. A lot has been written about the late-Victorian attraction to historical dress, especially in the context of fancy-dress balls and the Gothic revival in social events as well as art and music. Part of the appeal has to have been the way those costumes could just be beautiful clothing beautifully made. Historical dress provided an opportunity for some elite women to wear less-structured but still beautiful and influential clothing. ['''Calvert'''<ref>Calvert, Robyne Erica. ''Fashioning the Artist: Artistic Dress in Victorian Britain 1848-1900''. Ph.D. thesis, University of Glasgow, 2012. <nowiki>https://theses.gla.ac.uk/3279/</nowiki></ref>] The standards for beauty, then, with historical dress were Victorian, with the added benefit of possibly less structure. So, at the Duchess of Devonshire's ball, "while some attendees tried to hew closely to historical precedent, many rendered their historical or mythological personage in the sartorial vocabulary they knew best. The [photographs of people in their costumes at the ball offer] a glimpse into how Victorians understood history, not a glimpse into the costume of an authentic historical past."<ref>Mitchell, Rebecca N. "The Victorian Fancy Dress Ball, 1870–1900." ''Fashion Theory'' 2017 (21: 3): 291–315. DOI: 10.1080/1362704X.2016.1172817.</ref> (294) * historical dress: beautiful clothing. * the range at the ball, from Minnie Paget to Gwladys * "In light of such efforts, the ball remains to this day one of the best documented outings of the period, and a quick glance at the album shows that ..." Women had more choices about their waists than the simple opposition between no corset and tightlacing can accommodate. The range of choices is illustrated in Frith's painting, with a woman locating herself on it at a particular moment for particular reasons. Much analysis of 19th-century corsetry focuses on its sexualizing effects — corsets dominated Victorian photographic pornography ['''citations'''] and at the same time, the absence of a corset was sexual because it suggested nudity.['''citations'''] A great deal of analysis of 19th-century corsetry, on the other hand, assumes that women wore corsets for the male gaze ['''citations'''] or that they tightened their waists to compete with other women.['''citations'''] But as we can see in Frith's painting, the sexualizing effect was not universal or sweeping, and these analyses do not account for the choices women had in which corset to wear or how tightly to lace it. Especially given the way that some photographic portraits were mechanically altered to make the waist appear smaller, the size of a woman's waist had to do with how she was presenting herself to the world. That is, the fact that women made choices about the size of or emphasis on their waists suggests that they had agency that needs to be taken into account. As they navigated the complex social world, women's fashion choices had meaning. Society or political hostesses had agency not only in their clothing but generally in that complex social world. They had roles managing social events of the upper classes, especially of the upper aristocracy and oligarchy, like the Duchess of Devonshire's ball. Their class and rank, then, were essential to their agency, including to some degree their freedom to choose what kind of corset to wear and how to wear it. Also, by the end of the century lots of different kinds of corsets were available for lots of different purposes. Special corsets existed for pregnancy, sports (like tennis, bicycling, horseback riding, golf, fencing, archery, stalking and hunting), theatre and dance and, of course, for these women corsets could be made to support the special dress worn over it. Women's choices in how they presented themselves to the world included more than just their foundation garments, of course. "Every cap, bow, streamer, ruffle, fringe, bustle, glove," that is, the trim and decorations on their garments, their jewelry and accessories — which Davidoff calls "elaborations"<ref name=":1">Davidoff, Leonore. ''The Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig 1973).</ref>{{rp|93}} — pointed to a host of status categories, like class, rank, wealth, age, marital status, engagement with the empire, how sexual they wanted to seem, political alignment and purpose at the social event. For example, when women were being presented to the monarch, they were expected to wear three ostrich plumes, often called the [[Social Victorians/Terminology#Prince of Wales's Feathers or White Plumes|Prince of Wales's feathers]]. Like all fashions, the corset, which was quite long-lasting in all its various forms, eventually went out of style. Of the many factors that might have influenced its demise, perhaps most important was the women's movement, in which women's rights, freedom, employment and access to their own money and children were less slogan-worthy but at least as essential as votes for women. The activities of the animal-rights movements drew attention not only to the profligate use of the bodies and feathers of birds but also to the looming extinction of the baleen whale, which made whale bone scarce and expensive. Perhaps the century's debates over corseting and especially tightlacing were relevant to some decisions not to be corseted. And, of course, perhaps no other reason is required than that the nature of fashion is to change. == Cinque Cento == According to the ''Oxford English Dictionary'', ''Cinque Cento'' is a shortening of ''mil cinque cento'', or 1500.<ref>"cinquecento, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/33143. Accessed 7 February 2023.</ref> The term, then would refer, perhaps informally, to the sixteenth century. == Crevé == ''Creve'', without the accent, is an old word in English (c. 1450) for burst or split.<ref>"creve, v." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44339. Accessed 8 February 2023.</ref> ['''With the acute accent, it looks like a past participle in French.'''] == Elastic == Elastic had been invented and was in use by the end of the 19th century. For the sense of "Elastic cord or string, usually woven with india-rubber,"<ref name=":6">“elastic, adj. & n.”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1199670313>.</ref> the ''Oxford English Dictionary'' has usage examples beginning in 1847. The example for 1886 is vivid: "The thorough-going prim man will always place a circle of elastic round his hair previous to putting on his college cap."<ref name=":6" /> == Elaborations == In her 1973 ''The Best Circles: Society, Etiquette and the Season'', Leonore Davidoff notes that women’s status was indicated by dress and especially ornament: “Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration,” she says, “symbolised some status category for the female wearer.”<ref name=":1" />{{rp|93}} Looking at these elaborations as meaningful rather than dismissing them as failed attempts at "historical accuracy" reveals a great deal about the individual women who wore or carried them — and about the society women and political hostesses in their roles as managers of the social world. In her review of ''The House of Worth: Portrait of an Archive'', Mary Frances Gormally says,<blockquote>In a socially regulated year, garments custom made with a Worth label provided women with total reassurance, whatever the season, time of day or occasion, setting them apart as members of the “Best Circles” dressed in luxurious, fashionable and always appropriate attire (Davidoff 1973). The woman with a Worth wardrobe was a woman of elegance, lineage, status, extreme wealth and faultless taste.<ref>Gormally, Mary Frances. Review essay of ''The House of Worth: Portrait of an Archive'', by Amy de la Haye and Valerie D. Mendes (V&A Publishing, 2014). ''Fashion Theory'' 2017 (21, 1): 109–126. DOI: 10.1080/1362704X.2016.1179400.</ref> (117)</blockquote> === Aglet, Aiglet === Historically, an aglet is a "point or metal piece that capped a string [or ribbon] used to attach two pieces of the garment together, i.e., sleeve and bodice."<ref name=":7" />{{rp|4}} Although they were decorative, they were not always visible on the outside of the clothing. They were often stuffed inside the layers at the waist (for example, attaching the bodice to a skirt or breeches). === Frou-frou === In French, ''frou-frou'' or, spelled as ''froufrou'', is the sound of the rustling of silk or sometimes of fabrics in general.<ref>{{Cite journal|date=2023-07-25|title=frou-frou|url=https://fr.wiktionary.org/w/index.php?title=frou-frou&oldid=32508509|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/frou-frou.</ref> The first use the French ''Wiktionnaire'' lists is Honoré Balzac, ''La Cousine Bette'', 1846.<ref>{{Cite journal|date=2023-06-03|title=froufrou|url=https://fr.wiktionary.org/w/index.php?title=froufrou&oldid=32330124|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/froufrou.</ref> ''Frou-frou'' is a term clothing historians use to describe decorative additions to an article of clothing; often the term has a slight negative connotation, suggesting that the additions are superficial. === Pouf, Puff, Poof === According to the French ''Wikipédia'', a pouf was, beginning in 1744, a "kind of women's hairstyle":<blockquote>The hairstyle in question, known as the “pouf”, had launched the reputation of the enterprising Rose Bertin, owner of the Grand Mogol, a very prominent fashion accessories boutique on Rue Saint-Honoré in Paris in 1774. Created in collaboration with the famous hairdresser, Monsieur Léonard, the pouf was built on a scaffolding of wire, fabric, gauze, horsehair, fake hair, and the client's own hair held up in an almost vertical position. — (Marie-Antoinette, ''Queen of Fashion'', translated from the American by Sylvie Lévy, in ''The Rules of the Game'', n° 40, 2009)</blockquote>''Puff'' and ''poof'' are used to describe clothing. === Shirring === ''Shirring'' is the gathering of fabric to make poufs or puffs. The 19th century is known for its use of this decorative technique. Even men's clothing had shirring: at the shoulder seam. === Sequins === Sequins, paillettes, spangles Sequins — or paillettes — are "small, scalelike glittering disks."<ref name=":7" />(216) The French ''Wiktionnaire'' defines ''paillette'' as "Lamelle de métal, brillante, mince, percée au milieu, ordinairement ronde, et qu’on applique sur une étoffe pour l’orner [A strip of metal, shiny, thin, pierced in the middle, usually round, and which is applied to a fabric in order to decorate it.]"<ref name=":8">{{Cite journal|date=2024-03-18|title=paillette|url=https://fr.wiktionary.org/w/index.php?title=paillette&oldid=33809572|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/paillette.</ref> According to the ''OED'', the use of ''sequin'' as a decorative device for clothing (as opposed to gold coins minted and used for international trade) goes back to the 1850s.<ref>“Sequin, N.” ''Oxford English Dictionary'', Oxford UP, September 2023, https://doi.org/10.1093/OED/4074851670.</ref> The first instance of ''spangle'' as "A small round thin piece of glittering metal (usually brass) with a hole in the centre to pass a thread through, used for the decoration of textile fabrics and other materials of various sorts" is from c. 1420.<ref>“Spangle, N. (1).” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/4727197141.</ref> The first use of ''paillette'' listed in the French ''Wiktionnaire'' is in Jules Verne in 1873 to describe colored spots on icy walls.<ref name=":8" /> Currently many distinguish between sequins (which are smaller) and paillettes (which are larger). Before the 20th century, sequins were metal discs or foil leaves, and so of course if they were silver or copper, they tarnished. It is not until well into the 20th century that plastics were invented and used for sequins. === Trim and Lace === ''A History of Feminine Fashion'', published sometime before 1927 and probably commissioned by [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|the Maison Worth]], describes Charles Frederick Worth's contributions to the development of embroidery and [[Social Victorians/Terminology#Passementerie|passementerie]] (trim) from about the middle of the 19th century:<blockquote>For it must be remembered that one of M. Worth's most important and lasting contributions to the prosperity of those who cater for women's needs, as well as to the variety and elegance of his clients' garments, was his insistence on new fabrics, new trimmings, new materials of every description. In his endeavours to restore in Paris the splendours of the days of La Pompadour, and of Marie Antoinette, he found himself confronted at the outset with a grave difficulty, which would have proved unsurmountable to a man of less energy, resource and initiative. The magnificent materials of those days were no longer to be had! The Revolution had destroyed the market for beautiful materials of this, type, and the Restoration and regime of Louis Philippe had left a dour aspect in the City of Light. ... On parallel lines [to his development of better [[Social Victorians/Terminology#Satin|satin]]], [Worth] stimulated also the manufacture of embroidery and ''passementerie''. It was he who first started the manufacture of laces copied from the designs of the real old laces. He was the / first dressmaker to use fur in the trimming of light materials — but he employed only the richer furs, such as sable and ermine, and had no use whatever for the inferior varieties of skins.<ref name=":9" />{{rp|6–7}}</blockquote> ==== Gold and Silver Fabric and Lace ==== The ''Encyclopaedia Britannica'' (9th edition) has an article on gold and silver fabric, threads and lace attached to the article on gold. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>GOLD AND SILVER LACE. Under this heading a general account may be given of the use of the precious metals in textiles of all descriptions into which they enter. That these metals were used largely in the sumptuous textiles of the earliest periods of civilization there is abundant testimony; and to this day, in the Oriental centres whence a knowledge and the use of fabrics inwoven, ornamented, and embroidered with gold and silver first spread, the passion for such brilliant and costly textiles is still most strongly and generally prevalent. The earliest mention of the use of gold in a woven fabric occurs in the description of the ephod made for Aaron (Exod. xxxix. 2, 3) — "And he made the ephod of gold, blue, and purple, and scarlet, and fine twined linen. And they did beat the gold into thin plates, and cut it into wires (strips), to work it in the blue, and in the purple, and in the scarlet, and in the fine linen, with cunning work." In both the ''Iliad'' and the ''Odyssey'' distinct allusion is frequently made to inwoven and embroidered golden textiles. Many circumstances point to the conclusion that the art of weaving and embroidering with gold and silver originated in India, where it is still principally prosecuted, and that from one great city to another the practice travelled westward, — Babylon, Tarsus, Baghdad, Damascus, the islands of Cyprus and Sicily, Con- / stantinople and Venice, all in the process of time becoming famous centres of these much prized manufactures. Alexander the Great found Indian kings and princes arrayed in robes of gold and purple; and the Persian monarch Darius, we are told, wore a war mantle of cloth of gold, on which were figured two golden hawks as if pecking at each other. There is reason, according to Josephus, to believe that the “royal apparel" worn by Herod on the day of his death (Acts xii. 21) was a tissue of silver. Agrippina, the wife of the emperor Claudius, had a robe woven entirely of gold, and from that period downwards royal personages and high ecclesiastical dignitaries used cloth and tissues of gold and silver for their state and ceremonial robes, as well as for costly hangings and decorations. In England, at different periods, various names were applied to cloths of gold, as ciclatoun, tartarium, naques or nac, baudekiu or baldachin, Cyprus damask, and twssewys or tissue. The thin flimsy paper known as tissue paper, is so called because it originally was placed between the folds of gold "tissue" to prevent the contiguous surfaces from fraying each other. At what time the drawing of gold wire for the preparation of these textiles was first practised is not accurately known. The art was probably introduced and applied in different localities at widely different dates, but down till mediaeval times the method graphically described in the Pentateuch continued to be practised with both gold and silver. Fabrics woven with gold and silver continue to be used on the largest scale to this day in India; and there the preparation of the varieties of wire, and the working of the various forms of lace, brocade, and embroidery, is at once an important and peculiar art. The basis of all modern fabrics of this kind is wire, the "gold wire" of the manufacturer being in all cases silver gilt wire, and silver wire being, of course, composed of pure silver. In India the wire is drawn by means of simple draw-plates, with rude and simple appliances, from rounded bars of silver, or gold-plated silver, as the case may be. The wire is flattened into the strip or ribbon-like form it generally assumes by passing it, fourteen or fifteen strands simultaneously, over a fine, smooth, round-topped anvil, and beating it as it passes with a heavy hammer having a slightly convex surface. From wire so flattened there is made in India soniri, a tissue or cloth of gold, the web or warp being composed entirely of golden strips, and ruperi, a similar tissue of silver. Gold lace is also made on a warp of thick yellow silk with a weft of flat wire, and in the case of ribbons the warp or web is composed of the metal. The flattened wires are twisted around orange (in the case of silver, white) coloured silk thread, so as completely to cover the thread and present the appearance of a continuous wire; and in this form it is chiefly employed for weaving into the rich brocades known as kincobs or kinkhábs. Wires flattened, or partially flattened, are also twisted into exceedingly fine spirals, and in this form they are the basis of numerous ornamental applications. Such spirals drawn out till they present a waved appearance, and in that state flattened, are much used for rich heavy embroideries termed karchobs. Spangles for embroideries, &c., are made from spirals of comparatively stout wire, by cutting them down ring by ring, laying each C-like ring on an anvil, and by a smart blow with a hammer flattening it out into a thin round disk with a slit extending from the centre to one edge. Fine spirals are also used for general embroidery purposes. The demand for various kinds of loom-woven and embroidered gold and silver work in India is immense; and the variety of textiles so ornamented is also very great. "Gold and silver," says Dr Birdwood in his ''Handbook to the British-Indian Section, Paris Exhibition'', 1878, "are worked into the decoration of all the more costly loom-made garments and Indian piece goods, either on the borders only, or in stripes throughout, or in diapered figures. The gold-bordered loom embroideries are made chiefly at Sattara, and the gold or silver striped at Tanjore; the gold figured ''mashrus'' at Tanjore, Trichinopoly, and Hyderabad in the Deccau; and the highly ornamented gold-figured silks and gold and silver tissues principally at Ahmedabad, Benares, Murshedabad, and Trichinopoly." Among the Western communities the demand for gold and silver lace and embroideries arises chiefly in connexion with naval and military uniforms, court costumes, public and private liveries, ecclesiastical robes and draperies, theatrical dresses, and the badges and insignia of various orders. To a limited extent there is a trade in gold wire and lace to India and China. The metallic basis of the various fabrics is wire round and flattened, the wire being of three kinds — 1st, gold wire, which is invariably silver gilt wire; 2d, copper gilt wire, used for common liveries and theatrical purposes; and 3d, silver wire. These wires are drawn by the ordinary processes, and the flattening, when done, is accomplished by passing the wire between a pair of revolving rollers of fine polished steel. The various qualities of wire are prepared and used in precisely the same way as in India, — round wire, flat wire, thread made of flat gold wire twisted round orange-coloured silk or cotton, known in the trade as "orris," fine spirals and spangles, all being in use in the West as in the East. The lace is woven in the same manner as ribbons, and there are very numerous varieties in richness, pattern, and quality. Cloth of gold, and brocades rich in gold and silver, are woven for ecclesiastical vestments and draperies. The proportions of gold and silver in the gold thread for the lace trade varies, but in all cases the proportion of gold is exceedingly small. An ordinary gold lace wire is drawn from a bar containing 90 parts of silver and 7 of copper, coated with 3 parts of gold. On an average each ounce troy of a bar so plated is drawn into 1500 yards of wire; and therefore about 16 grains of gold cover a mile of wire. It is estimated that about 250,000 ounces of gold wire are made annually in Great Britain, of which about 20 per cent, is used for the headings of calico, muslin, &c., and the remainder is worked up in the gold lace trade.<ref>William Chandler Roberts-Austen and H. Bauerman [W.C.R. — H.B.]. "Gold and Silver Lace." In "Gold." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. 10 (X). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2010%20%28G-GOT%29%20193592738.23/page/753/mode/1up (accessed January 2023): 753, Col. 2c – 754, Cols. 1a–b – 2a–b.</ref></blockquote> ==== Honiton Lace ==== Kate Stradsin says,<blockquote>Honiton lace was the finest English equivalent of Brussels bobbin lace and was constructed in small ‘sprigs, in the cottages of lacemakers[.'] These sprigs were then joined together and bleached to form the large white flounces that were so sought after in the mid-nineteenth century.<ref>Strasdin, Kate. "Rediscovering Queen Alexandra’s Wardrobe: The Challenges and Rewards of Object-Based Research." ''The Court Historian'' 24.2 (2019): 181-196. Rpt http://repository.falmouth.ac.uk/3762/15/Rediscovering%20Queen%20Alexandra%27s%20Wardrobe.pdf: 13, and (for the little quotation) n. 37, which reads "Margaret Tomlinson, ''Three Generations in the Honiton Lace Trade: A Family History'', self-published, 1983."</ref></blockquote> [[File:Strook in Alençon naaldkant, 1750-1775.jpg|thumb|alt=A long piece of complex white lace with garlands, flowers and bows|Point d'Alençon lace, 1750-1775]] ==== Passementerie ==== ''Passementerie'' is the French term for trim on clothing or furniture. The 19th century (especially during the First and Second Empire) was a time of great "''exubérance''" in passementerie in French design, including the development and widespread use of the Jacquard loom.<ref>{{Cite journal|date=2023-06-10|title=Passementerie|url=https://fr.wikipedia.org/w/index.php?title=Passementerie&oldid=205068926|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Passementerie.</ref> ==== Point d'Alençon Lace ==== A lace made by hand using a number of complex steps and layers. The lacemakers build the point d'Alençon design on some kind of mesh and sometimes leave some of the mesh in as part of the lace and perhaps to provide structure. Elizabeth Lewandowski defines point d'Alençon lace and Alençon lace separately. Point lace is needlepoint lace,<ref name=":7">Lewandowski, Elizabeth J. ''The Complete Costume Dictionary''. Scarecrow Press, 2011.</ref>{{rp|233}} so Alençon point is "a two thread [needlepoint] lace."<ref name=":7" />{{rp|7}} Alençon lace has a "floral design on [a] fine net ground [and is] referred to as [the] queen of French handmade needlepoint laces. The original handmade Alençon was a fine needlepoint lace made of linen thread."<ref name=":7" />{{rp|7}} The sample of point d'Alençon lace (right), from 1750–1775, shows the linen mesh that the lace was constructed on.<ref>{{Cite web|url=http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689|title=MoMu - Open Fashion|website=openfashion.momu.be|access-date=2024-02-26}} ModeMuseum Antwerpen. http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689.</ref> The consistency in this sample suggests it may have been made by machine. == Fabric == === Brocatelle === Brocatelle is a kind of brocade, more simple than most brocades because it uses fewer warp and weft threads and fewer colors to form the design. The article in the French ''Wikipédia'' defines it like this:<blockquote>La '''brocatelle''' est un type de tissu datant du <abbr>xvi^e</abbr> siècle qui comporte deux chaînes et deux trames, au minimum. Il est composé pour que le dessin ressorte avec un relief prononcé, grâce à la chaîne sur un fond en sergé. Les brocatelles les plus anciennes sont toujours fabriquées avec une des trames en lin.<ref>{{Cite journal|date=2023-06-01|title=Brocatelle|url=https://fr.wikipedia.org/w/index.php?title=Brocatelle&oldid=204796410|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Brocatelle.</ref></blockquote>Which translates to this:<blockquote>Brocatelle is a type of fabric dating from the 16th century that has two warps and two wefts, at a minimum. It is composed so that the design stands out with a pronounced relief, thanks to the weft threads on a twill background. The oldest brocades were always made with one of the wefts being linen.</blockquote>The ''Oxford English Dictionary'' says, brocatelle is an "imitation of brocade, usually made of silk or wool, used for tapestry, upholstery, etc., now also for dresses. Both the nature and the use of the stuff have changed" between the late 17th century and 1888, the last time this definition was revised.<ref>"brocatelle, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/23550. Accessed 4 July 2023.</ref> === Broché === === Ciselé === === Crépe de Chine === The ''Oxford English Dictionary'' distinguishes the use of ''crêpe'' (using a circumflex rather than an acute accent over the first ''e'') from ''crape'' in textiles, saying ''crêpe'' is "often borrowed [from the French] as a term for all crapy fabrics other than ordinary black mourning crape,"<ref>"crêpe, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44242. Accessed 10 February 2023.</ref> with usage examples ranging from 1797 to the mid 20th century. Crêpe de chine, it says is "a white or other coloured crape made of raw silk." === Épinglé Velvet === Often spelled ''épingle'' rather than ''épinglé'', this term appears to have been used for a fabric made of wool, or at least wool along with linen or cotton, that was heavier and stiffer than silk velvet. It was associated with outer garments and men's clothing. Nowadays, épinglé velvet is an upholstery fabric in which the pile is cut into designs and patterns, and the portrait of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary, Duchess of Hamilton]] shows a mantle described as épinglé velvet that does seem to be a velvet with a woven pattern perhaps cut into the pile. === Lace === While lace also functioned sometimes as fabric — at the décolletage, for example, on the stomacher or as a veil — here we organize it as a [[Social Victorians/Terminology#Trim and Lace|part of the elaboration of clothing]]. === Liberty Fabrics === === Lisse === According to the ''Oxford English Dictionary'', the term ''lisse'' as a "kind of silk gauze" was used in the 19th-century UK and US.<ref>"lisse, n.1." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/108978. Accessed 4 July 2023.</ref> === Satin === The pre-1927 ''History of Feminine Fashion'', probably commissioned by Charles Frederick Worth's sons, describes Worth's "insistence on new fabrics, new trimmings, new materials of every description" at the beginning of his career in the mid 19th century:<blockquote>When Worth first entered the business of dressmaking, the only materials of the richer sort used for woman's dress were velvet, faille, and watered silk. Satin, for example, was never used. M. Worth desired to use satin very extensively in the gowns he designed, but he was not satisfied with what could be had at the time; he wanted something very much richer than was produced by the mills at Lyons. That his requirements entailed the reconstruction of mills mattered little — the mills were reconstructed under his directions, and the Lyons looms turned out a richer satin than ever, and the manufacturers prospered accordingly.<ref name=":9">[Worth, House of.] {{Cite book|url=http://archive.org/details/AHistoryOfFeminineFashion|title=A History Of Feminine Fashion (1800s to 1920s)}} Before 1927. [Likely commissioned by Worth. Link is to Archive.org; info from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Worth_Biarritz_salon.jpg.]</ref>{{rp|6 in printed, 26 in digital book}}</blockquote> === Selesia === According to the ''Oxford English Dictionary'', ''silesia'' is "A fine linen or cotton fabric originally manufactured in Silesia in what is now Germany (''Schlesien'').<ref>"Silesia, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/179664. Accessed 9 February 2023.</ref> It may have been used as a lining — for pockets, for example — in garments made of more luxurious or more expensive cloth. The word ''sleazy'' — "Of textile fabrics or materials: Thin or flimsy in texture; having little substance or body."<ref>"sleazy, adj." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/181563. Accessed 9 February 2023.</ref> — may be related. === Shot Fabric === According to the ''Oxford English Dictionary'', "Of a textile fabric: Woven with warp-threads of one colour and weft-threads of another, so that the fabric (usually silk) changes in tint when viewed from different points."<ref>“Shot, ''Adj.''”  ''Oxford English Dictionary'', Oxford UP,  July 2023, https://doi.org/10.1093/OED/2977164390.</ref> A shot fabric might also be made of silk and cotton fibers. === Tissue === A lightly woven fabric like gauze or chiffon. The light weave can make the fabric translucent and make pleating and gathering flatter and less bulky. Tissue can be woven to be shot, sheer, stiff or soft. Historically, the term in English was used for a "rich kind of cloth, often interwoven with gold or silver" or "various rich or fine fabrics of delicate or gauzy texture."<ref>“Tissue, N.” ''Oxford English Dictionary'', Oxford UP, March 2024, https://doi.org/10.1093/OED/5896731814.</ref> == Fan == The ''Encyclopaedia Britannica'' (9th edition) has an article on the fan. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>FAN (Latin, ''vannus''; French, ''éventail''), a light implement used for giving motion to the air. ''Ventilabrum'' and ''flabellum'' are names under which ecclesiastical fans are mentioned in old inventories. Fans for cooling the face have been in use in hot climates from remote ages. A bas-relief in the British Museum represents Sennacherib with female figures carrying feather fans. They were attributes of royalty along with horse-hair fly-flappers and umbrellas. Examples may be seen in plates of the Egyptian sculptures at Thebes and other places, and also in the ruins of Persepolis. In the museum of Boulak, near Cairo, a wooden fan handle showing holes for feathers is still preserved. It is from the tomb of Amen-hotep, of the 18th dynasty, 17th century <small>B</small>.<small>C</small>. In India fans were also attributes of men in authority, and sometimes sacred emblems. A heartshaped fan, with an ivory handle, of unknown age, and held in great veneration by the Hindus, was given to the prince of Wales. Large punkahs or screens, moved by a servant who does nothing else, are in common use by Europeans in India at this day. Fans were used in the early Middle Ages to keep flies from the sacred elements during the celebrations of the Christian mysteries. Sometimes they were round, with bells attached — of silver, or silver gilt. Notices of such fans in the ancient records of St Paul’s, London, Salisbury cathedral, and many other churches, exist still. For these purposes they are no longer used in the Western church, though they are retained in some Oriental rites. The large feather fans, however, are still carried in the state processions of the supreme pontiff in Rome, though not used during the celebration of the mass. The fan of Queen Theodolinda (7th century) is still preserved in the treasury of the cathedral of Monza. Fans made part of the bridal outfit, or ''mundus muliebris'', of ancient Roman ladies. Folding fans had their origin in Japan, and were imported thence to China. They were in the shape still used—a segment of a circle of paper pasted on a light radiating frame-work of bamboo, and variously decorated, some in colours, others of white paper on which verses or sentences are written. It is a compliment in China to invite a friend or distinguished guest to write some sentiment on your fan as a memento of any special occasion, and this practice has continued. A fan that has some celebrity in France was presented by the Chinese ambassador to the Comtesse de Clauzel at the coronation of Napoleon I. in 1804. When a site was given in 1635, on an artificial island, for the settlement of Portuguese merchants in Nippo in Japan, the space was laid out in the form of a fan as emblematic of an object agreeable for general use. Men and women of every rank both in China and Japan carry fans, even artisans using them with one hand while working with the other. In China they are often made of carved ivory, the sticks being plates very thin and sometimes carved on both sides, the intervals between the carved parts pierced with astonishing delicacy, and the plates held together by a ribbon. The Japanese make the two outer guards of the stick, which cover the others, occasionally of beaten iron, extremely thin and light, damascened with gold and other metals. Fans were used by Portuguese ladies in the 14th century, and were well known in England before the close of the reign of Richard II. In France the inventory of Charles V. at the end of the 14th century mentions a folding ivory fan. They were brought into general use in that country by Catherine de’ Medici, probably from Italy, then in advance of other countries in all matters of personal luxury. The court ladies of Henry VIII.’s reign in England were used to handling fans, A lady in the Dance of Death by Holbein holds a fan. Queen Elizabeth is painted with a round leather fan in her portrait at Gorhambury; and as many as twenty-seven are enumerated in her inventory (1606). Coryat, an English traveller, in 1608 describes them as common in Italy. They also became of general use from that time in Spain. In Italy, France, and Spain fans had special conventional uses, and various actions in handling them grew into a code of signals, by which ladies were supposed to convey hints or signals to admirers or to rivals in society. A paper in the ''Spectator'' humorously proposes to establish a regular drill for these purposes. The chief seat of the European manufacture of fans during the 17th century was Paris, where the sticks or frames, whether of wood or ivory, were made, and the decorations painted on mounts of very carefully prepared vellum (called latterly ''chicken skin'', but not correctly), — a material stronger and tougher than paper, which breaks at the folds. Paris makers exported fans unpainted to Madrid and other Spanish cities, where they were decorated by native artists. Many were exported complete; of old fans called Spanish a great number were in fact made in France. Louis XIV. issued edicts at various times to regulate the manufacture. Besides fans mounted with parchment, Dutch fans of ivory were imported into Paris, and decorated by the heraldic painters in the process called “Vernis Martin,” after a famous carriage painter and inventor of colourless lac varnish. Fans of this kind belonging to the Queen and to the late baroness de Rothschild were exhibited in 1870 at Kensington. A fan of the date of 1660, representing sacred subjects, is attributed to Philippe de Champagne, another to Peter Oliver in England in the / 17th century. Cano de Arevalo, a Spanish painter of the 17th century devoted himself to fan painting. Some harsh expressions of Queen Christina to the young ladies of the French court are said to have caused an increased ostentation in the splendour of their fans, which were set with jewels and mounted in gold. Rosalba Carriera was the name of a fan painter of celebrity in the 17th century. Lebrun and Romanelli were much employed during the same period. Klingstet, a Dutch artist, enjoyed a considerable reputation for his fans from the latter part of the 17th and the first thirty years of the 18th century. The revocation of the edict of Nantes drove many fan-makers out of France to Holland and England. The trade in England was well established under the Stuart sovereigns. Petitions were addressed by the fan-makers to Charles II. against the importation of fans from India, and a duty was levied upon such fans in consequence. This importation of Indian fans, according to Savary, extended also to France. During the reign of Louis XV. carved Indian and China fans displaced to some extent those formerly imported from Italy, which had been painted on swanskin parchment prepared with various perfumes. During the 18th century all the luxurious ornamentation of the day was bestowed on fans as far as they could display it. The sticks were made of mother-of-pearl or ivory, carved with extraordinary skill in France, Italy, England, and other countries. They were painted from designs of Boucher, Watteau, Lancret, and other "genre" painters, Hébert, Rau, Chevalier, Jean Boquet, Mad. Verité, are known as fan painters. These fashions were followed in most countries of Europe, with certain national differences. Taffeta and silk, as well as fine parchment, were used for the mounts. Little circles of glass were let into the stick to be looked through, and small telescopic glasses were sometimes contrived at the pivot of the stick. They were occasionally mounted with the finest point lace. An interesting fan (belonging to Madame de Thiac in France), the work of Le Flamand, was presented by the municipality of Dieppe to Marie Antoinette on the birth of her son the dauphin. From the time of the Revolution the old luxury expended on fans died out. Fine examples ceased to be exported to England and other countries. The painting on them represented scenes or personages connected with political events. At a later period fan mounts were often prints coloured by hand. The events of the day mark the date of many examples found in modern collections. Amongst the fanmakers of the present time the names of Alexandre, Duvelleroy, Fayet, Vanier, may be mentioned as well known in Paris. The sticks are chiefly made in the department of Oise, at Le Déluge, Crèvecœur, Méry, Ste Geneviève, and other villages, where whole families are engaged in preparing them; ivory sticks are carved at Dieppe. Water-colour painters of distinction often design and paint the mounts, the best designs being figure subjects. A great impulse has been given to the manufacture and painting of fans in England since the exhibition which took place at South Kensington in 1870. Other exhibitions have since been held, and competitive prizes offered, one of which was gained by the Princess Louise. Modern collections of fans take their date from the emigration of many noble families from France at the time of the Revolution. Such objects were given as souvenirs and occasionally sold by families in straitened circumstances. A large number of fans of all sorts, principally those of the 18th century, French, English, German, Italian Spanish, &c., have been lately bequeathed to the South Kensington Museum. Regarding the different parts of folding fans it may be well to state that the sticks are called in French ''brins'', the two outer guards ''panaches'', and the mount ''feuille''.<ref>J. H. Pollen [J.H.P.]. "Fan." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. '''10''' ('''X'''). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%209%20%28FAL-FYZ%29%20193323016.23/page/26/mode/2up (accessed January 2023): 27, Col. 1b – 28, Col. 1c.</ref></blockquote> == Fancy-dress Ball == Fancy-dress (or costume) balls were popular and frequent in the U.K. and France as well as the rest of Europe during the 19th century. The themes and styles of the fancy-dress balls influenced those that followed. At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the guests came dressed in costume from times before 1820, as instructed on '''the invitation''', but their clothing was much more about late-Victorian standards of beauty and fashion than the standards of whatever time period the portraits they were copying or basing their costumes on. ''The Queen'' published dress and fashion information and advice under the byline of Ardern Holt, who regularly answered questions from readers about fashion as well as about fancy dress. (More about Ardern Holt, which is almost certainly a pseudonym, can be found on the [[Social Victorians/People/Working in Publishing#Journalists|People Working in Publishing]] page.) Holt also ran wrote entire articles with suggestions for what might make an appealing fancy-dress costume as well as pointing readers away from costumes that had been worn too frequently. The suggestions for costumes are based on familiar types or portraits available to readers, similar to Holt's books on fancy dress, which ran through a number of editions in the 1880s and 1890s. Fancy-dress questions sometimes asked for details about costumes worn in theatrical or operatic productions, which Holt provides. In November 1897, Holt refers to the Duchess of Devonshire's 2 July ball: "Since the famous fancy ball, given at Devonshire House during this year, historical fancy dresses have assumed a prominence that they had not hitherto known."<ref>Holt, Ardern. "Fancy Dress a la Mode." The ''Queen'' 27 November 1897, Saturday: 94 [of 145 in BNA; print p. 1026], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971127/459/0094.</ref> Holt goes on to provide a number of ideas for costumes for historical fancy dress, as always with a strong leaning toward Victorian standards of beauty and style and away from any concern for historical accuracy. Ardern Holt published books on fancy dress as well as writing for the ''Queen'' and other periodicals, but not all of them were about fancy dress. # ''Gentlemen's Fancy Dress: How to Choose It''. Wyman & Sons, 1882. (''Google Books'' has this: https://books.google.com/books/about/Gentlemen_s_Fancy_Dress.html?id=ED8CAAAAQAAJ.) Later editions: 1898 (HathiTrust) # ''Fancy Dresses Described; Or, What to Wear at Fancy Balls''. Debenham & Freebody, 1882. Illustr., Lillian Young. (HathiTrust has this.) Later editions: 4th ed — 1884; 1887 (HathiTrust); 6th ed. — 1896 (HathiTrust) As Leonore Davidoff says, "Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration symbolised some status category for the female wearer."<ref name=":1" />{{rp|93}} [handled under Elaborations] === Historical Accuracy === Many of the costumes at the ball were based on portraits, especially when the guest was dressed as a historical figure. If possible, we have found the portraits likely to have been the originals, or we have found, if possible, portraits that show the subjects from the two time periods at similar ages. The way clothing was cut changed quite a bit between the 18th and 19th centuries. We think of Victorian clothing — particularly women's clothing, and particularly at the end of the century — as inflexible and restrictive, especially compared to 20th- and 21st-century customs permitting freedom of movement. The difference is generally evolutionary rather than absolute — that is, as time has passed since the 18th century, clothing has allowed an increasingly greater range of movement, especially for people who did not do manual labor. By the end of the 19th century, garments like women's bodices and men's coats were made fitted and smooth by attention to the grain of the fabric and by the use of darts (rather than techniques that assembled many small, individual pieces of fabric). * clothing construction and flat-pattern techniques * Generally, the further back in time we go, the more 2-dimensional the clothing itself was. ==== Women's Versions of Historical Accuracy at the Ball ==== As always with this ball, whatever historical accuracy might be present in a woman's costume is altered so that the wearer is still a fashionable Victorian lady. What makes the costumes look "Victorian" to our eyes is the line of the silhouette caused by the foundation undergarments as well as the many "elaborations"<ref name=":1" />{{rp|93}}, mostly in the decorations, trim and accessories. Also, the clothing hangs and drapes differently because the fabric was cut on grain and the shoulders were freed by the way the sleeves were set in. ==== Men's Versions of Historical Accuracy at the Ball ==== Because men were not wearing a Victorian foundation garment at the end of the century, the men's costumes at the ball are more historically accurate in some ways. * Trim * Mixing neck treatments * Hair * Breeches * Shoes and boots * Military uniforms, arms, gloves, boots == Feathers and Plumes == === Aigrette === Elizabeth Lewandowski defines ''aigrette'' as "France. Feather or plume from an egret or heron."<ref name=":7" />(5) Sometimes the newspapers use the term to refer to an accessory (like a fan or ornament on a hat) that includes such a feather or plume. The straight and tapered feathers in an aigrette are in a bundle. === Prince of Wales's Feathers or White Plumes === The feathers in an aigrette came from egrets and herons; Prince of Wales's feathers came from ostriches. A fuller discussion of Prince of Wales's feathers and the white ostrich plumes worn at court appears on [[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|Victorian Things]]. For much of the late 18th and 19th centuries, white ostrich plumes were central to fashion at court, and at a certain point in the late 18th century they became required for women being presented to the monarch and for their sponsors. Our purpose here is to understand why women were wearing plumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] as part of their costumes. First published in 1893, [[Social Victorians/People/Lady Colin Campbell|Lady Colin Campbell]]'s ''Manners and Rules of Good Society'' (1911 edition) says that<blockquote>It was compulsory for both Married and Unmarried Ladies to Wear Plumes. The married lady’s Court plume consisted of three white feathers. An unmarried lady’s of two white feathers. The three white feathers should be mounted as a Prince of Wales plume and worn towards the left hand side of the head. Colored feathers may not be worn. In deep mourning, white feathers must be worn, black feathers are inadmissible.<p> White veils or lace lappets must be worn with the feathers. The veils should not be longer than 45 inches.<ref>{{Cite web|url=https://www.edwardianpromenade.com/etiquette/the-court-presentation/|title=The Court Presentation|last=Holl|first=Evangeline|date=2007-12-07|website=Edwardian Promenade|language=en-US|access-date=2022-12-18}} https://www.edwardianpromenade.com/etiquette/the-court-presentation/.</ref></blockquote>[[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|This fashion was imported from France]] in the mid 1770s.<ref>"Abstract" for Blackwell, Caitlin. "'<nowiki/>''The Feather'd Fair in a Fright''': The Emblem of the Feather in Graphic Satire of 1776." ''Journal for Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. ''Wiley Online'' DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x (accessed November 2022).</ref> Separately, a secondary heraldic emblem of the Prince of Wales has been a specific arrangement of 3 ostrich feathers in a gold coronet<ref>{{Cite journal|date=2022-11-07|title=Prince of Wales's feathers|url=https://en.wikipedia.org/w/index.php?title=Prince_of_Wales%27s_feathers&oldid=1120556015|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Prince_of_Wales's_feathers.</ref> since King Edward III (1312–1377<ref>{{Cite journal|date=2022-12-14|title=Edward III of England|url=https://en.wikipedia.org/w/index.php?title=Edward_III_of_England&oldid=1127343221|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_III_of_England.</ref>). Some women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] wore white ostrich feathers in their hair, but most of them are not Prince of Wales's feathers. Most of the plumes in these portraits are arrangements of some kind of headdress to accompany the costume. A few, wearing what looks like the Princes of Wales's feathers, might be signaling that their character is royal or has royal ancestry. '''One of the women [which one?] was presented to the royals at this ball?''' Here is the list of women who are wearing white ostrich plumes in their portraits in the [[Social Victorians/1897 Fancy Dress Ball/Photographs|''Diamond Jubilee Fancy Dress Ball'' album of 286 photogravure portraits]]: # Kathleen Pelham-Clinton, the [[Social Victorians/People/Newcastle|Duchess of Newcastle]] # [[Social Victorians/People/Louisa Montagu Cavendish|Luise Cavendish]], the Duchess of Devonshire # Jesusa Murrieta del Campo Mello y Urritio (née Bellido), [[Social Victorians/People/Santurce|Marquisa de Santurce]] # Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]] # Princess (Laura Williamina Seymour) Victor of  [[Social Victorians/People/Gleichen#Laura%20Williamina%20Seymour%20of%20Hohenlohe-Langenburg|Hohenlohe Langenburg]] # Louisa Acheson, [[Social Victorians/People/Gosford|Lady Gosford]] # Alice Emily White Coke, [[Social Victorians/People/Leicester|Viscountess Coke]] # Lady Mary Stewart, Helen Mary Theresa [[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]] #[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], Duchess of [[Social Victorians/People/Marlborough|Marlborough]], dressed as the wife of the French Ambassador at the Court of Catherine of Russia (not white, but some color that reads dark in the black-and-white photograph) #Mrs. Mary [[Social Victorians/People/Chamberlain|Chamberlain]] (at 491), wearing white plumes, as Madame d'Epinay #Lady Clementine [[Social Victorians/People/Tweeddale|Hay]] (at 629), wearing white plumes, as St. Bris (''Les Huguenots'') #[[Social Victorians/People/Meysey-Thompson|Lady Meysey-Thompson]] (at 391), wearing white plumes, as Elizabeth, Queen of Bohemia #Mrs. [[Social Victorians/People/Grosvenor|Algernon (Catherine) Grosvenor]] (at 510), wearing white plumes, as Marie Louise #Lady [[Social Victorians/People/Ancaster|Evelyn Ewart]], at 401), wearing white plumes, as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson #[[Social Victorians/People/Lyttelton|Edith Sophy Balfour Lyttelton]] (at 580), wearing what might be white plumes on a large-brimmed white hat, after a picture by Romney #[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), wearing what might be white plumes, as Cydalise of the Comedie Italienne from the time of Louis XV #Lady [[Social Victorians/People/Ilchester|Muriel Fox Strangways]] (at 403), wearing what might be two smallish white plumes, as Lady Sarah Lennox, one of the bridesmaids of Queen Charlotte A.D. 1761 #Lady [[Social Victorians/People/Lucan|Violet Bingham]] (at 586), wearing perhaps one white plume in a headdress not related to the Prince of Wales's feathers #Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), wearing a headdress that includes some white plumes, as Lady Burleigh #[[Social Victorians/People/Dupplin|Agnes Blanche Marie Hay-Drummond]] (at 682), in a big headdress topped with white plumes, as Mademoiselle Andrée de Taverney A.D. 1775 #Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]] (at 336), wearing what looks like Prince of Wales's plumes #[[Social Victorians/People/Suffolk|Marguerite Hyde "Daisy" Leiter]] (at 684), wearing what looks like Prince of Wales's plumes #Lady [[Social Victorians/People/Spicer|Margaret Spicer]] (at 281), wearing one smallish white and one black plume, as Countess Zinotriff, Lady-in-Waiting to the Empress Catherine of Russia #Mrs. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 318), wearing what looks like Prince of Wales's plumes, as Elizabeth Cavendish, daughter of Bess of Hardwick #Nellie, [[Social Victorians/People/Kilmorey|Countess of Kilmorey]] (at 207), wearing three tall plumes, 2 white and one dark, as Comtesse du Barri #Daisy, [[Social Victorians/People/Warwick|Countess of Warwick]] (at 53), wearing at least 1 white plume, as Marie Antoinette More men than women were wearing plumes reminiscent of the Prince of Wales's feathers: * ==== Bibliography for Plumes and Prince of Wales's Feathers ==== * Blackwell, Caitlin. "'''The Feather'd Fair in a Fright'<nowiki/>'': The Emblem of the Feather in Graphic Satire of 1776." Journal for ''Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. Wiley Online DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x. * "Prince of Wales's feathers." ''Wikipedia'' https://en.wikipedia.org/wiki/Prince_of_Wales%27s_feathers (accessed November 2022). ['''Add women to this page'''] * Simpson, William. "On the Origin of the Prince of Wales' Feathers." ''Fraser's magazine'' 617 (1881): 637-649. Hathi Trust https://babel.hathitrust.org/cgi/pt?id=chi.79253140&view=1up&seq=643&q1=feathers (accessed December 2022). Deals mostly with use of feathers in other cultures and in antiquity; makes brief mention of feathers and plumes in signs and pub names that may not be associated with the Prince of Wales. No mention of the use of plumes in women's headdresses or court dress. == Honors == === The Bath === The Most Honourable Order of the Bath (GCB, Knight or Dame Grand Cross; KCB or DCB, Knight or Dame Commander; CB, Companion) === The Garter === The Most Noble Order of the Knights of the Garter (KG, Knight Companion; LG, Lady Companion) [[File:The Golden Fleece - collar exhibited at MET, NYC.jpg|thumb|The Golden Fleece collar and pendant for the 2019 "Last Knight" exhibition at the MET, NYC.|alt=Recent photograph of a gold necklace on a wide band, with a gold skin of a sheep hanging from it as a pendant]] === The Golden Fleece === To wear the golden fleece is to wear the insignia of the Order of the Golden Fleece, said to be "the most prestigious and historic order of chivalry in the world" because of its long history and strict limitations on membership.<ref name=":10">{{Cite journal|date=2020-09-25|title=Order of the Golden Fleece|url=https://en.wikipedia.org/w/index.php?title=Order_of_the_Golden_Fleece&oldid=980340875|journal=Wikipedia|language=en}}</ref> The monarchs of the U.K. were members of the originally Spanish order, as were others who could afford it, like the Duke of Wellington,<ref name=":12">Thompson, R[obert]. H[ugh]. "The Golden Fleece in Britain." Publication of the ''British Numismatic Society''. 2009 https://www.britnumsoc.org/publications/Digital%20BNJ/pdfs/2009_BNJ_79_8.pdf (accessed January 2023).</ref> the first Protestant to be admitted to the order.<ref name=":10" /> Founded in 1429/30 by Philip the Good, Duke of Burgundy, the order separated into two branches in 1714, one Spanish and the other Austrian, still led by the House of Habsburg.<ref name=":10" /> [[File:Prince Albert - Franz Xaver Winterhalter 1842.jpg|thumb|1842 Winterhalter portrait of Prince Albert wearing the insignia of the Order of the Golden Fleece, 1842|left|alt=1842 Portrait of Prince Albert by Winterhalter, wearing the insignia of the Golden Fleece]] The photograph (upper right) is of a Polish badge dating from the "turn of the XV and XVI centuries."<ref>{{Citation|title=Polski: Kolana orderowa orderu Złotego Runa, przełom XV i XVI wieku.|url=https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg|date=2019-11-10|accessdate=2023-01-10|last=Wulfstan}}. https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg.</ref> The collar to this Golden Fleece might be similar to the one the [[Social Victorians/People/Spencer Compton Cavendish#The Insignia of the Order of the Golden Fleece|Duke of Devonshire is wearing in the 1897 Lafayette portrait]]. The badges and collars that Knights of the Order actually wore vary quite a bit. The 1842 Franz Xaver Winterhalter portrait (left) of Prince Consort Albert, Victoria's husband and father of the Prince of Wales, shows him wearing the Golden Fleece on a red ribbon around his neck and the star of the Garter on the front of his coat.<ref>Winterhalter, Franz Xaver. ''Prince Albert''. {{Cite web|url=https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61|title=Explore the Royal Collection Online|website=www.rct.uk|access-date=2023-01-16}} https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61.</ref> === Royal Victorian Order === (GCVO, Knight or Dame Grand Cross; KCVO or DCVO, Knight or Dame Commander; CVO, Commander; LVO, Lieutenant; MVO, Member) === St. John === The Order of the Knights of St. John === Star of India === Most Exalted Order of the Star of India (GCSI, Knight Grand Commander; KCSI, Knight Commander; CSI, Companion) === Thistle === The Most Ancient and Most Noble Order of the Thistle == Jewelry and Stones == === Cabochon === This term describes both the treatment and shape of a precious or semiprecious stone. A cabochon treatment does not facet the stone but merely polishes it, removing "the rough parts" and the parts that are not the right stone.<ref>"cabochon, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/25778. Accessed 7 February 2023.</ref> A cabochon shape is often flat on one side and oval or round, forming a mound in the setting. === Jet === === ''Orfèvrerie'' === Sometimes misspelled in the newspapers as ''orvfèvrerie''. ''Orfèvrerie'' is the artistic work of a goldsmith, silversmith, or jeweler. === Turquoises === == Military == Several men from the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]] were dressed in military uniforms, some historical and some, possibly, not. === Baldric === According to the ''Oxford English Dictionary'', the primary sense of ''baldric'' is "A belt or girdle, usually of leather and richly ornamented, worn pendent from one shoulder across the breast and under the opposite arm, and used to support the wearer's sword, bugle, etc."<ref>"baldric, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/14849. Accessed 17 May 2023.</ref> This sense has been in existence since c. 1300. === Cuirass === According to the ''Oxford English Dictionary'', the primary sense of ''cuirass'' is "A piece of armour for the body (originally of leather); ''spec.'' a piece reaching down to the waist, and consisting of a breast-plate and a back-plate, buckled or otherwise fastened together ...."<ref>"cuirass, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/45604. Accessed 17 May 2023.</ref> [[File:Knötel IV, 04.jpg|thumb|alt=An Old drawing in color of British soldiers on horses brandishing swords in 1815.|1890 illustration of the Household Cavalry (Life Guard, left; Horse Guard, right) at the Battle of Waterloo, 1815]] === Household Cavalry === The Royal Household contains the Household Cavalry, a corps of British Army units assigned to the monarch. It is made up of 2 regiments, the Life Guards and what is now called The Blues and Royals, which were formed around the time of "the Restoration of the Monarchy in 1660."<ref name=":3">Joll, Christopher. "Tales of the Household Cavalry, No. 1. Roles." The Household Cavalry Museum, https://householdcavalry.co.uk/app/uploads/sites/2/2021/06/Household-Cavalry-Museum-video-series-large-print-text-Tales-episode-01.pdf.</ref>{{rp|1}} Regimental Historian Christopher Joll says, "the original Life Guards were formed as a mounted bodyguard for the exiled King Charles II, The Blues were raised as Cromwellian cavalry and The Royals were established to defend Tangier."<ref name=":3" />{{rp|1–2}} The 1st and 2nd Life Guards were formed from "the Troops of Horse and Horse Grenadier Guards ... in 1788."<ref name=":3" />{{rp|3}} The Life Guards were and are still official bodyguards of the queen or king, but through history they have been required to do quite a bit more than serve as bodyguards for the monarch. The Household Cavalry fought in the Battle of Waterloo on Sunday, 18 June 1815 as heavy cavalry.<ref name=":3" />{{rp|3}} Besides arresting the Cato Steet conspirators in 1820 "and guarding their subsequent execution," the Household Cavalry contributed to the "the expedition to rescue General Gordon, who was trapped in Khartoum by The Mahdi and his army of insurgents" in 1884.<ref name=":3" />{{rp|3}} In 1887 they "were involved ... in the suppression of rioters in Trafalgar Square on Bloody Sunday."<ref name=":3" />{{rp|3}} ==== Grenadier Guards ==== Three men — [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Lord Algernon Gordon-Lennox]], [[Social Victorians/People/Stanley#Edward George Villiers Stanley, Lord Stanley|Lord Stanley]], and [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|Hon. F. C. Stanley]] — attended the ball as officers of the Grenadier Guards, wearing "scarlet tunics, ... full blue breeches, scarlet hose and shoes, lappet wigs" as well as items associated with weapons and armor.<ref name=":14">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 34, Col. 2a}} Founded in England in 1656 as Foot Guards, this infantry regiment "was granted the 'Grenadier' designation by a Royal Proclamation" at the end of the Napoleonic Wars.<ref>{{Cite journal|date=2023-04-22|title=Grenadier Guards|url=https://en.wikipedia.org/w/index.php?title=Grenadier_Guards&oldid=1151238350|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Grenadier_Guards.</ref> They were not called Grenadier Guards, then, before about 1815. In 1660, the Stuart Restoration, they were called Lord Wentworth's Regiment, because they were under the command of Thomas Wentworth, 5th Baron Wentworth.<ref>{{Cite journal|date=2022-07-24|title=Lord Wentworth's Regiment|url=https://en.wikipedia.org/w/index.php?title=Lord_Wentworth%27s_Regiment&oldid=1100069077|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Wentworth%27s_Regiment.</ref> At the time of Lord Wentworth's Regiment, the style of the French cavalier had begun to influence wealthy British royalists. In the British military, a Cavalier was a wealthy follower of Charles I and Charles II — a commander, perhaps, or a field officer, but probably not a soldier.<ref>{{Cite journal|date=2023-04-22|title=Cavalier|url=https://en.wikipedia.org/w/index.php?title=Cavalier&oldid=1151166569|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier.</ref> The Guards were busy as infantry in the 17th century, engaging in a number of armed conflicts for Great Britain, but they also served the sovereign. According to the Guards Museum,<blockquote>In 1678 the Guards were ordered to form Grenadier Companies, these men were the strongest and tallest of the regiment, they carried axes, hatches and grenades, they were the shock troops of their day. Instead of wearing tri-corn hats they wore a mitre shaped cap.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/|title=Service to the Crown|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/.</ref></blockquote>The name comes from ''grenades'', then, and we are accustomed to seeing them in front of Buckingham Palace, with their tall mitre hats. The Guard fought in the American Revolution, and in the 19th century, the Grenadier Guards fought in the Crimean War, Sudan and the Boer War. They have roles as front-line troops and as ceremonial for the sovereign, which makes them elite:<blockquote>Queen Victoria decreed that she did not want to see a single chevron soldier within her Guards. Other then [sic] the two senior Warrant Officers of the British Army, the senior Warrant Officers of the Foot Guards wear a large Sovereigns personal coat of arms badge on their upper arm. No other regiments of the British Army are allowed to do so; all the others wear a small coat of arms of their lower arms. Up until 1871 all officers in the Foot Guards had the privilege of having double rankings. An Ensign was ranked as an Ensign and Lieutenant, a Lieutenant as Lieutenant and Captain and a Captain as Captain and Lieutenant Colonel. This was because at the time officers purchased their own ranks and it cost more to purchase a commission in the Foot Guards than any other regiments in the British Army. For example if it cost an officer in the Foot Guards £1,000 for his first rank, in the rest of the Army it would be £500 so if he transferred to another regiment he would loose [sic] £500, hence the higher rank, if he was an Ensign in the Guards and he transferred to a Line Regiment he went in at the higher rank of Lieutenant.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/|title=Formation and role of the Regiments|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/.</ref></blockquote> ==== Life Guards ==== [[Social Victorians/People/Shrewsbury#Reginald Talbot's Costume|General the Hon. Reginald Talbot]], a member of the 1st Life Guards, attended the Duchess of Devonshire's ball dressed in the uniform of his regiment during the Battle of Waterloo.<ref name=":14" />{{rp|p. 36, Col. 3b}} At the Battle of Waterloo the 1st Life Guards were part of the 1st Brigade — the Household Brigade — and were commanded by Major-General Lord Edward Somerset.<ref name=":4">{{Cite journal|date=2023-09-30|title=Battle of Waterloo|url=https://en.wikipedia.org/w/index.php?title=Battle_of_Waterloo&oldid=1177893566|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Battle_of_Waterloo.</ref> The 1st Life Guards were on "the extreme right" of a French countercharge and "kept their cohesion and consequently suffered significantly fewer casualties."<ref name=":4" /> == Photography == == Footnotes == {{reflist}} 2d6bnzq0h686akcm12pg8jai6idvdyf 2690311 2690306 2024-12-04T21:19:48Z Scogdill 1331941 /* Hoops */ 2690311 wikitext text/x-wiki Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. The definitions on this page are to provide a sense of what someone in the late 19th century might have meant by the term rather than a definition of what we might mean by it today. In the absence of a specialized glossary from the end of the 19th century in the U.K., we use the ''Oxford English Dictionary'' because the senses of a word are illustrated with examples that have dates so we can be sure that the senses we pick are appropriate for when they are used in the quotations we have. We also sometimes use the French ''Wikipédia'' to define a word because many technical terms of fashion were borrowings from the French. Also, often the French ''Wikipédia'' provides historical context for the uses of a word similar to the way the OED does. == Articles or Parts of Clothing: Non-gender-specific == === Mantle, Cloak, Cape === In 19th-century newspaper accounts, these terms are sometimes used without precision as synonyms. These are all outer garments. '''Mantle''' A mantle — often a long outer garment — might have elements like a train, sleeves, collars, revers, fur, and a cape. A late-19th-century writer making a distinction between a mantle and a cloak might use ''mantle'' if the garment is more voluminous. '''Cloak''' '''Cape''' === Peplum === According to the French ''Wiktionnaire'', a peplum is a "Short skirt or flared flounce layered at the waist of a jacket, blouse or dress" [translation by Google Translate].<ref>{{Cite journal|date=2021-07-02|title=péplum|url=https://fr.wiktionary.org/w/index.php?title=p%C3%A9plum&oldid=29547727|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/p%C3%A9plum.</ref> The ''Oxford English Dictionary'' has a fuller definition, although, it focuses on women's clothing because the sense is written for the present day:<blockquote>''Fashion''. ... a kind of overskirt resembling the ancient peplos (''obsolete''). Hence (now usually) in modern use: a short flared, gathered, or pleated strip of fabric attached at the waist of a woman's jacket, dress, or blouse to create a hanging frill or flounce.<ref name=":5">“peplum, n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1832614702>.</ref></blockquote>Men haven't worn peplums since the 18th century, except when wearing costumes based on historical portraits. The ''Daily News'' reported in 1896 that peplums had been revived as a fashion item for women.<ref name=":5" /> === Revers === According to the ''Oxford English Dictionary'', ''revers'' are the "edge[s] of a garment turned back to reveal the undersurface (often at the lapel or cuff) (chiefly in ''plural''); the material covering such an edge."<ref>"revers, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/164777. Accessed 17 April 2023.</ref> The term is French and was used this way in the 19th century (according to the ''Wiktionnaire'').<ref>{{Cite journal|date=2023-03-07|title=revers|url=https://fr.wiktionary.org/w/index.php?title=revers&oldid=31706560|journal=Wiktionnaire|language=fr}} https://fr.wiktionary.org/wiki/revers.</ref> == Articles or Parts of Clothing: Men's == [[Social Victorians/Terminology#Military|Men's military uniforms]] are discussed below. === À la Romaine === [[File:Johann Baptist Straub - Mars um 1772-1.jpg|thumb|left|alt=Old and damaged marble statue of a Roman god of war with flowing cloak, big helmet with a plume on top, and armor|Johann Baptist Straub's 1772 ''à la romaine'' ''Mars'']] A few people who attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball in 1897]] personated Roman gods or people. They were dressed not as Romans, however, but ''à la romaine'', which was a standardized style of depicting Roman figures that was used in paintings, sculpture and the theatre for historical dress from the 17th until the 20th century. The codification of the style was developed in France in the 17th century for theatre and ballet, when it became popular for masked balls. Women as well as men could be dressed ''à la romaine'', but much sculpture, portraiture and theatre offered opportunities for men to dress in Roman style — with armor and helmets — and so it was most common for men. In large part because of the codification of the style as well as the painting and sculpture, the style persisted and remained influential into the 20th century and can be found in museums and galleries and on monuments. For example, Johann Baptist Straub's 1772 statue of Mars (left), now in the Bayerisches Nationalmuseum, Munich, missing part of an arm, shows Mars ''à la romaine''. In London, an early 17th-century example of a figure of Mars ''à la romaine'', with a helmet, '''was''' "at the foot of the Buckingham tomb in Henry VII's Chapel at Westminster Abbey."<ref>Webb, Geoffrey. “Notes on Hubert Le Sueur-II.” ''The Burlington Magazine for Connoisseurs'' 52, no. 299 (1928): 81–89. http://www.jstor.org/stable/863535.</ref>{{rp|81, Col. 2c}} === Cavalier === [[File:Sir-Anthony-van-Dyck-Lord-John-Stuart-and-His-Brother-Lord-Bernard-Stuart.jpg|thumb|alt=Old painting of 2 men flamboyantly and stylishly dressed in colorful silk, with white lace, high-heeled boots and long hair|Van Dyck's c. 1638 painting of cavaliers Lord John Stuart and his brother Lord Bernard Stuart]] As a signifier in the form of clothing of a royalist political and social ideology begun in France in the early 17th century, the cavalier established France as the leader in fashion and taste. Adopted by [[Social Victorians/Terminology#Military|wealthy royalist British military officers]] during the time of the Restoration, the style signified a political and social position, both because of the loyalty to Charles I and II as well the wealth required to achieve the cavalier look. The style spread beyond the political, however, to become associated generally with dress as well as a style of poetry.<ref>{{Cite journal|date=2023-04-25|title=Cavalier poet|url=https://en.wikipedia.org/w/index.php?title=Cavalier_poet&oldid=1151690299|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier_poet.</ref> Van Dyck's 1638 painting of two brothers (right) emphasizes the cavalier style of dress. === Coats === ==== Doublet ==== * In the 19th-century newspaper accounts we have seen that use this word, doublet seems always to refer to a garment worn by a man, but historically women may have worn doublets. In fact, a doublet worn by Queen Elizabeth I exists and '''is somewhere'''. * Technically doublets were long sleeved, although we cannot be certain what this or that Victorian tailor would have done for a costume. For example, the [[Social Victorians/People/Spencer Compton Cavendish#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Duke of Devonshire's costume as Charles V]] shows long sleeves that may be part of the surcoat but should be the long sleeves of the doublet. ==== Pourpoint ==== A padded doublet worn under armor to protect the warrior from the metal chafing. A pourpoint could also be worn without the armor. ==== Surcoat ==== Sometimes just called ''coat''. [[File:Oscar Wilde by Sarony 1882 18.jpg|thumb|alt=Old photograph of a young man wearing a velvet jacket, knee breeches, silk hose and shiny pointed shoes with bows, seated on a sofa and leaning on his left hand and holding a book in his right| Oscar Wilde, 1882, by Napoleon Sarony]] === Hose, Stockings and Tights === Newspaper accounts from the late 19th century of men's clothing use the term ''hose'' for what we might call stockings or tights. In fact, the terminology is specific. ''Stockings'' is the more general term and could refer to hose or tights. With knee breeches men wore hose, which ended above the knee, and women wore hose under their dresses. The ''Oxford English Dictionary'' defines tights as "Tight-fitting breeches, worn by men in the 18th and early 19th centuries, and still forming part of court-dress."<ref>“Tights, N.” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2693287467.</ref> By 1897, the term was in use for women's stockings, which may have come up only to the knee. Tights were also worn by dancers and acrobats. This general sense of ''tights'' does not assume that they were knitted. ''Clocking'' is decorative embroidery on hose, usually, at the ankles on either the inside or the outside of the leg. It started at the ankle and went up the leg, sometimes as far as the knee. On women's hose, the clocking could be quite colorful and elaborate, while the clocking on men's hose was more inconspicuous. In many photographs men's hose are wrinkled, especially at the ankles and the knees, because they were shaped from woven fabric. Silk hose were knitted instead of woven, which gave them elasticity and reduced the wrinkling. The famous Sarony carte de visite photograph of Oscar Wilde (right) shows him in 1882 wearing knee breeches and silk hose, which are shiny and quite smoothly fitted although they show a few wrinkles at the ankles and knees. In the portraits of people in costume at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the men's hose are sometimes quite smooth, which means they were made of knitted silk and may have been smoothed for the portrait. In painted portraits the hose are almost always depicted as smooth, part of the artist's improvement of the appearance of the subject. === Shoes and Boots === == Articles or Parts of Clothing: Women's == === '''Chérusque''' === According to the French ''Wikipedia'', ''chérusque'' is a 19th-century term for the kind of standing collar like the ones worn by ladies in the Renaissance.<ref>{{Cite journal|date=2021-06-26|title=Collerette (costume)|url=https://fr.wikipedia.org/w/index.php?title=Collerette_(costume)&oldid=184136746|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Collerette_(costume)#Au+xixe+siècle+:+la+Chérusque.</ref> === Corsage === According to the ''Oxford English Dictionary'', the corsage is the "'body' of a woman's dress; a bodice."<ref>"corsage, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/42056. Accessed 7 February 2023.</ref> This sense is well documented in the ''OED'' for the mid and late 19th-century, used this way in fiction as well as in a publication like ''Godey's Lady's Book'', which would be expected to use appropriate terminology associated with fashion and dress making. The sense of "a bouquet worn on the bodice" is, according to the ''OED'', American. === Décolletage === === Girdle === === Mancheron === According to the ''Oxford English Dictionary'', a ''mancheron'' is a "historical" word for "A piece of trimming on the upper part of a sleeve on a woman's dress."<ref>"mancheron, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/113251. Accessed 17 April 2023.</ref> At the present, in French, a ''mancheron'' is a cap sleeve "cut directly on the bodice."<ref>{{Cite journal|date=2022-11-28|title=Manche (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manche_(v%C3%AAtement)&oldid=199054843|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Manche_(v%C3%AAtement).</ref> === Petticoat === According to the ''O.E.D.'', a petticoat is a <blockquote>skirt, as distinguished from a bodice, worn either externally or showing beneath a dress as part of the costume (often trimmed or ornamented); an outer skirt; a decorative underskirt. Frequently in ''plural'': a woman's or girl's upper skirts and underskirts collectively. Now ''archaic'' or ''historical''.<ref>“petticoat, n., sense 2.b”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1021034245></ref> </blockquote>This sense is, according to the ''O.E.D.'', "The usual sense between the 17th and 19th centuries." However, while petticoats belong in both outer- and undergarments — that is, meant to be seen or hidden, like underwear — they were always under another garment, for example, underneath an open overskirt. The primary sense seems to have shifted through the 19th century so that, by the end, petticoats were underwear and the term ''underskirt'' was used to describe what showed under an open overskirt. === Stomacher === According to the ''O.E.D.'', a stomacher is "An ornamental covering for the chest (often covered with jewels) worn by women under the lacing of the bodice,"<ref>“stomacher, n.¹, sense 3.a”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1169498955></ref> although by the end of the 19th century, the bodice did not often have visible laces. Some stomachers were so decorated that they were thought of as part of the jewelry. === Train === A train is The Length of the Train '''For the monarch [or a royal?]''' According to Debrett's,<blockquote>A peeress's coronation robe is a long-trained crimson velvet mantle, edged with miniver pure, with a miniver pure cape. The length of the train varies with the rank of the wearer: * Duchess: for rows of ermine; train to be six feet * Marchioness: three and a half rows of ermine; train to be three and three-quarters feet * Countess: three rows of ermine; train to be three and a half feet * Viscountess: two and a half rows of ermine; train to be three and a quarter feet * Baroness: two rows of ermine; train to be three feet<ref name=":2">{{Cite web|url=https://debretts.com/royal-family/dress-codes/|title=Dress Codes|website=debretts.com|language=en-US|access-date=2023-07-27}} https://debretts.com/royal-family/dress-codes/.</ref> </blockquote>The pattern on the coronet worn was also quite specific, similar but not exactly the same for peers and peeresses. Debrett's also distinguishes between coronets and tiaras, which were classified more like jewelry, which was regulated only in very general terms. Peeresses put on their coronets after the Queen or Queen Consort has been crowned. ['''peers?'''] === Foundation Garments === Unlike undergarments, Victorian women's foundation garments created the distinctive silhouette. Victorian undergarments included the chemise, the bloomers, the corset cover — articles that are not structural. The corset was an important element of the understructure of foundation garments — hoops, bustles, petticoats and so on — but it has never been the only important element. === Corset === [[File:Corset - MET 1972.209.49a, b.jpg|thumb|alt=Photograph of an old silk corset on a mannequin, showing the closure down the front, similar to a button, and channels in the fabric for the boning. It is wider at the top and bottom, creating smooth curves from the bust to the compressed waist to the hips, with a long point below the waist in front.|French 1890s corset, now in the Metropolitan Museum of Art, NYC]] The understructure of Victorian women's clothing is what makes the costumes worn by the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] so distinctly Victorian in appearance. An example of a corset that has the kind of structure often worn by fashionably dressed women in 1897 is the one at right. This corset exaggerated the shape of the women's bodies and made possible a bodice that looked and was fitted in the way that is so distinctive of the time — very controlled and smooth. And, as a structural element, this foundation garment carried the weight of all those layers and all that fabric and decoration on the gowns, trains and mantles. (The trains and mantles could be attached directly to the corset itself.) * This foundation emphasizes the waist and the bust in particular, in part because of the contrast between the very small waist and the rounded fullness of the bust and hips. * The idealized waist is defined by its small span and the sexualizing point at the center-bottom of the bodice, which directs the eye downwards. Interestingly, the pointed waistline worn by Elizabethan men has become level in the Victorian age. Highly fashionable Victorian women wearing the traditional style, however, had extremely pointed waists. * The busk (a kind of boning in the front of a corset that is less flexible than the rest) smoothed the bodice, flattened the abdomen and prevented the point on the bodice from curling up. * The sharp definition of the waist was caused by ** length of the corset (especially on the sides) ** the stiffness of the boning ** the layers of fabric ** the lacing (especially if the woman used tightlacing) ** the over-all shape, which was so much wider at the top and the bottom ** the contrast between the waist and the wider top and bottom * The late-19th-century corset was long, ending below the waist even on the sides and back. * The boning and the top edge of the late 19th-century fashion corset pushed up the bust, rounding (rather than flattening, as in earlier styles) the breasts, drawing attention to their exposed curves and creating cleavage. * The exaggerated bust was larger than the hips, whenever possible, an impression reinforced by the A-line of the skirt and the inverted Vs in the decorative trim near the waist and on the skirt. * This corset made the bodice very smooth with a very precise fit, that had no wrinkles, folds or loose drapery. The bodice was also trimmed or decorated, but the base was always a smooth bodice. More formal gowns would still have the fitted bodice and more elaborate trim made from lace, embroidery, appliqué, beading and possibly even jewels. The advantages and disadvantages of corseting and especially tight lacing were the subject of thousands of articles and opinions in the periodical press for a great part of the century, but the fetishistic and politicized tight lacing was practiced by very few women. And no single approach to corsetry was practiced by all women all the time. Most of the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 ball]] were not tightly laced, but the progressive style does not dominate either, even though all the costumes are technically historical dress. Part of what gives most of the costumes their distinctive 19th-century "look" is the more traditional corset beneath them. Even though this highly fashionable look was widely present in the historical costumes at the ball, some women's waists were obviously very small and others were hardly '''emphasized''' at all. Women's waists are never mentioned in the newspaper coverage of the ball — or, indeed, of any of the social events attended by the network at the ball — so it is only in photographs that we can see the effects of how they used their corsets. === Hoops === ''Hoops'' is a mid-19th-century term for a cage-like structure worn under a skirt to hold it away from the body. Striking for how long they lasted and '''the ways''' they evolved, hoops were the foundation undergarment for the bottom half of a woman's body, for a skirt and petticoat. Women wore this cage-like structure from the '''15th century''' through the late 19th century. The 16th-century Katherine of Aragon is credited with making it fashionable outside Spain The cage caused the silhouette of skirts to change shape over time and enabled the extreme distortions of 17th-and-18th-century panniers and the late 19th-century bustle. Early hoops circled the body symmetrically in a cone or drum shape, then were moved to the sides with panniers, then ballooned around the body like the top half of a sphere, and finally were pulled to the rear with a bustle. That is, like corsets, the distorted shapes of high fashion were made possible by hoops. High fashion demanded these shapes, which disguised women's bodies, '''especially below the waist, while corsets did their work above it'''. Besides the shape, the structure used to construct hoops evolved — from cane and wood to whalebone, then steel '''rods''' and wire. Add fabric structural stuff [[File:Pedro García de Benabarre St John Retable Detail.jpg|thumb|Pedro García de Benabarre St John Retable Detail]] [[File:Alonso Sánchez Coello 011.jpg|thumb|Alonso Sánchez Coello 011]] ==== 15th Century ==== Hoops first appeared in Spain in the 15th century and influenced European fashion for many years. Illustrations ==== 16th Century ==== [[File:Farthingale 2 (PSF).png|thumb|Farthingale 2 (PSF)]] In the 16th century, the garment we call ''hoops'' was called a farthingale. ''Vertugadin'' is a French term for ''farthingale'', a cage made of hoops supporting a skirt — "un élément essentiel de la mode Tudor en Angleterre [an essential element of Tudor fashion in England]."<ref name=":0">{{Cite journal|date=2022-03-12|title=Vertugadin|url=https://fr.wikipedia.org/w/index.php?title=Vertugadin&oldid=191825729|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Vertugadin.</ref> ''Farthingale'' is the term in English; in French, it's ''vertugadin'', and in Spanish ''vertugado''. Blanche Payne says,<blockquote>Katherine of Aragon is reputed to have introduced the Spanish farthingale (a hooped underskirt) into England early in the century. The result was to convert the columnar skirt of the fifteenth century into the cone shape of the sixteenth. ...<p> Spanish influence had introduced the hoop-supported skirt, smooth in contour, which was quite generally worn.<ref name=":11" /> (291)</blockquote>In fact, "The Spanish princess Catherine of Aragon brought the fashion to England for her marriage to Prince Arthur, eldest son of Henry VII in 1501 [La princesse espagnole Catherine d'Aragon amena la mode en Angleterre pour son mariage avec le prince Arthur, fils aîné d'Henri VII en 1501]."<ref name=":0" /> Catherine of Aragon, of course, married Henry VIII after Arthur's death. By the end of the 16th century the French and Spanish farthingales were not identical. The Spanish farthingale shaped the skirt into an A-line with a graduated series of hoops sewn to an undergarment. The French farthingale was a flattish "cartwheel" or platter of hoops worn below the waist and above the hips held the skirt out more or less horizontally. Once past the vertugadin, the skirt then fell straight to the floor, shaping the skirt into a kind of drum. The shoes show in the portraits of women wearing the French farthingale. <blockquote>"FARTHINGALE:  Renaissance (1450-1550 C.E. to Elizabethan (1550-1625 C.E.). Linen underskirt with wire supports which, when shaped, produced a variety of dome, bell, and oblong shapes."<ref name=":7" /> (105)</blockquote> ==== 17th Century ==== [[File:Panniers 1.jpg|thumb|alt=Photograph of the wooden and fabric skeleton of an 18th-century women's foundation garment|Panniers 1]] Generally, panniers were a kind of undergarment worn in the 17th and 18th centuries. Their design evolved during the century. Made of hoops of wood, they were "baskets" or cages worn on either side of the waist to broaden the skirts to the sides. Illustration ==== 18th Century ==== By the 18th century, the farthingale was called hoops, which were at this point were made of wood. Blanche Payne outlines the evolution of hoops, and thus the shape of the skirt, in the 18th century:<blockquote>SKIRT FASHIONS. Since skirts experienced the greatest alterations, a brief summary of the successive silhouettes should help to place individual costumes in their proper niches. Six basic forms appeared during the century, in the following order: # The bustle was a continuation of the 1690 mode. # The bell or dome shape resulted from the reintroduction of hoops; in England by 1710, in France by 1720. # The ellipse, the second phase of the hoop skirt, was achieved by broadening the support from side to side and compressing it from front to back. It had a long run of popularity, from 1740 to 1770, the extreme width being retained in court costumes. In France it persisted until the revolution, except that skirts were allowed to curve outward in [the] back again. English court costume followed this fashion well into the nineteenth century. # The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, which were used to form the three great ‘poufs’ known as the polonaise (see Fig. 452). These diversions appeared in the late sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period. # The return of the bustle in the 1780s. # The tubular form, drawn from classic art, in the 1790s.<ref name=":11" /> (411–413) </blockquote>While we think of the bustle as a 19th-century look, it can be found in the 18th century, as Payne says. Polonaise may need its own section. ==== 19th Century ==== [[File:Cutaway sketch of crinoline.gif|thumb|Cutaway sketch of crinoline]] [[File:Paris voulant englober la banlieue.JPG|thumb|Paris voulant englober la banlieue]] [[File:Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723.jpg|thumb|Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723]] In the 19th century, the hoops were made of wire and became lighter. By the 1860s, hoops caused skirts to be huge and round. In Laura Ingalls Wilder's 1941 ''Little Town on the Prairie'', the '''16-year-old Laura wears hoops. Laura Ingalls Wilder and thus the character Laura were born in 1862, so this moment is set in 1883'''.<ref>Hill, Pamela Smith, ed. ''Pioneer Girl: The Annotated Autobiography''.</ref><blockquote>“Well,” Laura began; then she stopped and spun round and round, for the strong wind blowing against her always made the wires of her hoop skirt creep slowly upward under her skirts until they bunched around her knees. Then she must whirl around and around until the wires shook loose and spiraled down to the bottom of her skirts where they should be. “As she and Carrie hurried '''ob''' she began again. “I think it was silly, the way they dressed when Ma was a girl, don’t you? Drat this wind!” she exclaimed as the hoops began creeping upward again. “Quietly Carrie stood by while Laura whirled. “I’m glad I’m not old enough to have to wear hoops,” she said. “They’d make me dizzy.” “They are rather a nuisance,” Laura admitted. “But they are stylish, and when you’re my age you’ll want to be in style.”<ref>Wilder, Laura Ingalls. ''Little Town on the Prairie.'' Harper and Row, 1941. Pp. 272–273.</ref></blockquote>Laura makes the comment that she wants to be in style, but that would be on the prairie, far from a large city, and not necessarily the latest Parisian style. This description of the way the wind could make hoops creep — and the solution of spinning to get the hoops to go back down — is very unusual. It must have been happening to other women wearing hoops at the time, but no other writer addresses this. == '''Traditional vs Progressive Style''' == === Progressive Style === The terms ''artistic dress'' and ''aesthetic dress'' are not synonymous and were in use at different times to refer to different groups of people in different contexts, but we recognize them as referring to a similar kind of personal style in clothing, a style we call progressive dress or the progressive style. Used in a very precise way, ''artistic dress'' is associated with the Pre-Raphaelite artists and the women in their circle beginning in the 1860s. Similarly, ''aesthetic dress'' is associated with the 1880s and 1890s and dress reform movements. In general, the progressive style is characterized by its resistance to the highly structured fashion of its day, especially corseting, aniline dyes and an extremely close fit. === Traditional Style === By the end of the century designs from the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]] (or Maison Worth) define what we think of as the traditional Victorian look, which was very stylish and expensive. Blanche Payne describes an example of the 1895 "high style" in a gown by Worth with "the idiosyncrasies of the [1890s] full blown":<blockquote>The dress is white silk with wine-red stripes. Sleeves, collars, bows, bag, hat, and hem border match the stripes. The sleeve has reached its maximum volume; the bosom full and emphasized with added lace; the waistline is elongated, pointed, and laced to the point of distress; the skirt is smooth over the hips, gradually swinging out to sweep the floor. This is the much vaunted hourglass figure.<ref name=":11">Payne, Blanche. ''History of Costume from the Ancient Egyptians to the Twentieth Century''. Harper & Row, 1965.</ref>{{rp|530}}</blockquote> The Victorian-looking gowns at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] are stylish in a way that recalls the designs of the House of Worth. The elements that make their look so Victorian are anachronisms on the costumes representing fashion of earlier eras. The women wearing these gowns preferred the standards of beauty from their own day to a more-or-less historically accurate look. The style competing at the very end of the century with the Worth look was not the historical, however, but a progressive style called at the time ''artistic'' or ''aesthetic''. William Powell Frith's 1883 painting ''A Private View at the Royal Academy, 1881'' (discussion below) pits this kind of traditional style against the progressive or artistic style. === The Styles === [[File:Frith A Private View.jpg|thumb|William Powell Frith, ''A Private View at the Royal Academy, 1881'']] We typically think of the late-Victorian silhouette as universal but, in the periods in which corsets dominated women's dress, not all women wore corsets and not all corsets were the same, as William Powell Frith's 1883 ''A Private View at the Royal Academy, 1881'' (right) illustrates. Frith is clear in his memoir that this painting — "recording for posterity the aesthetic craze as regards dress" — deliberately contrasts what he calls the "folly" of the Artistic Dress movement and the look of the traditional corseted waist.<ref>Frith, William Powell. ''My Autobiography and Reminiscences''. 1887.</ref> Frith considered the Artistic Movement and Artistic Dress "ephemeral," but its rejection of corsetry looks far more consequential to us in hindsight than it did in the 19th century. As Frith sees it, his painting critiques the "craze" associated with the women in this set of identifiable portraits who are not corseted, but his commitment to realism shows us a spectrum, a range, of conservatism and if not political then at least stylistic progressivism among the women. The progressives, oddly, are the women wearing artistic (that is, somewhat historical) dress, because they’re not corseted. It is a misreading to see the presentation of the women’s fashion as a simple opposition. Constance, Countess of Lonsdale — situated at the center of this painting with Frederick Leighton, president of the Royal Academy of Art — is the most conservatively dressed of the women depicted, with her narrow sleeves, tight waist and almost perfectly smooth bodice, which tells us that her corset has eyelets so that it can be laced precisely and tightly, and it has stays (or "bones") to prevent wrinkles or natural folds in the overclothing. Lillie Langtry, in the white dress, with her stylish narrow sleeves, does not have such a tightly bound waist or smooth bodice, suggesting she may not be corseted at all, as we know she sometimes was not.['''citation'''] Jenny Trip, a painter’s model, is the woman in the green dress in the aesthetic group being inspected by Anthony Trollope, who may be taking notes. She looks like she is not wearing a corset. Both Langtry and Trip are toward the middle of this spectrum: neither is dressed in the more extreme artistic dress of, say, the two figures between Trip and Trollope. A lot has been written about the late-Victorian attraction to historical dress, especially in the context of fancy-dress balls and the Gothic revival in social events as well as art and music. Part of the appeal has to have been the way those costumes could just be beautiful clothing beautifully made. Historical dress provided an opportunity for some elite women to wear less-structured but still beautiful and influential clothing. ['''Calvert'''<ref>Calvert, Robyne Erica. ''Fashioning the Artist: Artistic Dress in Victorian Britain 1848-1900''. Ph.D. thesis, University of Glasgow, 2012. <nowiki>https://theses.gla.ac.uk/3279/</nowiki></ref>] The standards for beauty, then, with historical dress were Victorian, with the added benefit of possibly less structure. So, at the Duchess of Devonshire's ball, "while some attendees tried to hew closely to historical precedent, many rendered their historical or mythological personage in the sartorial vocabulary they knew best. The [photographs of people in their costumes at the ball offer] a glimpse into how Victorians understood history, not a glimpse into the costume of an authentic historical past."<ref>Mitchell, Rebecca N. "The Victorian Fancy Dress Ball, 1870–1900." ''Fashion Theory'' 2017 (21: 3): 291–315. DOI: 10.1080/1362704X.2016.1172817.</ref> (294) * historical dress: beautiful clothing. * the range at the ball, from Minnie Paget to Gwladys * "In light of such efforts, the ball remains to this day one of the best documented outings of the period, and a quick glance at the album shows that ..." Women had more choices about their waists than the simple opposition between no corset and tightlacing can accommodate. The range of choices is illustrated in Frith's painting, with a woman locating herself on it at a particular moment for particular reasons. Much analysis of 19th-century corsetry focuses on its sexualizing effects — corsets dominated Victorian photographic pornography ['''citations'''] and at the same time, the absence of a corset was sexual because it suggested nudity.['''citations'''] A great deal of analysis of 19th-century corsetry, on the other hand, assumes that women wore corsets for the male gaze ['''citations'''] or that they tightened their waists to compete with other women.['''citations'''] But as we can see in Frith's painting, the sexualizing effect was not universal or sweeping, and these analyses do not account for the choices women had in which corset to wear or how tightly to lace it. Especially given the way that some photographic portraits were mechanically altered to make the waist appear smaller, the size of a woman's waist had to do with how she was presenting herself to the world. That is, the fact that women made choices about the size of or emphasis on their waists suggests that they had agency that needs to be taken into account. As they navigated the complex social world, women's fashion choices had meaning. Society or political hostesses had agency not only in their clothing but generally in that complex social world. They had roles managing social events of the upper classes, especially of the upper aristocracy and oligarchy, like the Duchess of Devonshire's ball. Their class and rank, then, were essential to their agency, including to some degree their freedom to choose what kind of corset to wear and how to wear it. Also, by the end of the century lots of different kinds of corsets were available for lots of different purposes. Special corsets existed for pregnancy, sports (like tennis, bicycling, horseback riding, golf, fencing, archery, stalking and hunting), theatre and dance and, of course, for these women corsets could be made to support the special dress worn over it. Women's choices in how they presented themselves to the world included more than just their foundation garments, of course. "Every cap, bow, streamer, ruffle, fringe, bustle, glove," that is, the trim and decorations on their garments, their jewelry and accessories — which Davidoff calls "elaborations"<ref name=":1">Davidoff, Leonore. ''The Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig 1973).</ref>{{rp|93}} — pointed to a host of status categories, like class, rank, wealth, age, marital status, engagement with the empire, how sexual they wanted to seem, political alignment and purpose at the social event. For example, when women were being presented to the monarch, they were expected to wear three ostrich plumes, often called the [[Social Victorians/Terminology#Prince of Wales's Feathers or White Plumes|Prince of Wales's feathers]]. Like all fashions, the corset, which was quite long-lasting in all its various forms, eventually went out of style. Of the many factors that might have influenced its demise, perhaps most important was the women's movement, in which women's rights, freedom, employment and access to their own money and children were less slogan-worthy but at least as essential as votes for women. The activities of the animal-rights movements drew attention not only to the profligate use of the bodies and feathers of birds but also to the looming extinction of the baleen whale, which made whale bone scarce and expensive. Perhaps the century's debates over corseting and especially tightlacing were relevant to some decisions not to be corseted. And, of course, perhaps no other reason is required than that the nature of fashion is to change. == Cinque Cento == According to the ''Oxford English Dictionary'', ''Cinque Cento'' is a shortening of ''mil cinque cento'', or 1500.<ref>"cinquecento, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/33143. Accessed 7 February 2023.</ref> The term, then would refer, perhaps informally, to the sixteenth century. == Crevé == ''Creve'', without the accent, is an old word in English (c. 1450) for burst or split.<ref>"creve, v." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44339. Accessed 8 February 2023.</ref> ['''With the acute accent, it looks like a past participle in French.'''] == Elastic == Elastic had been invented and was in use by the end of the 19th century. For the sense of "Elastic cord or string, usually woven with india-rubber,"<ref name=":6">“elastic, adj. & n.”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1199670313>.</ref> the ''Oxford English Dictionary'' has usage examples beginning in 1847. The example for 1886 is vivid: "The thorough-going prim man will always place a circle of elastic round his hair previous to putting on his college cap."<ref name=":6" /> == Elaborations == In her 1973 ''The Best Circles: Society, Etiquette and the Season'', Leonore Davidoff notes that women’s status was indicated by dress and especially ornament: “Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration,” she says, “symbolised some status category for the female wearer.”<ref name=":1" />{{rp|93}} Looking at these elaborations as meaningful rather than dismissing them as failed attempts at "historical accuracy" reveals a great deal about the individual women who wore or carried them — and about the society women and political hostesses in their roles as managers of the social world. In her review of ''The House of Worth: Portrait of an Archive'', Mary Frances Gormally says,<blockquote>In a socially regulated year, garments custom made with a Worth label provided women with total reassurance, whatever the season, time of day or occasion, setting them apart as members of the “Best Circles” dressed in luxurious, fashionable and always appropriate attire (Davidoff 1973). The woman with a Worth wardrobe was a woman of elegance, lineage, status, extreme wealth and faultless taste.<ref>Gormally, Mary Frances. Review essay of ''The House of Worth: Portrait of an Archive'', by Amy de la Haye and Valerie D. Mendes (V&A Publishing, 2014). ''Fashion Theory'' 2017 (21, 1): 109–126. DOI: 10.1080/1362704X.2016.1179400.</ref> (117)</blockquote> === Aglet, Aiglet === Historically, an aglet is a "point or metal piece that capped a string [or ribbon] used to attach two pieces of the garment together, i.e., sleeve and bodice."<ref name=":7" />{{rp|4}} Although they were decorative, they were not always visible on the outside of the clothing. They were often stuffed inside the layers at the waist (for example, attaching the bodice to a skirt or breeches). === Frou-frou === In French, ''frou-frou'' or, spelled as ''froufrou'', is the sound of the rustling of silk or sometimes of fabrics in general.<ref>{{Cite journal|date=2023-07-25|title=frou-frou|url=https://fr.wiktionary.org/w/index.php?title=frou-frou&oldid=32508509|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/frou-frou.</ref> The first use the French ''Wiktionnaire'' lists is Honoré Balzac, ''La Cousine Bette'', 1846.<ref>{{Cite journal|date=2023-06-03|title=froufrou|url=https://fr.wiktionary.org/w/index.php?title=froufrou&oldid=32330124|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/froufrou.</ref> ''Frou-frou'' is a term clothing historians use to describe decorative additions to an article of clothing; often the term has a slight negative connotation, suggesting that the additions are superficial. === Pouf, Puff, Poof === According to the French ''Wikipédia'', a pouf was, beginning in 1744, a "kind of women's hairstyle":<blockquote>The hairstyle in question, known as the “pouf”, had launched the reputation of the enterprising Rose Bertin, owner of the Grand Mogol, a very prominent fashion accessories boutique on Rue Saint-Honoré in Paris in 1774. Created in collaboration with the famous hairdresser, Monsieur Léonard, the pouf was built on a scaffolding of wire, fabric, gauze, horsehair, fake hair, and the client's own hair held up in an almost vertical position. — (Marie-Antoinette, ''Queen of Fashion'', translated from the American by Sylvie Lévy, in ''The Rules of the Game'', n° 40, 2009)</blockquote>''Puff'' and ''poof'' are used to describe clothing. === Shirring === ''Shirring'' is the gathering of fabric to make poufs or puffs. The 19th century is known for its use of this decorative technique. Even men's clothing had shirring: at the shoulder seam. === Sequins === Sequins, paillettes, spangles Sequins — or paillettes — are "small, scalelike glittering disks."<ref name=":7" />(216) The French ''Wiktionnaire'' defines ''paillette'' as "Lamelle de métal, brillante, mince, percée au milieu, ordinairement ronde, et qu’on applique sur une étoffe pour l’orner [A strip of metal, shiny, thin, pierced in the middle, usually round, and which is applied to a fabric in order to decorate it.]"<ref name=":8">{{Cite journal|date=2024-03-18|title=paillette|url=https://fr.wiktionary.org/w/index.php?title=paillette&oldid=33809572|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/paillette.</ref> According to the ''OED'', the use of ''sequin'' as a decorative device for clothing (as opposed to gold coins minted and used for international trade) goes back to the 1850s.<ref>“Sequin, N.” ''Oxford English Dictionary'', Oxford UP, September 2023, https://doi.org/10.1093/OED/4074851670.</ref> The first instance of ''spangle'' as "A small round thin piece of glittering metal (usually brass) with a hole in the centre to pass a thread through, used for the decoration of textile fabrics and other materials of various sorts" is from c. 1420.<ref>“Spangle, N. (1).” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/4727197141.</ref> The first use of ''paillette'' listed in the French ''Wiktionnaire'' is in Jules Verne in 1873 to describe colored spots on icy walls.<ref name=":8" /> Currently many distinguish between sequins (which are smaller) and paillettes (which are larger). Before the 20th century, sequins were metal discs or foil leaves, and so of course if they were silver or copper, they tarnished. It is not until well into the 20th century that plastics were invented and used for sequins. === Trim and Lace === ''A History of Feminine Fashion'', published sometime before 1927 and probably commissioned by [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|the Maison Worth]], describes Charles Frederick Worth's contributions to the development of embroidery and [[Social Victorians/Terminology#Passementerie|passementerie]] (trim) from about the middle of the 19th century:<blockquote>For it must be remembered that one of M. Worth's most important and lasting contributions to the prosperity of those who cater for women's needs, as well as to the variety and elegance of his clients' garments, was his insistence on new fabrics, new trimmings, new materials of every description. In his endeavours to restore in Paris the splendours of the days of La Pompadour, and of Marie Antoinette, he found himself confronted at the outset with a grave difficulty, which would have proved unsurmountable to a man of less energy, resource and initiative. The magnificent materials of those days were no longer to be had! The Revolution had destroyed the market for beautiful materials of this, type, and the Restoration and regime of Louis Philippe had left a dour aspect in the City of Light. ... On parallel lines [to his development of better [[Social Victorians/Terminology#Satin|satin]]], [Worth] stimulated also the manufacture of embroidery and ''passementerie''. It was he who first started the manufacture of laces copied from the designs of the real old laces. He was the / first dressmaker to use fur in the trimming of light materials — but he employed only the richer furs, such as sable and ermine, and had no use whatever for the inferior varieties of skins.<ref name=":9" />{{rp|6–7}}</blockquote> ==== Gold and Silver Fabric and Lace ==== The ''Encyclopaedia Britannica'' (9th edition) has an article on gold and silver fabric, threads and lace attached to the article on gold. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>GOLD AND SILVER LACE. Under this heading a general account may be given of the use of the precious metals in textiles of all descriptions into which they enter. That these metals were used largely in the sumptuous textiles of the earliest periods of civilization there is abundant testimony; and to this day, in the Oriental centres whence a knowledge and the use of fabrics inwoven, ornamented, and embroidered with gold and silver first spread, the passion for such brilliant and costly textiles is still most strongly and generally prevalent. The earliest mention of the use of gold in a woven fabric occurs in the description of the ephod made for Aaron (Exod. xxxix. 2, 3) — "And he made the ephod of gold, blue, and purple, and scarlet, and fine twined linen. And they did beat the gold into thin plates, and cut it into wires (strips), to work it in the blue, and in the purple, and in the scarlet, and in the fine linen, with cunning work." In both the ''Iliad'' and the ''Odyssey'' distinct allusion is frequently made to inwoven and embroidered golden textiles. Many circumstances point to the conclusion that the art of weaving and embroidering with gold and silver originated in India, where it is still principally prosecuted, and that from one great city to another the practice travelled westward, — Babylon, Tarsus, Baghdad, Damascus, the islands of Cyprus and Sicily, Con- / stantinople and Venice, all in the process of time becoming famous centres of these much prized manufactures. Alexander the Great found Indian kings and princes arrayed in robes of gold and purple; and the Persian monarch Darius, we are told, wore a war mantle of cloth of gold, on which were figured two golden hawks as if pecking at each other. There is reason, according to Josephus, to believe that the “royal apparel" worn by Herod on the day of his death (Acts xii. 21) was a tissue of silver. Agrippina, the wife of the emperor Claudius, had a robe woven entirely of gold, and from that period downwards royal personages and high ecclesiastical dignitaries used cloth and tissues of gold and silver for their state and ceremonial robes, as well as for costly hangings and decorations. In England, at different periods, various names were applied to cloths of gold, as ciclatoun, tartarium, naques or nac, baudekiu or baldachin, Cyprus damask, and twssewys or tissue. The thin flimsy paper known as tissue paper, is so called because it originally was placed between the folds of gold "tissue" to prevent the contiguous surfaces from fraying each other. At what time the drawing of gold wire for the preparation of these textiles was first practised is not accurately known. The art was probably introduced and applied in different localities at widely different dates, but down till mediaeval times the method graphically described in the Pentateuch continued to be practised with both gold and silver. Fabrics woven with gold and silver continue to be used on the largest scale to this day in India; and there the preparation of the varieties of wire, and the working of the various forms of lace, brocade, and embroidery, is at once an important and peculiar art. The basis of all modern fabrics of this kind is wire, the "gold wire" of the manufacturer being in all cases silver gilt wire, and silver wire being, of course, composed of pure silver. In India the wire is drawn by means of simple draw-plates, with rude and simple appliances, from rounded bars of silver, or gold-plated silver, as the case may be. The wire is flattened into the strip or ribbon-like form it generally assumes by passing it, fourteen or fifteen strands simultaneously, over a fine, smooth, round-topped anvil, and beating it as it passes with a heavy hammer having a slightly convex surface. From wire so flattened there is made in India soniri, a tissue or cloth of gold, the web or warp being composed entirely of golden strips, and ruperi, a similar tissue of silver. Gold lace is also made on a warp of thick yellow silk with a weft of flat wire, and in the case of ribbons the warp or web is composed of the metal. The flattened wires are twisted around orange (in the case of silver, white) coloured silk thread, so as completely to cover the thread and present the appearance of a continuous wire; and in this form it is chiefly employed for weaving into the rich brocades known as kincobs or kinkhábs. Wires flattened, or partially flattened, are also twisted into exceedingly fine spirals, and in this form they are the basis of numerous ornamental applications. Such spirals drawn out till they present a waved appearance, and in that state flattened, are much used for rich heavy embroideries termed karchobs. Spangles for embroideries, &c., are made from spirals of comparatively stout wire, by cutting them down ring by ring, laying each C-like ring on an anvil, and by a smart blow with a hammer flattening it out into a thin round disk with a slit extending from the centre to one edge. Fine spirals are also used for general embroidery purposes. The demand for various kinds of loom-woven and embroidered gold and silver work in India is immense; and the variety of textiles so ornamented is also very great. "Gold and silver," says Dr Birdwood in his ''Handbook to the British-Indian Section, Paris Exhibition'', 1878, "are worked into the decoration of all the more costly loom-made garments and Indian piece goods, either on the borders only, or in stripes throughout, or in diapered figures. The gold-bordered loom embroideries are made chiefly at Sattara, and the gold or silver striped at Tanjore; the gold figured ''mashrus'' at Tanjore, Trichinopoly, and Hyderabad in the Deccau; and the highly ornamented gold-figured silks and gold and silver tissues principally at Ahmedabad, Benares, Murshedabad, and Trichinopoly." Among the Western communities the demand for gold and silver lace and embroideries arises chiefly in connexion with naval and military uniforms, court costumes, public and private liveries, ecclesiastical robes and draperies, theatrical dresses, and the badges and insignia of various orders. To a limited extent there is a trade in gold wire and lace to India and China. The metallic basis of the various fabrics is wire round and flattened, the wire being of three kinds — 1st, gold wire, which is invariably silver gilt wire; 2d, copper gilt wire, used for common liveries and theatrical purposes; and 3d, silver wire. These wires are drawn by the ordinary processes, and the flattening, when done, is accomplished by passing the wire between a pair of revolving rollers of fine polished steel. The various qualities of wire are prepared and used in precisely the same way as in India, — round wire, flat wire, thread made of flat gold wire twisted round orange-coloured silk or cotton, known in the trade as "orris," fine spirals and spangles, all being in use in the West as in the East. The lace is woven in the same manner as ribbons, and there are very numerous varieties in richness, pattern, and quality. Cloth of gold, and brocades rich in gold and silver, are woven for ecclesiastical vestments and draperies. The proportions of gold and silver in the gold thread for the lace trade varies, but in all cases the proportion of gold is exceedingly small. An ordinary gold lace wire is drawn from a bar containing 90 parts of silver and 7 of copper, coated with 3 parts of gold. On an average each ounce troy of a bar so plated is drawn into 1500 yards of wire; and therefore about 16 grains of gold cover a mile of wire. It is estimated that about 250,000 ounces of gold wire are made annually in Great Britain, of which about 20 per cent, is used for the headings of calico, muslin, &c., and the remainder is worked up in the gold lace trade.<ref>William Chandler Roberts-Austen and H. Bauerman [W.C.R. — H.B.]. "Gold and Silver Lace." In "Gold." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. 10 (X). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2010%20%28G-GOT%29%20193592738.23/page/753/mode/1up (accessed January 2023): 753, Col. 2c – 754, Cols. 1a–b – 2a–b.</ref></blockquote> ==== Honiton Lace ==== Kate Stradsin says,<blockquote>Honiton lace was the finest English equivalent of Brussels bobbin lace and was constructed in small ‘sprigs, in the cottages of lacemakers[.'] These sprigs were then joined together and bleached to form the large white flounces that were so sought after in the mid-nineteenth century.<ref>Strasdin, Kate. "Rediscovering Queen Alexandra’s Wardrobe: The Challenges and Rewards of Object-Based Research." ''The Court Historian'' 24.2 (2019): 181-196. Rpt http://repository.falmouth.ac.uk/3762/15/Rediscovering%20Queen%20Alexandra%27s%20Wardrobe.pdf: 13, and (for the little quotation) n. 37, which reads "Margaret Tomlinson, ''Three Generations in the Honiton Lace Trade: A Family History'', self-published, 1983."</ref></blockquote> [[File:Strook in Alençon naaldkant, 1750-1775.jpg|thumb|alt=A long piece of complex white lace with garlands, flowers and bows|Point d'Alençon lace, 1750-1775]] ==== Passementerie ==== ''Passementerie'' is the French term for trim on clothing or furniture. The 19th century (especially during the First and Second Empire) was a time of great "''exubérance''" in passementerie in French design, including the development and widespread use of the Jacquard loom.<ref>{{Cite journal|date=2023-06-10|title=Passementerie|url=https://fr.wikipedia.org/w/index.php?title=Passementerie&oldid=205068926|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Passementerie.</ref> ==== Point d'Alençon Lace ==== A lace made by hand using a number of complex steps and layers. The lacemakers build the point d'Alençon design on some kind of mesh and sometimes leave some of the mesh in as part of the lace and perhaps to provide structure. Elizabeth Lewandowski defines point d'Alençon lace and Alençon lace separately. Point lace is needlepoint lace,<ref name=":7">Lewandowski, Elizabeth J. ''The Complete Costume Dictionary''. Scarecrow Press, 2011.</ref>{{rp|233}} so Alençon point is "a two thread [needlepoint] lace."<ref name=":7" />{{rp|7}} Alençon lace has a "floral design on [a] fine net ground [and is] referred to as [the] queen of French handmade needlepoint laces. The original handmade Alençon was a fine needlepoint lace made of linen thread."<ref name=":7" />{{rp|7}} The sample of point d'Alençon lace (right), from 1750–1775, shows the linen mesh that the lace was constructed on.<ref>{{Cite web|url=http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689|title=MoMu - Open Fashion|website=openfashion.momu.be|access-date=2024-02-26}} ModeMuseum Antwerpen. http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689.</ref> The consistency in this sample suggests it may have been made by machine. == Fabric == === Brocatelle === Brocatelle is a kind of brocade, more simple than most brocades because it uses fewer warp and weft threads and fewer colors to form the design. The article in the French ''Wikipédia'' defines it like this:<blockquote>La '''brocatelle''' est un type de tissu datant du <abbr>xvi^e</abbr> siècle qui comporte deux chaînes et deux trames, au minimum. Il est composé pour que le dessin ressorte avec un relief prononcé, grâce à la chaîne sur un fond en sergé. Les brocatelles les plus anciennes sont toujours fabriquées avec une des trames en lin.<ref>{{Cite journal|date=2023-06-01|title=Brocatelle|url=https://fr.wikipedia.org/w/index.php?title=Brocatelle&oldid=204796410|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Brocatelle.</ref></blockquote>Which translates to this:<blockquote>Brocatelle is a type of fabric dating from the 16th century that has two warps and two wefts, at a minimum. It is composed so that the design stands out with a pronounced relief, thanks to the weft threads on a twill background. The oldest brocades were always made with one of the wefts being linen.</blockquote>The ''Oxford English Dictionary'' says, brocatelle is an "imitation of brocade, usually made of silk or wool, used for tapestry, upholstery, etc., now also for dresses. Both the nature and the use of the stuff have changed" between the late 17th century and 1888, the last time this definition was revised.<ref>"brocatelle, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/23550. Accessed 4 July 2023.</ref> === Broché === === Ciselé === === Crépe de Chine === The ''Oxford English Dictionary'' distinguishes the use of ''crêpe'' (using a circumflex rather than an acute accent over the first ''e'') from ''crape'' in textiles, saying ''crêpe'' is "often borrowed [from the French] as a term for all crapy fabrics other than ordinary black mourning crape,"<ref>"crêpe, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44242. Accessed 10 February 2023.</ref> with usage examples ranging from 1797 to the mid 20th century. Crêpe de chine, it says is "a white or other coloured crape made of raw silk." === Épinglé Velvet === Often spelled ''épingle'' rather than ''épinglé'', this term appears to have been used for a fabric made of wool, or at least wool along with linen or cotton, that was heavier and stiffer than silk velvet. It was associated with outer garments and men's clothing. Nowadays, épinglé velvet is an upholstery fabric in which the pile is cut into designs and patterns, and the portrait of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary, Duchess of Hamilton]] shows a mantle described as épinglé velvet that does seem to be a velvet with a woven pattern perhaps cut into the pile. === Lace === While lace also functioned sometimes as fabric — at the décolletage, for example, on the stomacher or as a veil — here we organize it as a [[Social Victorians/Terminology#Trim and Lace|part of the elaboration of clothing]]. === Liberty Fabrics === === Lisse === According to the ''Oxford English Dictionary'', the term ''lisse'' as a "kind of silk gauze" was used in the 19th-century UK and US.<ref>"lisse, n.1." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/108978. Accessed 4 July 2023.</ref> === Satin === The pre-1927 ''History of Feminine Fashion'', probably commissioned by Charles Frederick Worth's sons, describes Worth's "insistence on new fabrics, new trimmings, new materials of every description" at the beginning of his career in the mid 19th century:<blockquote>When Worth first entered the business of dressmaking, the only materials of the richer sort used for woman's dress were velvet, faille, and watered silk. Satin, for example, was never used. M. Worth desired to use satin very extensively in the gowns he designed, but he was not satisfied with what could be had at the time; he wanted something very much richer than was produced by the mills at Lyons. That his requirements entailed the reconstruction of mills mattered little — the mills were reconstructed under his directions, and the Lyons looms turned out a richer satin than ever, and the manufacturers prospered accordingly.<ref name=":9">[Worth, House of.] {{Cite book|url=http://archive.org/details/AHistoryOfFeminineFashion|title=A History Of Feminine Fashion (1800s to 1920s)}} Before 1927. [Likely commissioned by Worth. Link is to Archive.org; info from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Worth_Biarritz_salon.jpg.]</ref>{{rp|6 in printed, 26 in digital book}}</blockquote> === Selesia === According to the ''Oxford English Dictionary'', ''silesia'' is "A fine linen or cotton fabric originally manufactured in Silesia in what is now Germany (''Schlesien'').<ref>"Silesia, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/179664. Accessed 9 February 2023.</ref> It may have been used as a lining — for pockets, for example — in garments made of more luxurious or more expensive cloth. The word ''sleazy'' — "Of textile fabrics or materials: Thin or flimsy in texture; having little substance or body."<ref>"sleazy, adj." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/181563. Accessed 9 February 2023.</ref> — may be related. === Shot Fabric === According to the ''Oxford English Dictionary'', "Of a textile fabric: Woven with warp-threads of one colour and weft-threads of another, so that the fabric (usually silk) changes in tint when viewed from different points."<ref>“Shot, ''Adj.''”  ''Oxford English Dictionary'', Oxford UP,  July 2023, https://doi.org/10.1093/OED/2977164390.</ref> A shot fabric might also be made of silk and cotton fibers. === Tissue === A lightly woven fabric like gauze or chiffon. The light weave can make the fabric translucent and make pleating and gathering flatter and less bulky. Tissue can be woven to be shot, sheer, stiff or soft. Historically, the term in English was used for a "rich kind of cloth, often interwoven with gold or silver" or "various rich or fine fabrics of delicate or gauzy texture."<ref>“Tissue, N.” ''Oxford English Dictionary'', Oxford UP, March 2024, https://doi.org/10.1093/OED/5896731814.</ref> == Fan == The ''Encyclopaedia Britannica'' (9th edition) has an article on the fan. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>FAN (Latin, ''vannus''; French, ''éventail''), a light implement used for giving motion to the air. ''Ventilabrum'' and ''flabellum'' are names under which ecclesiastical fans are mentioned in old inventories. Fans for cooling the face have been in use in hot climates from remote ages. A bas-relief in the British Museum represents Sennacherib with female figures carrying feather fans. They were attributes of royalty along with horse-hair fly-flappers and umbrellas. Examples may be seen in plates of the Egyptian sculptures at Thebes and other places, and also in the ruins of Persepolis. In the museum of Boulak, near Cairo, a wooden fan handle showing holes for feathers is still preserved. It is from the tomb of Amen-hotep, of the 18th dynasty, 17th century <small>B</small>.<small>C</small>. In India fans were also attributes of men in authority, and sometimes sacred emblems. A heartshaped fan, with an ivory handle, of unknown age, and held in great veneration by the Hindus, was given to the prince of Wales. Large punkahs or screens, moved by a servant who does nothing else, are in common use by Europeans in India at this day. Fans were used in the early Middle Ages to keep flies from the sacred elements during the celebrations of the Christian mysteries. Sometimes they were round, with bells attached — of silver, or silver gilt. Notices of such fans in the ancient records of St Paul’s, London, Salisbury cathedral, and many other churches, exist still. For these purposes they are no longer used in the Western church, though they are retained in some Oriental rites. The large feather fans, however, are still carried in the state processions of the supreme pontiff in Rome, though not used during the celebration of the mass. The fan of Queen Theodolinda (7th century) is still preserved in the treasury of the cathedral of Monza. Fans made part of the bridal outfit, or ''mundus muliebris'', of ancient Roman ladies. Folding fans had their origin in Japan, and were imported thence to China. They were in the shape still used—a segment of a circle of paper pasted on a light radiating frame-work of bamboo, and variously decorated, some in colours, others of white paper on which verses or sentences are written. It is a compliment in China to invite a friend or distinguished guest to write some sentiment on your fan as a memento of any special occasion, and this practice has continued. A fan that has some celebrity in France was presented by the Chinese ambassador to the Comtesse de Clauzel at the coronation of Napoleon I. in 1804. When a site was given in 1635, on an artificial island, for the settlement of Portuguese merchants in Nippo in Japan, the space was laid out in the form of a fan as emblematic of an object agreeable for general use. Men and women of every rank both in China and Japan carry fans, even artisans using them with one hand while working with the other. In China they are often made of carved ivory, the sticks being plates very thin and sometimes carved on both sides, the intervals between the carved parts pierced with astonishing delicacy, and the plates held together by a ribbon. The Japanese make the two outer guards of the stick, which cover the others, occasionally of beaten iron, extremely thin and light, damascened with gold and other metals. Fans were used by Portuguese ladies in the 14th century, and were well known in England before the close of the reign of Richard II. In France the inventory of Charles V. at the end of the 14th century mentions a folding ivory fan. They were brought into general use in that country by Catherine de’ Medici, probably from Italy, then in advance of other countries in all matters of personal luxury. The court ladies of Henry VIII.’s reign in England were used to handling fans, A lady in the Dance of Death by Holbein holds a fan. Queen Elizabeth is painted with a round leather fan in her portrait at Gorhambury; and as many as twenty-seven are enumerated in her inventory (1606). Coryat, an English traveller, in 1608 describes them as common in Italy. They also became of general use from that time in Spain. In Italy, France, and Spain fans had special conventional uses, and various actions in handling them grew into a code of signals, by which ladies were supposed to convey hints or signals to admirers or to rivals in society. A paper in the ''Spectator'' humorously proposes to establish a regular drill for these purposes. The chief seat of the European manufacture of fans during the 17th century was Paris, where the sticks or frames, whether of wood or ivory, were made, and the decorations painted on mounts of very carefully prepared vellum (called latterly ''chicken skin'', but not correctly), — a material stronger and tougher than paper, which breaks at the folds. Paris makers exported fans unpainted to Madrid and other Spanish cities, where they were decorated by native artists. Many were exported complete; of old fans called Spanish a great number were in fact made in France. Louis XIV. issued edicts at various times to regulate the manufacture. Besides fans mounted with parchment, Dutch fans of ivory were imported into Paris, and decorated by the heraldic painters in the process called “Vernis Martin,” after a famous carriage painter and inventor of colourless lac varnish. Fans of this kind belonging to the Queen and to the late baroness de Rothschild were exhibited in 1870 at Kensington. A fan of the date of 1660, representing sacred subjects, is attributed to Philippe de Champagne, another to Peter Oliver in England in the / 17th century. Cano de Arevalo, a Spanish painter of the 17th century devoted himself to fan painting. Some harsh expressions of Queen Christina to the young ladies of the French court are said to have caused an increased ostentation in the splendour of their fans, which were set with jewels and mounted in gold. Rosalba Carriera was the name of a fan painter of celebrity in the 17th century. Lebrun and Romanelli were much employed during the same period. Klingstet, a Dutch artist, enjoyed a considerable reputation for his fans from the latter part of the 17th and the first thirty years of the 18th century. The revocation of the edict of Nantes drove many fan-makers out of France to Holland and England. The trade in England was well established under the Stuart sovereigns. Petitions were addressed by the fan-makers to Charles II. against the importation of fans from India, and a duty was levied upon such fans in consequence. This importation of Indian fans, according to Savary, extended also to France. During the reign of Louis XV. carved Indian and China fans displaced to some extent those formerly imported from Italy, which had been painted on swanskin parchment prepared with various perfumes. During the 18th century all the luxurious ornamentation of the day was bestowed on fans as far as they could display it. The sticks were made of mother-of-pearl or ivory, carved with extraordinary skill in France, Italy, England, and other countries. They were painted from designs of Boucher, Watteau, Lancret, and other "genre" painters, Hébert, Rau, Chevalier, Jean Boquet, Mad. Verité, are known as fan painters. These fashions were followed in most countries of Europe, with certain national differences. Taffeta and silk, as well as fine parchment, were used for the mounts. Little circles of glass were let into the stick to be looked through, and small telescopic glasses were sometimes contrived at the pivot of the stick. They were occasionally mounted with the finest point lace. An interesting fan (belonging to Madame de Thiac in France), the work of Le Flamand, was presented by the municipality of Dieppe to Marie Antoinette on the birth of her son the dauphin. From the time of the Revolution the old luxury expended on fans died out. Fine examples ceased to be exported to England and other countries. The painting on them represented scenes or personages connected with political events. At a later period fan mounts were often prints coloured by hand. The events of the day mark the date of many examples found in modern collections. Amongst the fanmakers of the present time the names of Alexandre, Duvelleroy, Fayet, Vanier, may be mentioned as well known in Paris. The sticks are chiefly made in the department of Oise, at Le Déluge, Crèvecœur, Méry, Ste Geneviève, and other villages, where whole families are engaged in preparing them; ivory sticks are carved at Dieppe. Water-colour painters of distinction often design and paint the mounts, the best designs being figure subjects. A great impulse has been given to the manufacture and painting of fans in England since the exhibition which took place at South Kensington in 1870. Other exhibitions have since been held, and competitive prizes offered, one of which was gained by the Princess Louise. Modern collections of fans take their date from the emigration of many noble families from France at the time of the Revolution. Such objects were given as souvenirs and occasionally sold by families in straitened circumstances. A large number of fans of all sorts, principally those of the 18th century, French, English, German, Italian Spanish, &c., have been lately bequeathed to the South Kensington Museum. Regarding the different parts of folding fans it may be well to state that the sticks are called in French ''brins'', the two outer guards ''panaches'', and the mount ''feuille''.<ref>J. H. Pollen [J.H.P.]. "Fan." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. '''10''' ('''X'''). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%209%20%28FAL-FYZ%29%20193323016.23/page/26/mode/2up (accessed January 2023): 27, Col. 1b – 28, Col. 1c.</ref></blockquote> == Fancy-dress Ball == Fancy-dress (or costume) balls were popular and frequent in the U.K. and France as well as the rest of Europe during the 19th century. The themes and styles of the fancy-dress balls influenced those that followed. At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the guests came dressed in costume from times before 1820, as instructed on '''the invitation''', but their clothing was much more about late-Victorian standards of beauty and fashion than the standards of whatever time period the portraits they were copying or basing their costumes on. ''The Queen'' published dress and fashion information and advice under the byline of Ardern Holt, who regularly answered questions from readers about fashion as well as about fancy dress. (More about Ardern Holt, which is almost certainly a pseudonym, can be found on the [[Social Victorians/People/Working in Publishing#Journalists|People Working in Publishing]] page.) Holt also ran wrote entire articles with suggestions for what might make an appealing fancy-dress costume as well as pointing readers away from costumes that had been worn too frequently. The suggestions for costumes are based on familiar types or portraits available to readers, similar to Holt's books on fancy dress, which ran through a number of editions in the 1880s and 1890s. Fancy-dress questions sometimes asked for details about costumes worn in theatrical or operatic productions, which Holt provides. In November 1897, Holt refers to the Duchess of Devonshire's 2 July ball: "Since the famous fancy ball, given at Devonshire House during this year, historical fancy dresses have assumed a prominence that they had not hitherto known."<ref>Holt, Ardern. "Fancy Dress a la Mode." The ''Queen'' 27 November 1897, Saturday: 94 [of 145 in BNA; print p. 1026], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971127/459/0094.</ref> Holt goes on to provide a number of ideas for costumes for historical fancy dress, as always with a strong leaning toward Victorian standards of beauty and style and away from any concern for historical accuracy. Ardern Holt published books on fancy dress as well as writing for the ''Queen'' and other periodicals, but not all of them were about fancy dress. # ''Gentlemen's Fancy Dress: How to Choose It''. Wyman & Sons, 1882. (''Google Books'' has this: https://books.google.com/books/about/Gentlemen_s_Fancy_Dress.html?id=ED8CAAAAQAAJ.) Later editions: 1898 (HathiTrust) # ''Fancy Dresses Described; Or, What to Wear at Fancy Balls''. Debenham & Freebody, 1882. Illustr., Lillian Young. (HathiTrust has this.) Later editions: 4th ed — 1884; 1887 (HathiTrust); 6th ed. — 1896 (HathiTrust) As Leonore Davidoff says, "Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration symbolised some status category for the female wearer."<ref name=":1" />{{rp|93}} [handled under Elaborations] === Historical Accuracy === Many of the costumes at the ball were based on portraits, especially when the guest was dressed as a historical figure. If possible, we have found the portraits likely to have been the originals, or we have found, if possible, portraits that show the subjects from the two time periods at similar ages. The way clothing was cut changed quite a bit between the 18th and 19th centuries. We think of Victorian clothing — particularly women's clothing, and particularly at the end of the century — as inflexible and restrictive, especially compared to 20th- and 21st-century customs permitting freedom of movement. The difference is generally evolutionary rather than absolute — that is, as time has passed since the 18th century, clothing has allowed an increasingly greater range of movement, especially for people who did not do manual labor. By the end of the 19th century, garments like women's bodices and men's coats were made fitted and smooth by attention to the grain of the fabric and by the use of darts (rather than techniques that assembled many small, individual pieces of fabric). * clothing construction and flat-pattern techniques * Generally, the further back in time we go, the more 2-dimensional the clothing itself was. ==== Women's Versions of Historical Accuracy at the Ball ==== As always with this ball, whatever historical accuracy might be present in a woman's costume is altered so that the wearer is still a fashionable Victorian lady. What makes the costumes look "Victorian" to our eyes is the line of the silhouette caused by the foundation undergarments as well as the many "elaborations"<ref name=":1" />{{rp|93}}, mostly in the decorations, trim and accessories. Also, the clothing hangs and drapes differently because the fabric was cut on grain and the shoulders were freed by the way the sleeves were set in. ==== Men's Versions of Historical Accuracy at the Ball ==== Because men were not wearing a Victorian foundation garment at the end of the century, the men's costumes at the ball are more historically accurate in some ways. * Trim * Mixing neck treatments * Hair * Breeches * Shoes and boots * Military uniforms, arms, gloves, boots == Feathers and Plumes == === Aigrette === Elizabeth Lewandowski defines ''aigrette'' as "France. Feather or plume from an egret or heron."<ref name=":7" />(5) Sometimes the newspapers use the term to refer to an accessory (like a fan or ornament on a hat) that includes such a feather or plume. The straight and tapered feathers in an aigrette are in a bundle. === Prince of Wales's Feathers or White Plumes === The feathers in an aigrette came from egrets and herons; Prince of Wales's feathers came from ostriches. A fuller discussion of Prince of Wales's feathers and the white ostrich plumes worn at court appears on [[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|Victorian Things]]. For much of the late 18th and 19th centuries, white ostrich plumes were central to fashion at court, and at a certain point in the late 18th century they became required for women being presented to the monarch and for their sponsors. Our purpose here is to understand why women were wearing plumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] as part of their costumes. First published in 1893, [[Social Victorians/People/Lady Colin Campbell|Lady Colin Campbell]]'s ''Manners and Rules of Good Society'' (1911 edition) says that<blockquote>It was compulsory for both Married and Unmarried Ladies to Wear Plumes. The married lady’s Court plume consisted of three white feathers. An unmarried lady’s of two white feathers. The three white feathers should be mounted as a Prince of Wales plume and worn towards the left hand side of the head. Colored feathers may not be worn. In deep mourning, white feathers must be worn, black feathers are inadmissible.<p> White veils or lace lappets must be worn with the feathers. The veils should not be longer than 45 inches.<ref>{{Cite web|url=https://www.edwardianpromenade.com/etiquette/the-court-presentation/|title=The Court Presentation|last=Holl|first=Evangeline|date=2007-12-07|website=Edwardian Promenade|language=en-US|access-date=2022-12-18}} https://www.edwardianpromenade.com/etiquette/the-court-presentation/.</ref></blockquote>[[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|This fashion was imported from France]] in the mid 1770s.<ref>"Abstract" for Blackwell, Caitlin. "'<nowiki/>''The Feather'd Fair in a Fright''': The Emblem of the Feather in Graphic Satire of 1776." ''Journal for Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. ''Wiley Online'' DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x (accessed November 2022).</ref> Separately, a secondary heraldic emblem of the Prince of Wales has been a specific arrangement of 3 ostrich feathers in a gold coronet<ref>{{Cite journal|date=2022-11-07|title=Prince of Wales's feathers|url=https://en.wikipedia.org/w/index.php?title=Prince_of_Wales%27s_feathers&oldid=1120556015|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Prince_of_Wales's_feathers.</ref> since King Edward III (1312–1377<ref>{{Cite journal|date=2022-12-14|title=Edward III of England|url=https://en.wikipedia.org/w/index.php?title=Edward_III_of_England&oldid=1127343221|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_III_of_England.</ref>). Some women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] wore white ostrich feathers in their hair, but most of them are not Prince of Wales's feathers. Most of the plumes in these portraits are arrangements of some kind of headdress to accompany the costume. A few, wearing what looks like the Princes of Wales's feathers, might be signaling that their character is royal or has royal ancestry. '''One of the women [which one?] was presented to the royals at this ball?''' Here is the list of women who are wearing white ostrich plumes in their portraits in the [[Social Victorians/1897 Fancy Dress Ball/Photographs|''Diamond Jubilee Fancy Dress Ball'' album of 286 photogravure portraits]]: # Kathleen Pelham-Clinton, the [[Social Victorians/People/Newcastle|Duchess of Newcastle]] # [[Social Victorians/People/Louisa Montagu Cavendish|Luise Cavendish]], the Duchess of Devonshire # Jesusa Murrieta del Campo Mello y Urritio (née Bellido), [[Social Victorians/People/Santurce|Marquisa de Santurce]] # Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]] # Princess (Laura Williamina Seymour) Victor of  [[Social Victorians/People/Gleichen#Laura%20Williamina%20Seymour%20of%20Hohenlohe-Langenburg|Hohenlohe Langenburg]] # Louisa Acheson, [[Social Victorians/People/Gosford|Lady Gosford]] # Alice Emily White Coke, [[Social Victorians/People/Leicester|Viscountess Coke]] # Lady Mary Stewart, Helen Mary Theresa [[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]] #[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], Duchess of [[Social Victorians/People/Marlborough|Marlborough]], dressed as the wife of the French Ambassador at the Court of Catherine of Russia (not white, but some color that reads dark in the black-and-white photograph) #Mrs. Mary [[Social Victorians/People/Chamberlain|Chamberlain]] (at 491), wearing white plumes, as Madame d'Epinay #Lady Clementine [[Social Victorians/People/Tweeddale|Hay]] (at 629), wearing white plumes, as St. Bris (''Les Huguenots'') #[[Social Victorians/People/Meysey-Thompson|Lady Meysey-Thompson]] (at 391), wearing white plumes, as Elizabeth, Queen of Bohemia #Mrs. [[Social Victorians/People/Grosvenor|Algernon (Catherine) Grosvenor]] (at 510), wearing white plumes, as Marie Louise #Lady [[Social Victorians/People/Ancaster|Evelyn Ewart]], at 401), wearing white plumes, as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson #[[Social Victorians/People/Lyttelton|Edith Sophy Balfour Lyttelton]] (at 580), wearing what might be white plumes on a large-brimmed white hat, after a picture by Romney #[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), wearing what might be white plumes, as Cydalise of the Comedie Italienne from the time of Louis XV #Lady [[Social Victorians/People/Ilchester|Muriel Fox Strangways]] (at 403), wearing what might be two smallish white plumes, as Lady Sarah Lennox, one of the bridesmaids of Queen Charlotte A.D. 1761 #Lady [[Social Victorians/People/Lucan|Violet Bingham]] (at 586), wearing perhaps one white plume in a headdress not related to the Prince of Wales's feathers #Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), wearing a headdress that includes some white plumes, as Lady Burleigh #[[Social Victorians/People/Dupplin|Agnes Blanche Marie Hay-Drummond]] (at 682), in a big headdress topped with white plumes, as Mademoiselle Andrée de Taverney A.D. 1775 #Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]] (at 336), wearing what looks like Prince of Wales's plumes #[[Social Victorians/People/Suffolk|Marguerite Hyde "Daisy" Leiter]] (at 684), wearing what looks like Prince of Wales's plumes #Lady [[Social Victorians/People/Spicer|Margaret Spicer]] (at 281), wearing one smallish white and one black plume, as Countess Zinotriff, Lady-in-Waiting to the Empress Catherine of Russia #Mrs. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 318), wearing what looks like Prince of Wales's plumes, as Elizabeth Cavendish, daughter of Bess of Hardwick #Nellie, [[Social Victorians/People/Kilmorey|Countess of Kilmorey]] (at 207), wearing three tall plumes, 2 white and one dark, as Comtesse du Barri #Daisy, [[Social Victorians/People/Warwick|Countess of Warwick]] (at 53), wearing at least 1 white plume, as Marie Antoinette More men than women were wearing plumes reminiscent of the Prince of Wales's feathers: * ==== Bibliography for Plumes and Prince of Wales's Feathers ==== * Blackwell, Caitlin. "'''The Feather'd Fair in a Fright'<nowiki/>'': The Emblem of the Feather in Graphic Satire of 1776." Journal for ''Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. Wiley Online DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x. * "Prince of Wales's feathers." ''Wikipedia'' https://en.wikipedia.org/wiki/Prince_of_Wales%27s_feathers (accessed November 2022). ['''Add women to this page'''] * Simpson, William. "On the Origin of the Prince of Wales' Feathers." ''Fraser's magazine'' 617 (1881): 637-649. Hathi Trust https://babel.hathitrust.org/cgi/pt?id=chi.79253140&view=1up&seq=643&q1=feathers (accessed December 2022). Deals mostly with use of feathers in other cultures and in antiquity; makes brief mention of feathers and plumes in signs and pub names that may not be associated with the Prince of Wales. No mention of the use of plumes in women's headdresses or court dress. == Honors == === The Bath === The Most Honourable Order of the Bath (GCB, Knight or Dame Grand Cross; KCB or DCB, Knight or Dame Commander; CB, Companion) === The Garter === The Most Noble Order of the Knights of the Garter (KG, Knight Companion; LG, Lady Companion) [[File:The Golden Fleece - collar exhibited at MET, NYC.jpg|thumb|The Golden Fleece collar and pendant for the 2019 "Last Knight" exhibition at the MET, NYC.|alt=Recent photograph of a gold necklace on a wide band, with a gold skin of a sheep hanging from it as a pendant]] === The Golden Fleece === To wear the golden fleece is to wear the insignia of the Order of the Golden Fleece, said to be "the most prestigious and historic order of chivalry in the world" because of its long history and strict limitations on membership.<ref name=":10">{{Cite journal|date=2020-09-25|title=Order of the Golden Fleece|url=https://en.wikipedia.org/w/index.php?title=Order_of_the_Golden_Fleece&oldid=980340875|journal=Wikipedia|language=en}}</ref> The monarchs of the U.K. were members of the originally Spanish order, as were others who could afford it, like the Duke of Wellington,<ref name=":12">Thompson, R[obert]. H[ugh]. "The Golden Fleece in Britain." Publication of the ''British Numismatic Society''. 2009 https://www.britnumsoc.org/publications/Digital%20BNJ/pdfs/2009_BNJ_79_8.pdf (accessed January 2023).</ref> the first Protestant to be admitted to the order.<ref name=":10" /> Founded in 1429/30 by Philip the Good, Duke of Burgundy, the order separated into two branches in 1714, one Spanish and the other Austrian, still led by the House of Habsburg.<ref name=":10" /> [[File:Prince Albert - Franz Xaver Winterhalter 1842.jpg|thumb|1842 Winterhalter portrait of Prince Albert wearing the insignia of the Order of the Golden Fleece, 1842|left|alt=1842 Portrait of Prince Albert by Winterhalter, wearing the insignia of the Golden Fleece]] The photograph (upper right) is of a Polish badge dating from the "turn of the XV and XVI centuries."<ref>{{Citation|title=Polski: Kolana orderowa orderu Złotego Runa, przełom XV i XVI wieku.|url=https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg|date=2019-11-10|accessdate=2023-01-10|last=Wulfstan}}. https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg.</ref> The collar to this Golden Fleece might be similar to the one the [[Social Victorians/People/Spencer Compton Cavendish#The Insignia of the Order of the Golden Fleece|Duke of Devonshire is wearing in the 1897 Lafayette portrait]]. The badges and collars that Knights of the Order actually wore vary quite a bit. The 1842 Franz Xaver Winterhalter portrait (left) of Prince Consort Albert, Victoria's husband and father of the Prince of Wales, shows him wearing the Golden Fleece on a red ribbon around his neck and the star of the Garter on the front of his coat.<ref>Winterhalter, Franz Xaver. ''Prince Albert''. {{Cite web|url=https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61|title=Explore the Royal Collection Online|website=www.rct.uk|access-date=2023-01-16}} https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61.</ref> === Royal Victorian Order === (GCVO, Knight or Dame Grand Cross; KCVO or DCVO, Knight or Dame Commander; CVO, Commander; LVO, Lieutenant; MVO, Member) === St. John === The Order of the Knights of St. John === Star of India === Most Exalted Order of the Star of India (GCSI, Knight Grand Commander; KCSI, Knight Commander; CSI, Companion) === Thistle === The Most Ancient and Most Noble Order of the Thistle == Jewelry and Stones == === Cabochon === This term describes both the treatment and shape of a precious or semiprecious stone. A cabochon treatment does not facet the stone but merely polishes it, removing "the rough parts" and the parts that are not the right stone.<ref>"cabochon, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/25778. Accessed 7 February 2023.</ref> A cabochon shape is often flat on one side and oval or round, forming a mound in the setting. === Jet === === ''Orfèvrerie'' === Sometimes misspelled in the newspapers as ''orvfèvrerie''. ''Orfèvrerie'' is the artistic work of a goldsmith, silversmith, or jeweler. === Turquoises === == Military == Several men from the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]] were dressed in military uniforms, some historical and some, possibly, not. === Baldric === According to the ''Oxford English Dictionary'', the primary sense of ''baldric'' is "A belt or girdle, usually of leather and richly ornamented, worn pendent from one shoulder across the breast and under the opposite arm, and used to support the wearer's sword, bugle, etc."<ref>"baldric, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/14849. Accessed 17 May 2023.</ref> This sense has been in existence since c. 1300. === Cuirass === According to the ''Oxford English Dictionary'', the primary sense of ''cuirass'' is "A piece of armour for the body (originally of leather); ''spec.'' a piece reaching down to the waist, and consisting of a breast-plate and a back-plate, buckled or otherwise fastened together ...."<ref>"cuirass, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/45604. Accessed 17 May 2023.</ref> [[File:Knötel IV, 04.jpg|thumb|alt=An Old drawing in color of British soldiers on horses brandishing swords in 1815.|1890 illustration of the Household Cavalry (Life Guard, left; Horse Guard, right) at the Battle of Waterloo, 1815]] === Household Cavalry === The Royal Household contains the Household Cavalry, a corps of British Army units assigned to the monarch. It is made up of 2 regiments, the Life Guards and what is now called The Blues and Royals, which were formed around the time of "the Restoration of the Monarchy in 1660."<ref name=":3">Joll, Christopher. "Tales of the Household Cavalry, No. 1. Roles." The Household Cavalry Museum, https://householdcavalry.co.uk/app/uploads/sites/2/2021/06/Household-Cavalry-Museum-video-series-large-print-text-Tales-episode-01.pdf.</ref>{{rp|1}} Regimental Historian Christopher Joll says, "the original Life Guards were formed as a mounted bodyguard for the exiled King Charles II, The Blues were raised as Cromwellian cavalry and The Royals were established to defend Tangier."<ref name=":3" />{{rp|1–2}} The 1st and 2nd Life Guards were formed from "the Troops of Horse and Horse Grenadier Guards ... in 1788."<ref name=":3" />{{rp|3}} The Life Guards were and are still official bodyguards of the queen or king, but through history they have been required to do quite a bit more than serve as bodyguards for the monarch. The Household Cavalry fought in the Battle of Waterloo on Sunday, 18 June 1815 as heavy cavalry.<ref name=":3" />{{rp|3}} Besides arresting the Cato Steet conspirators in 1820 "and guarding their subsequent execution," the Household Cavalry contributed to the "the expedition to rescue General Gordon, who was trapped in Khartoum by The Mahdi and his army of insurgents" in 1884.<ref name=":3" />{{rp|3}} In 1887 they "were involved ... in the suppression of rioters in Trafalgar Square on Bloody Sunday."<ref name=":3" />{{rp|3}} ==== Grenadier Guards ==== Three men — [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Lord Algernon Gordon-Lennox]], [[Social Victorians/People/Stanley#Edward George Villiers Stanley, Lord Stanley|Lord Stanley]], and [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|Hon. F. C. Stanley]] — attended the ball as officers of the Grenadier Guards, wearing "scarlet tunics, ... full blue breeches, scarlet hose and shoes, lappet wigs" as well as items associated with weapons and armor.<ref name=":14">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 34, Col. 2a}} Founded in England in 1656 as Foot Guards, this infantry regiment "was granted the 'Grenadier' designation by a Royal Proclamation" at the end of the Napoleonic Wars.<ref>{{Cite journal|date=2023-04-22|title=Grenadier Guards|url=https://en.wikipedia.org/w/index.php?title=Grenadier_Guards&oldid=1151238350|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Grenadier_Guards.</ref> They were not called Grenadier Guards, then, before about 1815. In 1660, the Stuart Restoration, they were called Lord Wentworth's Regiment, because they were under the command of Thomas Wentworth, 5th Baron Wentworth.<ref>{{Cite journal|date=2022-07-24|title=Lord Wentworth's Regiment|url=https://en.wikipedia.org/w/index.php?title=Lord_Wentworth%27s_Regiment&oldid=1100069077|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Wentworth%27s_Regiment.</ref> At the time of Lord Wentworth's Regiment, the style of the French cavalier had begun to influence wealthy British royalists. In the British military, a Cavalier was a wealthy follower of Charles I and Charles II — a commander, perhaps, or a field officer, but probably not a soldier.<ref>{{Cite journal|date=2023-04-22|title=Cavalier|url=https://en.wikipedia.org/w/index.php?title=Cavalier&oldid=1151166569|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier.</ref> The Guards were busy as infantry in the 17th century, engaging in a number of armed conflicts for Great Britain, but they also served the sovereign. According to the Guards Museum,<blockquote>In 1678 the Guards were ordered to form Grenadier Companies, these men were the strongest and tallest of the regiment, they carried axes, hatches and grenades, they were the shock troops of their day. Instead of wearing tri-corn hats they wore a mitre shaped cap.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/|title=Service to the Crown|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/.</ref></blockquote>The name comes from ''grenades'', then, and we are accustomed to seeing them in front of Buckingham Palace, with their tall mitre hats. The Guard fought in the American Revolution, and in the 19th century, the Grenadier Guards fought in the Crimean War, Sudan and the Boer War. They have roles as front-line troops and as ceremonial for the sovereign, which makes them elite:<blockquote>Queen Victoria decreed that she did not want to see a single chevron soldier within her Guards. Other then [sic] the two senior Warrant Officers of the British Army, the senior Warrant Officers of the Foot Guards wear a large Sovereigns personal coat of arms badge on their upper arm. No other regiments of the British Army are allowed to do so; all the others wear a small coat of arms of their lower arms. Up until 1871 all officers in the Foot Guards had the privilege of having double rankings. An Ensign was ranked as an Ensign and Lieutenant, a Lieutenant as Lieutenant and Captain and a Captain as Captain and Lieutenant Colonel. This was because at the time officers purchased their own ranks and it cost more to purchase a commission in the Foot Guards than any other regiments in the British Army. For example if it cost an officer in the Foot Guards £1,000 for his first rank, in the rest of the Army it would be £500 so if he transferred to another regiment he would loose [sic] £500, hence the higher rank, if he was an Ensign in the Guards and he transferred to a Line Regiment he went in at the higher rank of Lieutenant.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/|title=Formation and role of the Regiments|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/.</ref></blockquote> ==== Life Guards ==== [[Social Victorians/People/Shrewsbury#Reginald Talbot's Costume|General the Hon. Reginald Talbot]], a member of the 1st Life Guards, attended the Duchess of Devonshire's ball dressed in the uniform of his regiment during the Battle of Waterloo.<ref name=":14" />{{rp|p. 36, Col. 3b}} At the Battle of Waterloo the 1st Life Guards were part of the 1st Brigade — the Household Brigade — and were commanded by Major-General Lord Edward Somerset.<ref name=":4">{{Cite journal|date=2023-09-30|title=Battle of Waterloo|url=https://en.wikipedia.org/w/index.php?title=Battle_of_Waterloo&oldid=1177893566|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Battle_of_Waterloo.</ref> The 1st Life Guards were on "the extreme right" of a French countercharge and "kept their cohesion and consequently suffered significantly fewer casualties."<ref name=":4" /> == Photography == == Footnotes == {{reflist}} drgw4jsvbv4mejft4t3ozf3eitn6a5i 2 286261 2690320 2647000 2024-12-04T23:20:46Z Jtneill 10242 /* Social contribution */ 2690320 wikitext text/x-wiki == About Me: == My name is Hassan Alsamara, I am a full time university student, who studies sport [[w:Psychology|psychology]] at the [https://www.canberra.edu.au/ University of Canberra.]. Semester 2, 2024 is my final semester of my university career where i am only studying my final unit of [[Motivation and emotion/Assessment/Chapter|Motivation and Emotion.]] === My hobbies === * [[w:Football|Football]] * Gym * Gaming == Book chapter: == During semester 2, 2024, in my motivation and emotion unit I was tasked to write a book chapter for my major assessment. My book chapter topic was based on [[Motivation and emotion/Book/2024/Comprehensive action determination model|Comprehensive action determination model]] with the key question being, What is the CADM and how can it be applied to understanding human motivation? == Social contribution == [[File:Yalta summit 1945 with Churchill, Roosevelt, Stalin.jpg|thumb|figure 1: random photo ]] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FEffort_justification&diff=2646940&oldid=2646104 Fixed minor capitalisation errors] * [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2025%2FYouth_environmental_activism_motivation#Some_research_to_look_at.. Made a discussion post on a current book chapter] * [[Talk:Motivation and emotion/Book/2024/Sleep onset optimisation#Providing an idea|Provided an idea to a peer writing a current book chapter]] ildu7p5a3u7rvegmx0g95m2eb7wjek3 0 287787 2690278 2679340 2024-12-04T15:55:33Z Bocardodarapti 289675 2690278 wikitext text/x-wiki {{ Mathematical text/Exercise{{{opt|}}} |Text= We consider the families of vectors {{ mathcor/display|term1= {{basis|v|}} = {{op:Column vector|7|-4}}, \, {{op:Column vector|8|1}} |and|term2= {{basis|u|}} = {{op:Column vector|4|6}}, \, {{op:Column vector|7|3}} |pm= }} in {{mat|term= \R^2|pm=.}} {{ Enumeration3/a |Show that {{mat|term= {{basis|v|}} |pm=}} and {{mat|term= {{basis|u|}} |pm=}} are both a {{ Definitionlink |basis| |Context=vs| |pm= }} of {{mat|term= \R^2|pm=.}} |Let {{ Relationchain | P |\in| \R^2 || || || |pm= }} denote the point that has the coordinates {{mathl|term= (-2,5) |pm=}} with respect to the basis {{mat|term= {{basis|v|}} |pm=.}} What are the coordinates of this point with respect to the basis {{mat|term= {{basis|u|}} |pm=?}} |Determine the {{ Definitionlink |transformation matrix| |Context=| |pm= }} that describes the {{ Definitionlink |change of bases| |Context=| |pm= }} from {{mat|term= {{basis|v}} |pm=}} to {{mat|term= {{basis|u}} |pm=.}} }} |Textform=Exercise |Wikidata= |Category= }} cj9omqbm4lqcn887winzsqtjfjatlmh 0 287793 2690279 2679343 2024-12-04T15:57:17Z Bocardodarapti 289675 2690279 wikitext text/x-wiki {{ Mathematical text/Exercise{{{opt|}}} |Text= We consider the families of vectors {{ mathcor/display|term1= {{basis|v|}} = {{op:Column vector|1|2|3}}, \, {{op:Column vector|4|7|1}}, \, {{op:Column vector|0|2|5}} |and|term2= {{basis|u|}} = {{op:Column vector|0|2|4}}, \, {{op:Column vector|6|6|1}}, \, {{op:Column vector|3|5|-2}} |pm= }} in {{mat|term= \R^3|pm=.}} {{ Enumeration3/a |Show that {{mat|term= {{basis|v|}} |pm=}} and {{mat|term= {{basis|u|}} |pm=}} are both a {{ Definitionlink |basis| |Context=vs| |pm= }} of {{mat|term= \R^3|pm=.}} |Let {{ Relationchain | P |\in| \R^3 || || || |pm= }} denote the point that has the coordinates {{mathl|term= (2,5,4)|pm=}} with respect to the basis {{mat|term= {{basis|v|}} |pm=.}} What are the coordinates of this point with respect to the basis {{mat|term= {{basis|u|}} |pm=?}} |Determine the {{ Definitionlink |transformation matrix| |Context=| |pm= }} that describes the {{ Definitionlink |change of basis| |Context=| |pm= }} from {{mat|term= {{basis|v}} |pm=}} to {{mat|term= {{basis|u}} |pm=.}} }} |Textform=Exercise |Marks=6 |m1=3 |m2=1 |m3=2 }} lbta98uq6eggjq5yj6agl9wtzphc2xx 0 287940 2690283 2436149 2024-12-04T16:30:06Z CommonsDelinker 9184 Removing [[:c:File:20040514_large_hail_5.25".jpg|20040514_large_hail_5.25".jpg]], it has been deleted from Commons by [[:c:User:Krd|Krd]] because: per [[:c:Commons:Deletion requests/File:20040514 large hail 5.25".jpg|]]. 2690283 wikitext text/x-wiki {{Article info | first1 = Henry A. | last1 = Hoff | affiliations = Wikiversity | first2 = | last2 = | first3 = | last3 = | first4 = <!-- up to 9 authors can be added in this above format --> | last4 = | et_al = <!-- if there are >9 authors, hyperlink to the list here --> | correspondence = henryhoff@rocketmail.com | journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> | license = <!-- default is CC-BY --> | abstract = A cryometeor is a meteor of variable size that has been radiated and is still moving composed of ice, e.g. water or methane ice. A cryometeor that has stopped moving has become a cryometeorite. This review follows the path of cryometeors from their origins within and occasionally above the atmosphere of the Earth and other planet-like objects to their recycling. | keywords = Cryometeor, Megacryometeor, glacier<!-- up to 6 keywords --> }} == Ices == [[Image:Eiszapfen Schwellenbach.jpg|thumb|left|250px|This is an image of columnar ice crystals. Credit: [[c:User:DrAlzheimer|DrAlzheimer]].{{tlx|free media}}]] '''Def.''' "any frozen volatile chemical, such as water, ammonia, or carbon dioxide"<ref name=IceWikt>{{ cite book |author=[[wikt:User:Długosz|Długosz]] |title=ice |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=23 April 2004 |url=https://en.wiktionary.org/wiki/ice |accessdate=2015-01-05 }}</ref> is called an '''ice'''. The discoveries of water ice on the Moon, Mars and Europa add an extraterrestrial component to the field, as in "astroglaciology".<ref name=RSWilliams>{{ cite journal |title=Annals of Glaciology |volume=9 |page=255 |author=Richard S. Williams, Jr. |url=http://www.igsoc.org/annals/9/igs_annals_vol09_year1987_pg254-255.pdf |year=1987 |publisher=International Glaciological Society |accessdate=7 February 2011}}</ref> ==Radiation== '''Def.''' the "shooting forth of anything from a point or surface, like the diverging rays of light; as, the radiation of heat"<ref name=RadiationWikt>{{ cite book |author=[[wikt:User:Długosz|Długosz]] |title=radiation |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=4 May 2004 |url=http://en.wiktionary.org/wiki/radiation |accessdate=2015-03-28 }}</ref> is called '''radiation'''. Here is a [[Dominant group/Theoretical definition|theoretical definition]]: '''Def.''' "an action or process of throwing or sending out (splitting) a ray in a line, beam, or stream of small cross section" is called '''radiation'''. Meteorites from the Moon (selenometeorites), Mars (arieometeorites) and the asteroids (astrometeorites) have been found on Earth. Acfer 049, an astrometeorite discovered in Algeria in 1990, was shown to have an ultraporous lithology (UPL) that could be formed by removal of ice from these pores, such that a UPL may "represent fossils of primordial ice".<ref name=Matsumoto>{{cite journal |last1=Matsumoto |first1=Megumi |last2=Tsuchiyama |first2=Akira |last3=Nakato |first3=Aiko |last4=Matsuno |first4=Junya |last5=Miyake |first5=Akira |last6=Kataoka |first6=Akimasa |last7=Ito |first7=Motoo |last8=Tomioka |first8=Naotaka |last9=Kodama |first9=Yu |last10=Uesugi |first10=Kentaro |last11=Takeuchi |first11=Akihisa |last12=Nakano |first12=Tsukasa |last13=Vaccaro |first13=Epifanio |title=Discovery of fossil asteroidal ice in primitive meteorite Acfer 094 |journal=Science Advances |date=November 2019 |volume=5 |issue=11 |pages=eaax5078 |doi=10.1126/sciadv.aax5078|pmid=31799392 |pmc=6867873 |bibcode=2019SciA....5.5078M }}</ref> Matter from the Moon, Mars and the asteroids have been radiated into the Earth perhaps including ice. Asteroids and larger bodies can be radiated through precession or irradiated through solar activity cycles. ==Carbon dioxide ices== [[Image:Dry Ice Vapor (17304510479).jpg|thumb|center|250px|Dry ice is sublimating to produce dry ice (carbon dioxide) vapor. Credit: [https://www.flickr.com/people/87296837@N00 Tony Webster from Minneapolis, Minnesota, United States].{{tlx|free media}}]] Dry ice sublimates at {{convert|194.7|K|C F}} at Earth atmospheric pressure. ==Methane or gas hydrate ices== [[Image:Gas hydrate from Indian Ocean.jpg|thumb|center|250px|Sample is gas hydrate (methane clathrate) from sediments under the Indian Ocean. Credit: USGS.{{tlx|free media}}]] Methane clathrate, also called methane hydrate, methane ice or "fire ice" is a solid clathrate hydrate in which methane is trapped within a crystal structure of water, forming a solid. ==Ammonia ices== [[Image:M0354 1951-23-102 2.jpg|thumb|left|250px|Flask contains (NH₄)₂CO₃ salts or smelling slats. Credit: MONNIN Jacques.{{tlx|free media}}]] Ammonia is a colourless gas with a characteristically pungent smell, that is lighter than air, its density being 0.589 times that of Earth's atmosphere. It is easily liquefied due to the strong hydrogen bonding between molecules; the liquid boils at {{convert|-33.1|°C|°F|2}}, and freezes to white crystals<ref name=Chisholm>Chisholm, Hugh, ed. (1911). "Ammonia". Encyclopædia Britannica. Vol. 1 (11th ed.). Cambridge University Press. pp. 861–863.</ref> at {{convert|-77.7|°C|°F|2}}. Solid ammonium carbonate and ammonium bicarbonate salts partly dissociate to form {{chem|NH|3}}, {{chem|CO|2}} and {{chem|H|2|O}} vapour as follows: :{{chem|({{chem|NH|4}})|2}}C{{chem|O|3}} → 2 N{{chem|H|3}} + {{chem|CO|2}} + {{chem|H|2|O}}. :{{chem|NH|4|HCO|3}} → N{{chem|H|3}} + {{chem|CO|2}} + {{chem|H|2|O}}. {{clear}} ==Meteors== [[Image:Hubble Sees Monstrous Cloud Boomerang Back to our Galaxy (24676686535).jpg|right|thumb|300px|The invisible cloud is plummeting toward our galaxy at nearly 700,000 miles per hour. Credit: Saxton/Lockman/NRAO/AUI/NSF/Mellinger.{{tlx|free media}}]] '''Def.''' "'''1 :''' a phenomenon or appearance in the atmosphere (as lightning, a rainbow, or a snowfall) '''2 a :''' one of the small particles of matter in the solar system observable directly only when it falls into the earth's atmosphere where friction may cause its temporary incandescence '''b :''' the streak of light produced by the passage of a meteor"<ref name=Gove>{{ cite book |author= |title=Webster's Seventh New Collegiate Dictionary |publisher=G. & C. Merriam Company |location=Springfield, Massachusetts |date=1963 |editor=Philip B. Gove |pages=1221 |bibcode= |doi= |pmid= |isbn= }}</ref> is called a '''meteor'''. '''Def.''' a "fast-moving streak of light in the night sky caused by the entry of extraterrestrial matter into the earth's atmosphere"<ref name=MeteorWikt>{{ cite book |author=[[wikt:User:Xed~enwiktionary|Xed~enwiktionary]] |title=meteor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=3 September 2004 |url=https://en.wiktionary.org/wiki/meteor |accessdate=26 June 2019 }}</ref> is called a '''meteor'''. '''Def.''' "any natural object radiating through a portion or all of the Earth's or another natural, astronomical object's atmosphere"<ref name=Marshallsumter>{{ cite book |author=[[User:Marshallsumter|Marshallsumter]] |title=meteor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California USA |date=September 24, 2011 |url=http://en.wikiversity.org/wiki/Radiation/Meteors |accessdate=2018-01-24 }}</ref> is called a '''meteor'''. {{clear}} ==Cryometeor theory== [[Image:Snowflake Detail.jpg|thumb|right|250px|Stellate snowflake was photographed in Vermont 2015. Credit: [[c:user:Charles Schmitt|Charles Schmitt]].{{tlx|free media}}]] '''Def.''' "the study of the atmosphere and its phenomena, especially with weather and weather forecasting"<ref name=MeteorologyWikt>{{ cite book |author=[[wikt:User:CORNELIUSSEON|CORNELIUSSEON]] |title=meteorology |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=15 June 2006 |url=http://en.wiktionary.org/wiki/meteorology |accessdate=2013-02-15 }}</ref> or the "atmospheric phenomena in a specific region or period"<ref name=MeteorologyWikt1>{{ cite book |author=[[wikt:User:DCDuring|DCDuring]] |title=meteorology |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=26 March 2009 |url=http://en.wiktionary.org/wiki/meteorology |accessdate=2013-02-15 }}</ref> is called '''meteorology'''. Here are two theoretical definitions: '''Def.''' a single ice crystal (such as a snowflake) or large ice object that is radiated and still moving is called a '''cryometeor'''. '''Def.''' a cryometeor that has been stopped from moving (such as by impacting the Earth) is called a '''cryometeorite'''. "Part two [of the book ''Comet/Asteroid Impacts and Human Society: An Interdisciplinary Approach''] contains contributions focused on the status of near-earth object (NEO) surveys, current knowledge of NEO populations in space, physical properties of NEOs, the quantitative risk of impacts and risk reduction scenarios, the physical terrestrial effects of impacts, the atmospheric and oceanic (tsunami) effects of impacts, case studies including the [[w:Kaali crater|Kaali meteorite]] and [[w:Tunguska event|Tunguska event]]s and cryometeors."<ref name=Bobrowsky>{{ cite book |author=Richard A. F. Grieve and David A. Kring |title=Preface, In: ''Comet/Asteroid Impacts and Human Society: An Interdisciplinary Approach'' |publisher=Springer |location=Berlin |date=November 2006 |editor=Peter T. Bobrowsky and Hans Rickman |pages=546 |url=https://pdfdrive.to/pdfs/cometasteroid-impacts-and-human-society-an-interdisciplinary-approach-pdf |arxiv= |bibcode= |doi= |pmid= |{{isbn|3-540-3270-6}} |accessdate=18 September 2022 }}</ref> "Isotope studies suggest that most of the water did not form on Earth but is the result of the impact of a huge cryometeor that impacted on Earth billions of years ago [Morbidelli ''et al''., 2000]."<ref name=Waitz>{{ cite book |author=Fritz Waitz |title=On the Discrimination and Interaction of Droplets and Ice in Mixed-Phase Clouds |publisher=Karlsruher Institut für Technologie |location= |date=19 November 2021 |editor= |pages=151 |url=https://scholar.archive.org/work/zcneh57gonehzhah7drh6bx33e/access/wayback/https://publikationen.bibliothek.kit.edu/1000140968/138549438 |arxiv= |bibcode= |doi= |pmid= |{{isbn|}} |accessdate=18 September 2022 }}</ref> "Schwerdtfeger (1970, p. 294) notes "With reference to Antarctica, the term ‘cryometeors’ might be more appropriate than "hydrometeors, but it is not used"."<ref name=Turner>{{ cite book |author=John Turner |title=Precipitation/Accumulation, In: ''The International Antarctic Weather Forecasting Handbook'' |publisher=British Antarctic Survey Natural Environment Research Council |location=High Cross, Madingley Road Cambridge, CB3 0ET, UK |date=16 June 2004 |editor=John Turner and Stephen Pendlebury |pages=663 |url=https://nora.nerc.ac.uk/id/eprint/17324/1/handbook_16june04.pdf |arxiv= |bibcode= |doi= |pmid= |{{isbn|1 85531 221 2}} |accessdate=18 September 2022 }}</ref> {{clear}} ==Cryomicrometeoroids== [[Image:PIA17172 Saturn eclipse mosaic bright crop.jpg|thumb|upright=1.5|right|250px|The full set of rings, is imaged as Saturn eclipsed the Sun from the vantage of the Cassini–Huygens orbiter, 1.2 million km distant, on 19 July 2013 (brightness is exaggerated). Earth appears as a pale blue dot at 4 o'clock, between the G and E rings.{{tlx|free media}}]] The rings of Saturn consist of countless small particles, ranging in size from micrometers to meters,<ref name="Questions">{{cite book | last=Porco | first=Carolyn | title=Questions around Saturn's rings|url=http://www.ciclops.org/sci/common_questions.php#ring | accessdate=2012-10-05 }}</ref> that are made almost entirely of water ice, with a trace component of rocky material. The light spectra [of the Upsilon Pegasid fireball], combined with trajectory and light curve measurements, have yielded various compositions and densities, ranging from fragile snowball-like objects with density about a quarter that of ice,<ref name=Povenmire>Povenmire, H. [http://www.lpi.usra.edu/meetings/lpsc2000/pdf/1183.pdf PHYSICAL DYNAMICS OF THE UPSILON PEGASID FIREBLL – EUROPEAN NETWORK 190882A]. Florida Institute of Technology</ref> to nickel-iron rich dense rocks. "It is empirically known that all cooling older stars that possess a global magnetic field have rings. This includes the Earth regardless if they are or not observed with the naked eye."<ref name=Wolynski>{{ cite journal |author=Jeffrey J Wolynski and Stephen Crothers |title=The Earth has Rings |journal=CiteSeer^x |date=27 December 2012 |volume= |issue= |pages=1 |url=https://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=AF51714874998EC06B72646A8C206E96?doi=10.1.1.671.7560&rep=rep1&type=pdf |arxiv= |bibcode= |doi=10.1.1.671.7560 |pmid= |accessdate=19 September 2022 }}</ref> "[W]ater/ice rings will always be oriented in the direction perpendicular to the magnetic field orientation of the cooling star, unless that said star is changing orbits and undergoing a magnetic reversal."<ref name=Wolynski/> Jupiter and Saturn have water ice rings.<ref name=Wolynski/> {{clear}} ==Megacryometeors== [[Image:Megacryometeor1.jpeg|thumb|right|250px|Megacryometeors are something very different, and they are still a mystery to science. Credit: Jesús Martínez-Frias.{{tlx|fairuse}}]] "A '''megacryometeor''' is a very large chunk of ice, weighing at least 10 kg, that are sometimes called huge hailstones, but do not [need] to form in Thunderstorms."<ref name=Megacryometeor>{{ cite book |author=[[w:User:207.61.87.226|207.61.87.226]] |title=Megacryometeor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=2 July 2004 |url=https://en.wikipedia.org/wiki/Megacryometeor |accessdate=16 September 2022 }}</ref> "Over the past decade, over fifty such objects have been recorded worldwide. Some have been as small as about one pound, but one monstrous mass of ice that fell in Brazil weighed about 400 pounds⁠— almost a quarter of a ton⁠— and crashed through the roof of a Mercedes-Benz factory. One recently made headlines in Oakland, California, weighing over 200 pounds and creating a dent in the Earth three feet deep. A similar event occurred in Chicago last February, crashing through the roof of a house."<ref name=Megacryometeor2>{{ cite book |author=Alan Bellows |title=The Peculiar Phenomenon of Megacryometeors |publisher=Damn Interesting |location= |date=April 2006 |url=https://www.damninteresting.com/the-peculiar-phenomenon-of-megacryometeors/ |accessdate=16 September 2022 }}</ref> "The mysterious ice blobs, like hail, have been found to contain air bubbles, onion-like layering, and traces of ammonia and silica. The icy objects also have isotopic distributions of oxygen-18 and deuterium similar to those found in hailstones. Aside from their surprising mass and their tendency to plunge one-at-a-time from clear skies, the ice balls are almost identical to hail."<ref name=Megacryometeor2/> "They are sometimes confused as meteors, because they can leave impact craters. The difference between a megacryometeor and a hailstone is not clearly defined, mostly because the process that creates megacryometeors is not fully understood, but they have been recorded falling out of a clear sky on a hot summer day. They are also not made from airplane toilets or exhaust streams. All analysis of the ice shows it matches normal rain for the region it fell on."<ref name=Megacryometeor/> "A '''megacryometeor''' is a very large chunk of ice which, despite sharing many textural, hydro-chemical and isotopic features detected in large hailstones, is formed under unusual atmospheric conditions which clearly differ from those of the cumulonimbus cloud scenario (i.e. clear-sky conditions). They are sometimes called huge hailstones, but do not need to form in thunderstorms."<ref name=Megacryometeor1>{{ cite book |author=[[w:User:213.0.212.100|213.0.212.100]] |title=Megacryometeor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=24 November 2005 |url=https://en.wikipedia.org/wiki/Megacryometeor |accessdate=16 September 2022 }}</ref> '''Def.''' "a very large water ice object that falls from the sky, similar in composition to hailstones"<ref name=MegacryometeorWikt>{{ cite book |author=[[wikt:User:76.66.203.138|76.66.203.138]] |title=megacryometeor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=14 November 2010 |url=https://en.wiktionary.org/wiki/megacryometeor |accessdate=10 March 2018 }}</ref> is called a '''megacryometeor'''. "LARGE icy conglomerates, occasionally falling from a clear sky even when there are no clouds or precipitation, have recently been termed as megacryometeors¹."<ref name=Deshpande>{{ cite journal |author=R. D. Deshpande, A. S. Maurya, R. C. Angasaria, Medha Dave, A. D. Shukla, N. Bhandari and S. K. Gupta |title=Isotopic studies of megacryometeors in western India |journal=Current Science |date=25 March 2013 |volume=104 |issue=6 |pages=728-737 |url=http://professional.thebabyspecialist.com.sg/wp-content/uploads/2015/01/07281.pdf |arxiv= |bibcode= |doi= |pmid= |accessdate=18 September 2022 }}</ref> "That large blocks of ice fall to the ground is evident enough; they are observed to fall and they are collected, but the central question here is did they enter the Earth’s atmosphere from interplanetary space?"<ref name=Beech>{{ cite journal |author=Martin Beech |title=The Problem of Ice Meteorites |journal=Meteorite Quarterly |date=November 2006 |volume=12 |issue=4 |pages=17-19 |url=https://web.archive.org/web/20110927074403/http://hyperion.cc.uregina.ca/~astro/Ice_Mets.pdf |arxiv= |bibcode= |doi= |pmid= |accessdate=19 September 2022 }}</ref> The "solar system contains numerous bodies that have water-ice as a major compositional component."<ref name=Beech/> It "is a certainty that ice-meteoroids exist. The recent outburst of comet 73P/ Schwassmann- Wachmann 3 [...] provides one example of an event that produced icy-nuclei many tens of meters in diameter, and no-doubt smaller icy meteoroids as well."<ref name=Beech/> ==Ice meteorites== '''Def.''' "a meteor that reaches the surface of the earth without being completely vaporized"<ref name=Gove/> is called a '''meteorite'''. '''Def.''' a "metallic or stony object or body that [is the remains of a meteor]<ref name=MeteoriteWikt1>{{ cite book |author=[[wikt:User:SemperBlotto|SemperBlotto]] |title=meteorite |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=28 December 2007 |url=http://en.wiktionary.org/wiki/meteorite |accessdate=2015-03-28 }}</ref>oid]<ref name=MeteoriteWikt2>{{ cite book |author=[[wikt:User:186.74.9.130|186.74.9.130]] |title=meteorite |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=28 May 2019 |url=http://en.wiktionary.org/wiki/meteorite |accessdate=2015-03-28 }}</ref> [or] has fallen to the surface of the Earth from outer space"<ref name=MeteoriteWikt>{{ cite book |author=[[wikt:User:SnoopY|SnoopY]] |title=meteorite |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 January 2006 |url=http://en.wiktionary.org/wiki/meteorite |accessdate=2015-03-28 }}</ref> is called a '''meteorite'''. "In addition, accepting for the moment that ice meteorites might fall to Earth, the question of their origin must also be addressed – literally, where are the ice fragments from."<ref name=Beech/> "One of the key factors in determining the delivery of a meteorite to the Earth’s surface is the meteoroids initial encounter speed: the lower the encounter speed the better. With respect to known cometary meteoroid streams, the smallest known Earth encounter speed is the 15 km/s of the occasionally active τ-Herculid meteor shower. The next lowest encounter speeds being those for the π-Puppid meteoroids (18 km/s) and the Draconid meteoroids (20 km/s)."<ref name=Beech/> "Firstly, “what is the lifetime of a pure water-ice fragment in the inner solar system”, and second “can water-ice meteoroids survive passage through the Earth’s atmosphere”?"<ref name=Beech/> "While ice-meteoroids must exist within our solar system the more important question at this stage is, how long do they exist for?"<ref name=Beech/> "Once any icy nucleus or ice-meteoroid approaches within about 2.5 AU of the Sun then sublimation will become important."<ref name=Beech/> "For a spherical ice-meteoroid moving in an orbit similar to, for example, Comet 73P/ Schwassmann-Wachmann 3 [Aphelion is 5.211 AU, Perihelion is 0.9722 AU] the radius would decrease due to sublimation at a rate of about 1.4 meters per orbit (or 0.25 m/yr). In other words, a 10-m diameter ice-block would disappear within about 4 orbits of the Sun – a timescale of about 20 years. The same sized meteoroid in an orbit similar to that of the Earth would disappear on an even more rapid timescale of about 2 years. Comet’s that move deep into the outer solar system spend much less time close in towards the Sun, and consequently any ice-meteoroids left in their wake will survive longer. A 10-m diameter ice-block with an orbit similar to that of comet C/1861 G1 (Thatcher) [Aphelion is 110 AU, Perihelion is 0.9207 AU], the parent comet to the April Lyrid meteor shower, which has an aphelion distance of about 109 AU, should survive for about 2000 years – but it would encounter the Earth with an initial speed of 48 km/s."<ref name=Beech/> "The problem with respect to the production of ice-meteorites therefore is that they must encounter the Earth within just a few years of being ejected from their parent body, and this dynamically speaking is highly unlikely to happen."<ref name=Beech/> "The lowest speed that any meteoroid can have at the top of the atmosphere is Earth’s escape velocity of 11.2 km/s."<ref name=Beech/> When "the initial velocity at the top of the atmosphere is 11.5 km/s an ice-meteoroid of mass ~50,000-kg (diameter ≈ 4.8-m) is required to produce a 2-kg meteorite on the ground."<ref name=Beech/> "When the initial velocity is 15 km/s, however, even a 1,000,000-kg (diameter ≈ 15-m) ice-meteoroid will only produce an ice meteorite of a few grams mass on the ground."<ref name=Beech/> If "the Earth did encounter a τ-Herculid fragment of several tens of meters in diameter it would probably produce an air-burst explosion similar to that of the 1908 Tunguska impact."<ref name=Beech/> "Catastrophic fragmentation of all large ice-meteoroids in the Earth’s upper atmosphere is almost inevitable, in fact, because the ram pressure due to the on-coming air flow will easily exceed the tensile strength of solid-ice or that of a cometary nucleus. The tensile strength of comet D/1993 F2 (Shoemaker-Levy 9) was estimated to be about 1000 Pa [Scotti and Melosh, 1993]; the tensile strength of water-ice falls between 10⁶ to 10⁷ Pa."<ref name=Beech/> "So, can an ice-meteoroid survive atmospheric passage to hit the ground? Well, the answer is perhaps yes – just maybe! If the encounter velocity is not much greater than the Earth’s escape velocity then a 5 to 10-m diameter ice-meteoroid might just produce a 1 to 10-kg ice-meteorite at the Earth’s surface (provided that the tensile strength of the ice-meteoroid is greater than ~10⁷ Pa)."<ref name=Beech/> "Two main factors argue against ice meteorites. Firstly the velocity restriction requires that the meteoroids must encounter the Earth with very low velocities – certainly less than 12 – 13 km/s. No currently known cometary meteoroid stream, therefore, can produce ice-meteorites."<ref name=Beech/> "The second reason why ice meteorites must, at best, be exceptionally rare relates to their survival lifetime in space. To get close to the Earth means that an ice-meteoroid must become heated, and once this happens lifetimes against mass-loss by sublimation are typically just a few tens of years. In other words an ice-meteoroid is ‘destroyed’ in space long before it might encounter the Earth to produce an ice-meteorite."<ref name=Beech/> It "has been occasionally noted that meteorite falls can precipitate distinct smells; most often described as sulfurous, or ‘metallic’. Berczi and Lukacs (1997) have picked-up on this point and suggested that odors of sulphuric and ammonia compounds might in fact be released by ‘freshly’ fallen ice-meteorites".<ref name=Beech/> Megacryometeors may "form under a rare, clear-sky variant of the nucleation process responsible for the production of ordinary hail (Bosch, 2002). The ‘meteor’ part of megacryometeors, it should be pointed out, relates to the idea that these objects are considered to be meteorological (that is atmospheric) in origin."<ref name=Beech/> ==Selenometeorites== [[Image:Lunar breccia Apollo sample 14321.jpg|right|thumb|300px|Lunar breccia Apollo sample 14321 formed somewhere between 4 and 4.1 billion years ago, about 12.4 miles beneath the Earth’s crust. Credit: David A. Kring/Center for Lunar Science and Exploration.{{tlx|fairuse}}]] '''Def.''' "a meteorite that is known to have originated on the Moon"<ref name=LunarMeteorite>{{ cite book |author=[[w:User:Robinh|Robinh]] |title=Lunar meteorite |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=18 August 2004 |url=https://en.wikipedia.org/wiki/Lunar_meteorite |accessdate=12 December 2021 }}</ref> is called a '''lunar meteorite''', or perhaps a '''selenometeorite'''. About 371 lunar meteorites have been discovered so far (as of July, 2019),<ref name="MeteorBulletin">{{cite book | url=https://www.lpi.usra.edu/meteor/metbull.php | title=Meteoritical Bulletin Database — Lunar Meteorite search results | publisher=The Meteoritical Society | work=Meteoritical Bulletin Database | date=10 July 2019 | accessdate=20 July 2019}}</ref> perhaps representing more than 30 separate meteorite falls (i.e., many of the stones are "paired" fragments of the same meteoroid).<ref name="wustl.edu">{{cite book|url=http://meteorites.wustl.edu/lunar/moon_meteorites_list_alumina.htm|title=List of Lunar Meteorites - Feldspathic to Basaltic Order|website=meteorites.wustl.edu|accessdate=8 April 2018}}</ref> The total mass is more than {{convert|190|kg}}.<ref name="wustl.edu"/> All lunar meteorites have been found in deserts; most have been found in Antarctica, northern Africa, and the Sultanate of Oman, but none have yet been found in North America, South America, or Europe.<ref>Washington University in St. Louis: [http://meteorites.wustl.edu/lunar/howdoweknow.htm How Do We Know That It's a Rock from the Moon?]</ref> Cosmic ray exposure history established with noble gas measurements has shown that all lunar meteorites were ejected from the Moon in the past 20 million years. Most left the Moon in the past 100,000 years. {{clear}} ==Extraterrestrial megacryometeorites== [[Image:Enceladus fountains.jpg|thumb|right|250px|Enceladus, Saturn's moon, spews out water vapor from its southern pole creating a halo of ice, gas, and dust. Credit: NASA/JPL/Space Science Institute.{{tlx|fairuse}}]] [[Image:Color polar maps of Enceladus PIA18435 Nov. 2014 full size.jpg|thumb|center|400px|These are the north and south polar hemispheres of Enceladus from left to right. Credit: NASA/JPL-Caltech/Space Science Institute/Lunar and Planetary Institute.{{tlx|free media}}]] "The theory of an origin [for megacryometeors] within the Troposphere [...] seems unlikely because there would be significant heating due to an increase in {{chem|CO|2}} concentration (Fu ''et al''. 2011)."<ref name=Snyder>{{ cite journal |author=Duane P Snyder and Rhawn Joseph |title=The Origins of Megacryometeors: Troposphere or Extraterrestrial? |journal=Cosmology |date=2015 |volume=19 |issue= |pages=70-86 |url=https://www.researchgate.net/profile/Rhawn-Joseph/publication/353002917_The_Origins_of_Megacryometeors_Troposphere_or_Extraterrestrial/links/60e37e7ba6fdccb7450ac9f2/The-Origins-of-Megacryometeors-Troposphere-or-Extraterrestrial.pdf |arxiv= |bibcode= |doi= |pmid= |accessdate=19 September 2022 }}</ref> "[M]egacryometeors have been observed and recorded in the mid 1800s, long before the invention of airplanes".<ref name=Snyder/> The "proliferation of reports may be due to increased access to the media, such that, it's not the number of meteors which has increased but the number of people reporting them."<ref name=Snyder/> "In March of 2000 [...] large chunks and a substantial amount of smashed ice [of the Pullman ice meteorite was discovered near a residence] on a clear, cloudless day. The ice was stratified ice, transitioning from clear transparent to translucent to opaque ice. This is indicative of laid down layers of frozen precipitation. The directionally increasing density is suggestive [of] glacial ice."<ref name=Snyder/> "In July of 2000, [...] two vials of melt-water from the suspected ice meteorite [were sent] to Geochronology Labortories, Cambridge, Massachusetts for stable isotope ratio and tritium analysis. Subsequently, high tritium levels were detected, the most likely source being exposure to cosmic radiation."<ref name=Snyder/> An "oval shaped sphere, approximately 300 nm in diameter [was transported] to the Ecloe Polytechnique Surface Analysis Laboratory (LASM), located at the Unversite de Montreal in Montreal, Canada. This sample was bombarded with a pulsed liquid metal ion source at energy of 25 KeV. Both polarities, positive and negative, were registered. The most intense element is the Na (sodium) in positive and Cl (chlorine) in negative [...]. This indicates the presence of sodium chloride salt. Also noticeable is the presence of Ca, K, Si, Al and known and unknown aluminum hydroxides."<ref name=Snyder/> "Melt-water from the suspected ice meteorite, was analyzed by the labs of EAG, [in] Raleigh, North Carolina. The melt-water was sonicated for 10 minutes then transferred to a copper mesh TEM grid. Imaging using STEM (Hitachi HD2700 scanning transmission electron microscope) provided various magnifications in atomic number contrast mode (ZC) and transmitted electron mode (TE). Chemical analysis was preformed with a Bruker Quantax EDS system."<ref name=Snyder/> "Mass spectra of 7 particles [...] indicates high levels of carbon, and Si and O as highly significant particle constituents, as well as Sodium (Na) and chlorine (Cl), being possible salts. When the carbon and the salts are taken into account, the elemental composition of these particles [is] in agreement with the hydrothermal nano-silica ({{chem|SiO|2}}) particles found in the E ring of Saturn. However, carbon was also the most abundant contaminate element found in Saturn’s E ring by the Cassini’s CDA. When the carbon is taken into account, the elemental composition of the particles are in agreement with the hydrothermal nano-silica ({{chem|SiO|2}}) particles found in the E ring of Saturn."<ref name=Snyder/> There "is no evidence megacryometeors are formed in the stratosphere. Moreover, it is a fact that ice chunks, weighing over tens of kilograms (22 pounds), do fall to Earth and it seems highly unlikely such large objects could develop in the stratosphere when there is no evidence that they were formed in the stratosphere in the first place."<ref name=Snyder/> Growth "and layering was [...] observed. Growth, however, requires a place to grow. Micro-Raman spectroscopy of band profiles has indicated that this growth takes place in a range of temperatures (Ruff ''et al''. (2010); and this suggests that the place where these megacryometeors must have been subject to a range of temperatures over a significant duration of time."<ref name=Snyder/> These "ice meteors are formed either in space or they are ejecta from stellar objects consisting of large amounts of water. Be they formed in space or ejecta, these ice meteors would break apart and melt as they enter Earth's atmosphere. Their origin, therefore, could include comets. However, if from a water world, or a planet or moon with ample amounts of water, then the moon Enceladus is one possible candidate."<ref name=Snyder/> "Enceladus, the six largest moon of Saturn has Cryovolcanic ice water vapor plumes that replenish the E ring of Saturn with material. The plumes contain ice particles, salts, organic compounds, water vapor and nano-silica. The gravitational return, to the surface of Enceladus, of some of the frozen precipitation, salts, organic compounds, and dust particles will lay down a glacial like ice surface."<ref name=Snyder/> The "dominant, if not the sole constituent of most E ring stream particles, are {{chem|SiO|2}} (nano-silica) (Hsu ''et al''. 2015)."<ref name=Snyder/> The "nano-silica particles with a radius of ~8 nm (~16 nm dia.), observed by the Cassini mission Cosmic Dust Analyzer (DCA) (Srama ''et al''. 2011) may have been formed over a period of months or years before being ejected into E ring (Hsu ''et al''. 2015)."<ref name=Snyder/> "These nano-silica particles, initially embedded in icy grains, are presumably emitted from Saturn's moon Enceladus’ subsurface waters. They are released by sputter erosion of the icy grains while in Saturns' E ring."<ref name=Snyder/> "Quantitative mass spectra analysis of Saturn’s E ring stream of particles detected by the Cassini mission Cosmic Dust Analyzer (CDA) (Srama ''et al''. 2011), indicates a diameter D_max = 12 to 18 nm for the largest observed stream particles. This is in agreement with the upper particle size limit independently inferred by simulations (R_max= 8 nm) (Hsu ''et al''. 2011)."<ref name=Snyder/> "The plumes of icy particles and water vapor ejected from the south pole of Enceladus have been shown to contain simple organic compounds (McKay ''et al''. 2008). Analysis of the composition of freshly ejected plume particles have found that salt-rich ice particles dominate the total mass flux of ejected particles (Postberg ''et al''. 2011). However, the salt-rich ice particles are depleted in the population escaping into Saturn’s E ring, due to sputter erosion."<ref name=Snyder/> "Salts are found in the mass spectra [..] of the particles found in the melt-water of this suspected ice meteorite. Sodium chloride and known and unknown aluminum hydroxides were found in this ice. The water in this ice is salt-water. Precipitation here on Earth does not contain salt due to the evaporation cycle of water here on earth."<ref name=Snyder/> It "is possible that this suspected ice meteorite [...], is a genuine extraterrestrial ice meteorite because the ice is frozen tritiated salt-water precipitation containing salts and hydrothermal nano-silica, the chemical footprints from the E ring of Saturn."<ref name=Snyder/> "The stratigraphic evolution of the south pole Tiger Stripe surface of Enceladus (Jaumann ''et al''. 2008) is indicative of material being laid down in a glacial like process. The suggested episodically active tectonic events and the proposed localized catastrophic overturn of the rigid ice surface (O’Neill & Nimmo 2010) of Enceladus, allows for the possibility of large bodies of ice to periodically be ejected from Enceladus. The surface of Enceladus and the E ring of Saturn are exposed to cosmic radiation that creates tritium in the exposed water."<ref name=Snyder/> "The data from the analysis of the Pullman ice meteorite is compatible with the possibility that this ice is a genuine ice meteorite. The data is compatible with the possibility that this ice is of extraterrestrial origin. [And] was formed on the surface of Enceladus and constitutes ejecta which eventually fell to Earth."<ref name=Snyder/> {{clear}} ==Hail== [[Image:Hagelkorn mit Anlagerungsschichten.jpg|thumb|right|250px|A large hailstone (clear and white) with concentric rings is shown. Credit: [[c:User:ERZ|ERZ]].{{tlx|free media}}]] [[Image:Small hail, fractured to show internal structure.jpg|thumb|left|250px|The image shows small hail that has been fractured to show internal structure. Credit: Erbe, Pooley: USDA, ARS, EMU.{{tlx|free media}}]] [[Image:Pital nevada.jpg|thumb|right|250px|On April 13, 2004, a blanket of hail fell during a storm in Cerro El Pital, El Salvador. Credit: [[c:User:Wanakoo|Wanakoo]].{{tlx|free media}}]] [[Image:Bogota hailstorm.jpg|thumb|left|250px|The image captures a hailstorm in progress in Bogotá, D.C., Colombia, on March 3, 2006. Credit: [[w:User:Ju98 5|Ju98 5]].{{tlx|free media}}]] [[Image:Nssl0098 - Flickr - NOAA Photo Library.jpg|thumb|right|250px|This is a very large hailstone from the NOAA Photo Library. Credit: NOAA Legacy Photo; OAR/ERL/Wave Propagation Laboratory.{{tlx|free media}}]] [[Image:Wea02251 - Flickr - NOAA Photo Library.jpg|thumb|left|250px|This hailstone was four inches in diameter and weighed seven ounces. Credit: Archival Photography by Steve Nicklas, NOS, NGS.{{tlx|free media}}]] [[Image:Hailstone.jpg|thumb|right|250px|As of June 22, 2003, this is the largest hailstone ever recovered. Credit: NOAA.{{tlx|free media}}]] [[Image:Record hailstone Vivian, SD.jpg|thumb|right|250px|This is a record-setting hailstone that fell in Vivian, South Dakota on July 23, 2010. Credit: NWS Aberdeen, SD.{{tlx|free media}}]] '''Hail''' is a form of solid [water] precipitation. It consists of balls or irregular lumps of ice, each of which is referred to as a '''hailstone'''.<ref name="webster">{{ cite book |title=Merriam-Webster definition of "hailstone" |publisher=Merriam-Webster |url=http://www.merriam-webster.com/dictionary/hailstone |accessdate=2013-01-23 }}</ref> Unlike graupel, which is made of rime, and ice pellets, which are smaller and translucent, hailstones – on Earth – consist mostly of water ice and measure between {{convert|5|and|200|mm|in}} in diameter. '''Def.''' "balls [or pieces]<ref name=HailWikt1>{{ cite book |author=[[wikt:User:Dcwinds~enwiktionary|Dcwinds~enwiktionary]] |title=hail |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=18 June 2006 |url=http://en.wiktionary.org/wiki/hail |accessdate=2013-02-15 }}</ref> of ice falling as precipitation from the sky [a thunderstorm]<ref name=HailWikt1/>"<ref name=HailWikt>{{ cite book |author=[[wikt:User:Paul G|Paul G]] |title=hail |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 April 2004 |url=http://en.wiktionary.org/wiki/hail |accessdate=2013-02-15 }}</ref> are called '''hail'''. '''Def.''' a "single ball of hail"<ref name=HailstoneWikt>{{ cite book |author=[[wikt:User:Newnoise|Newnoise]] |title=hailstone |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=29 December 2005 |url=http://en.wiktionary.org/wiki/hailstone |accessdate=2013-02-15 }}</ref> is called a '''hailstone'''. Terminal velocity of hail, or the speed at which hail is falling when it strikes the ground, varies by the diameter of the hail stones. A hail stone of 1 cm (0.39 in) in diameter falls at a rate of 9 metres per second (20 mph), while stones the size of 8 centimetres (3.1 in) in diameter fall at a rate of 48 metres per second (110 mph). Hail stone velocity is dependent on the size of the stone, friction with air it is falling through, the motion of wind it is falling through, collisions with raindrops or other hail stones, and melting as the stones fall through a warmer atmosphere.<ref name=NSSL>{{ cite book |url=http://www.nssl.noaa.gov/primer/hail/hail_basics.html |title=Hail Basics |author=National Severe Storms Laboratory |publisher=National Oceanic and Atmospheric Administration |date=2006-11-15 |accessdate=2009-08-28 }}</ref> Unlike ice pellets, hailstones are layered and can be irregular and clumped together. A cross-section through a large hailstone shows an onion-like structure. This means the hailstone is made of thick and translucent layers, alternating with layers that are thin, white and opaque. The image second down on the right shows a blanket of hail precipitated on the ground at Cerro El Pital, El Savador. {{lang|es|"Cerro El Pital se encuentra a 12 kilómetros de La Palma, con una altura de 2730 msnm es el punto más alto del territorio Salvadoreño. Es una montaña en medio de un bosque nebuloso que suele tener una temperatura aproximada de 10 ºC. El 13 de abril de 2004, las temperaturas bajaron tanto que el cerro fue cubierto por una escarcha de hielo que causó conmoción entre los pobladores, atribuyendo el fenómeno a una supuesta "nevada"."}} The third image at left shows a hailstone that fell at Washington, D. C., on May 26, 1953, that was 4 in in diameter and weighed 7 oz. In the fourth image at right is the largest hailstone ever recovered in the United States as of June 22, 2003. This hailstone fell in Aurora, Nebraska. It has a 7-inch (17.8 cm) diameter and an approximate circumference of 18.75 inches.<ref name=Leslie>{{ cite book |author=John Leslie |title=Central Plains Storm Produced Largest Hailstone in U.S. History |publisher=NOAA Satellites and Information |location=Maryland |date=2008 |url=http://www.noaanews.noaa.gov/stories/s2008.htm |accessdate=2012-10-14 }}</ref> The fourth down on the left hailstone image is one, approximately 133 mm (5 1/4 inches) in diameter, that fell in Harper, Kansas on May 14, 2004. After 2003, another record-setting hailstone fell in Vivian, South Dakota, on July 23, 2010. Its diameter is 8 inches with a weight of 1 pound 15 ounces. It's in the fifth image down on the right. {{clear}} ==Graupel== [[Image:Graupel encasing a snow crystal.jpg|thumb|right|300px|Graupel is shown encasing an unseen snow crystal. Credit: Erbe, Pooley: USDA, ARS, EMU.{{tlx|free media}}]] The METAR reporting code for hail {{convert|5|mm|in|abbr=on}} or greater is '''GR''', while smaller hailstones and graupel are coded '''GS'''. Hail has a diameter of {{convert|5|mm|in}} or more.<ref name="gloss">{{ cite book |url=http://amsglossary.allenpress.com/glossary/search?id=hail1 |title=Hail |date=2009 |accessdate=2009-07-15 |author=Glossary of Meteorology |publisher=American Meteorological Society }}</ref> Hailstones can grow to {{convert|15|cm|in|0}} and weigh more than {{convert|0.5|kg|lb|1}}.<ref name=NSSL2007>{{ cite book |url=http://www.photolib.noaa.gov/htmls/nssl0001.htm |title=Aggregate hailstone |author=National Severe Storms Laboratory |publisher=National Oceanic and Atmospheric Administration |date=2007-04-23 |accessdate=2009-07-15 }}</ref> '''Graupel''' also called '''soft hail''' or '''snow pellets''')<ref name=Webster>{{ cite book |url = http://www.merriam-webster.com/dictionary/graupel |title = Graupel - Definition, In: ''Merriam-Webster Dictionary'' |publisher = Merriam-Webster |accessdate = 15 Jan 2012 }}</ref> refers to precipitation that forms when supercooled droplets of water are collected and freeze on a falling snowflake, forming a {{convert|2|-|5|mm|in|3|abbr=on}} ball of rime. '''Def.''' a "precipitation that forms when supercooled droplets of water condense on a snowflake"<ref name=GraupelWikt>{{ cite book |author=[[wikt:User:Equinox|Equinox]] |title=graupel |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 June 2009 |url=https://en.wiktionary.org/wiki/graupel |accessdate=4 July 2019 }}</ref> is called '''graupel'''. Strictly speaking, graupel is not the same as hail or ice pellets, although it is sometimes referred to as '''small hail'''; however, the World Meteorological Organization defines ''small hail'' as snow pellets encapsulated by ice, a precipitation halfway between graupel and hail.<ref name=Secretariat>{{ cite book |title=International Cloud Atlas |date=1975 |publisher=Secretariat of the World Meteorological Organization |location=Geneva |url=https://books.google.com/books?id=hkTEMgAACAAJ |{{ISBN|92-63-10407-7}} }}</ref> "Under some atmospheric conditions, forming and descending snow crystals may encounter and pass through atmospheric supercooled cloud droplets. These droplets, which have a diameter of about 10 µm, can exist in the unfrozen state down to temperatures near -40° C. Contact between the snow crystal and the supercooled droplets results in freezing of the liquid droplets onto the surface of the crystals. This process of crystal growth is know[n] as accretion. Crystals that exhibit frozen droplets on their surfaces are referred to as rimed. When this process continues so that the shape of the original snow crystal is no longer identifiable, the resulting crystal is referred to as graupel. The frozen droplets on the surface of rimed crystals are hard to resolve and the topography of a graupel particle is not easy to record with a light microscope because of the limited resolution and depth of field in the instrument. However, observations of snow crystals with a low temperature LT-SEM clearly show cloud droplets measuring up to 50 µm on the surface of the crystals. The rime has been observed on all four basic forms of snow crystals, including plates [..]., dendrites [...], columns [...] and needles [...]. As the riming process continues, the mass of frozen, accumulated cloud droplets obscures the identity of the original snow crystal thereby giving rise to a graupel particle [...]."<ref name="emu">{{cite book | title = Rime and Graupel | website = U.S. Department of Agriculture Electron Microscopy Unit, Beltsville Agricultural Research Center | accessdate = 2020-03-23 |url = https://web.archive.org/web/20170711205706/https://sgil.ba.ars.usda.gov/snowsite/rimegraupel/rg.html }}</ref> "Graupel commonly forms in high altitude climates and is both denser and more granular than ordinary snow, due to its rimed exterior. Macroscopically, graupel resembles small beads of polystyrene. The combination of density and low viscosity makes fresh layers of graupel unstable on slopes, and layers of {{convert|20|-|30|cm|in|abbr=on}} present a high risk of dangerous slab avalanches." In addition, thinner layers of graupel falling at low temperatures can act as ball bearings below subsequent falls of more naturally stable snow, rendering them also liable to avalanche or otherwise making surfaces slippery.<ref name=LaChapelle>{{cite book |title=The Relation of Crystal Riming to Avalanche Formation in New Snow |first=Edward R. |last=LaChapelle |date=May 1966 |publisher=Department of Atmospheric Sciences, University of Washington |url=https://web.archive.org/web/20081206021903/http://www.avalanche.org/~moonstone/snowpack/the%20relation%20of%20crystal%20riming%20to%20avalanche%20formation%20in%20new%20snow.htm }}</ref> Graupel tends to compact and stabilise ("weld") approximately one or two days after falling, depending on the temperature and the properties of the graupel.<ref>{{ cite book |title=Graupel |publisher=American Avalanche Association |url=https://web.archive.org/web/20100504142118/http://www.avalanche.org/~uac/encyclopedia/graupel.htm |date=2010-05-04 }}</ref> {{clear}} ==Sleet== [[Image:Sleet (ice pellets).jpg|thumb|right|250px|The image shows ice pellets aka sleet in North America, with a United States penny for scale. Credit: [[c:User:Runningonbrains|Runningonbrains]].{{tlx|free media}}]] Ice pellets (also referred to as sleet by the United States National Weather Service<ref>{{ cite book |url=http://www.weather.gov/glossary/index.php?word=sleet |title= Sleet (glossary entry) |publisher= National Oceanic and Atmospheric Administration's National Weather Service |accessdate=2007-03-20 }}</ref>) are a form of precipitation consisting of small, translucent balls of ice. Ice pellets are usually smaller than hailstones<ref>{{ cite book |url=http://www.weather.gov/glossary/index.php?word=hail |title= Hail (glossary entry) |publisher= National Oceanic and Atmospheric Administration's National Weather Service |accessdate=2007-03-20 }}</ref> and are different from graupel, which is made of rime, or rain and snow mixed, which is soft. Ice pellets often bounce when they hit the ground, and generally do not freeze into a solid mass unless mixed with freezing rain. The METAR code for ice pellets is PL. '''Def.''' rain "which freezes before reaching the ground"<ref name=SleetWikt>{{ cite book |author=[[wikt:User:Dmh|Dmh]] |title=sleet |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 April 2001 |url=https://en.wiktionary.org/wiki/sleet |accessdate=4 July 2019 }}</ref> is called '''sleet'''. '''Def.''' "a single pellet of sleet"<ref name=IcePelletWikt>{{ cite book |author=[[wikt:User:WikiPedant|WikiPedant]] |title=ice pellet |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=17 December 2007 |url=https://en.wiktionary.org/wiki/ice_pellet |accessdate=4 July 2019 }}</ref> is called an '''ice pellet'''. {{clear}} ==Rime== [[Image:Snowflake 300um LTSEM, 13368.jpg|thumb|right|300px|Rime occurs on both ends of a columnar snow crystal. Credit: [[w:User:Brian0918|Brian0918]].{{tlx|free media}}]] [[Image:Mwrime.JPG|thumb|left|250px|Rime ice is shown after deposition on a window. Credit: [[c:User:Ws47|Ws47]].{{tlx|free media}}]] '''Def.''' "ice formed by the rapid freezing of cold water droplets of fog onto a cold surface"<ref name=RimeWikt>{{ cite book |author=[[wikt:User:Beobach972|Beobach972]] |title=rime |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=7 March 2008 |url=https://en.wiktionary.org/wiki/rime |accessdate=4 July 2019 }}</ref> is called '''rime'''. Hard rime is a white ice that forms when the water droplets in fog freeze to the outer surfaces of objects. It is often seen on trees atop mountains and ridges in winter, when low-hanging clouds cause freezing fog. This fog freezes to the windward (wind-facing) side of tree branches, buildings, or any other solid objects, usually with high wind velocities and air temperatures between {{convert|−2|and|−8|°C|°F|1}}. {{clear}} ==Snow== [[Image:Snow Clouds in Korea.jpg|thumb|right|250px|This image is a satellite photo of lake-effect snow bands near the Korean Peninsula. Credit: NASA.{{tlx|free media}}]] '''Snow''' is precipitation in the form of flakes of crystalline water ice that fall from clouds. Since snow is composed of small ice particles, it is a granular material. It has an open and therefore soft structure, unless subjected to external pressure. '''Def.''' a "crystal of snow, having approximate hexagonal symmetry"<ref name="OED">“[http://dictionary.oed.com/cgi/entry/50229479 snowflake]” listed in the '''Oxford English Dictionary''' [2^nd Ed.; 1989]</ref> is called a '''snowflake'''. Snowflakes come in a variety of sizes and shapes. Types that fall in the form of a ball due to melting and refreezing, rather than a flake, are known as hail, ice pellets or snow grains. '''Def.''' water ice crystals falling as light white flakes are called '''snow'''. '''Def.''' "[a]ny or all of the forms of water particles, whether liquid or solid, that fall from the atmosphere"<ref name=PrecipitationWikt>{{ cite book |title=precipitation |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=February 10, 2013 |url=http://en.wiktionary.org/wiki/precipitation |accessdate=2013-02-15 }}</ref> are called '''precipitation'''. "Condensation or sublimation of atmospheric water vapor produces a hydrometeor. It forms in the free atmosphere, or at the earth's surface, and includes frozen water lifted by the wind. Hydrometeors which can cause a surface visibility reduction, generally fall into one of the following two categories: # '''Precipitation'''. Precipitation includes all forms of water particles, both liquid and solid, which fall from the atmosphere and reach the ground; these include: liquid precipitation (drizzle and rain), freezing precipitation (freezing drizzle and freezing rain), and solid (frozen) precipitation (ice pellets, hail, snow, snow pellets, snow grains, and ice crystals). # '''Suspended (Liquid or Solid) Water Particles'''. Liquid or solid water particles that form and remain suspended in the air (damp haze, cloud, fog, ice fog, and mist), as well as liquid or solid water particles that are lifted by the wind from the earth’s surface (drifting snow, blowing snow, blowing spray) cause restrictions to visibility. One of the more unusual causes of reduced visibility due to suspended water/ice particles is whiteout, while the most common cause is fog."<ref name=Mireles>{{ cite book |author=Mark R. Mireles |author2=Kirth L. Pederson |author3=Charles H. Elford |title=Meteorologial Techniques |publisher=Air Force Weather Agency/DNT |location=Offutt Air Force Base, Nebraska, USA |date=February 21, 2007 |editor= |pages= |url=http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA466107 |arxiv= |bibcode= |doi= |pmid= |isbn= |accessdate=2013-02-17 }}</ref> '''Def.''' a "storm consisting of thunder and lightning produced by a cumulonimbus, usually accompanied with rain and sometimes hail, sleet, freezing rain, or snow"<ref name=ThunderstormWikt>{{ cite book |title=thunderstorm |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=June 30, 2013 |url=http://en.wiktionary.org/wiki/thunderstorm |accessdate=2013-08-03 }}</ref> is called a '''thunderstorm'''. {{clear}} ==Avalanches== [[Image:Avalanche by Armstrong.jpg|thumb|right|300px|This shows an avalanche in motion. Credit: Richard Armstrong, National Snow and Ice Data Center.]] [[Image:Lawine.jpg|center|thumb|300px|Avalanche is occurring in a USA national park. Credit: National Park Service.{{tlx|free media}}]] [[Image:Avalanche.jpg|thumb|left|300px|An avalanche is coming down the face of Mount Index, WA. Credit: [[commons:User:Josh Lewis|Josh Lewis]].]] [[Image:Avalanche on Everest.JPG|right|thumb|300px|An avalanche is occurring in May 2006 on Mount Everest. Credit: [[w:user:Chagai|Chagai]].{{tlx|free media}}]] [[Image:2007-02-15-CLB-Couloir2-1c.JPG|left|thumb|300px|This is the start of a powder snow avalanche. Credit: [[c:user:Scientif38|Scientif38]].{{tlx|free media}}]] [[Image:2007-02-15-CLB-Couloir2-c.JPG|center|thumb|300px|This is the small powder snow avalanche after the start of a powder snow avalanche. Credit: [[c:user:Scientif38|Scientif38]].{{tlx|free media}}]] '''Def.''' a "mass of snow which becomes detached and slides down a slope, often acquiring great bulk by fresh addition as it descends"<ref name=Beitler/> is called an '''avalanche'''. Both avalanches, left and right, top are avalanches in motion in the USA, the one on the left is in Washington. The one on the right was photographed by Richard Armstrong for the National Snow and Ice Data Center, probably in Colorado. {{clear}} ==Firns== [[Image:Taku glacier firn ice sampling.png|thumb|right|250px|Sampling the surface of the Taku Glacier in Alaska demonstrates that there is increasingly denser firn between surface snow and blue glacier ice. Credit: [[c:user:SEWilco|SEWilco]].{{tlx|free media}}]] While collecting snow and ice samples from the wall of a snow pit, fresh snow can be seen at the surface and glacier ice at the bottom of the pit wall. The snow layers are composed of progressively denser firn, Taku Glacier, Juneau Icefield. '''Def.''' "a type of old snow which has gone through multiple thaw and refreeze cycles and thus is made of numerous small icy grains, though it is not nearly as saturated with water as snow-cone slush is; can be hard or somewhat soft depending on recent and current weather conditions"<ref name=MartianBachelor>{{ cite book |author=[[wikt:User:MartianBachelor|MartianBachelor]] |title=firn |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=26 August 2006 |url=https://en.wiktionary.org/wiki/firn |accessdate=2014-09-17 }}</ref> is called '''firn'''. '''Def.''' "rounded, well-bonded snow that is older than one year; firn has a density greater than 550 kilograms per cubic-meter (35 pounds per cubic-foot); called névé during the first year"<ref name=Beitler>{{ cite book |author=Jane Beitler |title=Cryosphere Glossary |publisher=National Snow and Ice Data Center |location= |date= 2014 |url=http://nsidc.org/cryosphere/glossary/all?keys=&page=8 |accessdate=2014-09-18 }}</ref> is called '''firn'''. {{clear}} ==Icefalls== [[Image:Icefall Curtis Glacier 1995.jpg|thumb|right|250px|An Icefall is an area of rapid movement on a steep slope with extensive open crevassing. Credit: Mauri S. Pelto.{{tlx|fairuse}}]] "Crevasses [o]pen because of an acceleration of the glacier."<ref name=Pelto>{{ cite book |author=Mauri S. Pelto |title=North Cascade Glacier Climate Project |publisher=Nichols College |location=Dudley, Massachusetts USA |date=2008 |url=http://www.nichols.edu/departments/glacier/ |accessdate=2014-10-29 }}</ref> '''Def.''' a "part of a glacier with rapid flow and a chaotic crevassed surface; occurs where the glacier bed steepens or narrows"<ref name=Beitler/> is called an '''icefall'''. '''Def.''' "an area of rapid movement on a steep slope with extensive open crevassing"<ref name=Pelto/> is called an '''icefall'''. {{clear}} ==Ice streams== [[Image:FRicestreams.jpg|thumb|right|200px|Radarsat image is of ice streams flowing into the Filchner-Ronne Ice Shelf. Credit: [[w:User:Polargeo|Polargeo]].{{tlx|free media}}]] On the right is a radarsat image of ice streams flowing into the Filchner-Ronne ice shelf. Annotations outline the Rutford ice stream. '''Def.''' "a current of ice in an ice sheet or ice cap that flows faster than the surrounding ice"<ref name=Beitler/> is called an '''ice stream'''. {{clear}} ==Brittle ices== [[Image:Icecore-thinsection awi.jpg|thumb|right|250px|This is a thin section of an ice core from the Antarctic. Credit: Sepp Kipfstuhl/Alfred Wegener Institute (AWI).{{tlx|free media}}]] "It is well known that bulk brittle ice has a hexagonal stucture, while brittle ice that forms in pores may be cubic in structure [...]. Adjacent surfaces appear to further alter the dynamics and structure of confined liquids and their crystals, leading in the case of a water/ice system to a state of enhanced rotational motion (plastic ice) just below the confined freezing/melting transitions. This plastic ice layer appears to form at both the ice-silica interface and the ice-vapour surface, and reversibly transforms to brittle ice at lower temperatures."<ref name=Webber2010>{{ cite journal |author=J. Beau W. Webber |title=Studies of nano-structured liquids in confined geometries and at surfaces |journal=Progress in Nuclear Magnetic Resonance Spectroscopy |date= 2010 |volume=56 |issue=1 |pages=78-93 |url=http://kar.kent.ac.uk/id/eprint/25821 |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-08-16 }}</ref> Systems "with larger dimensions (∼10nm) contain brittle cubic ice and also some hexagonal ice (if a vapour interface is present); even larger systems (> ∼30nm) contain predominately hexagonal ice. It is conjectured that this layer of plastic ice at vapour surfaces may be present at the myriad of such interfaces in macroscopic systems, such as snow-packs, glaciers and icebergs".<ref name=Webber2010/> "The [Greenland] Dye 3 1979 core is not completely intact and is not undamaged."<ref name=Marshallsumter1>{{ cite book |author=[[User:Marshallsumter|Marshallsumter]] |title=Dye 3 |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=5 September 2009 |url=https://en.wikipedia.org/wiki/Dye_3 |accessdate=2014-08-16 }}</ref> “Below 600 m, the ice became brittle with increasing depth and badly fractured between 800 and 1,200 m. The physical property of the core progressively improved and below ~1,400 m was of excellent quality.”<ref name=Shoji>{{ cite journal |author=Shoji, Langway Jr CC |journal=Nature. |date=August |page=548 }}</ref> “The deep ice core drilling terminated in August 1981. The ice core is 2035 m long and has a diameter of 10 cm. It was drilled with less than 6° deviation from vertical, and less than 2 m is missing. The deepest 22 m consists of silty ice with an increasing concentration of pebbles downward. In the depth interval 800 to 1400 m the ice was extremely brittle, and even careful handling unavoidably damaged this part of the core, but the rest of the core is in good to excellent condition.”<ref name=Dansgaard>{{ cite journal |author=Dansgaard W |author2=Clausen HB |author3=Gundestrup NS |author4=Hammer CU |author5=Johnsen SJ |author6=Kristinsdottir PM |author7=Reeh |journal=Science. |date=December 1982 |pages=1273–77 [1274] | doi=10.1126/science.218.4579.1273 |pmid=17770148 |volume=218 |issue=4579 |title=A new greenland deep ice core |bibcode = 1982Sci...218.1273D }}</ref> The depth interval 800 to 1400 m would be a period approximately from about two thousand years ago to about five or six thousand years ago.<ref name=Rose43>{{ cite journal |author=Rose LE |title=Some preliminary remarks about ice cores |journal=Kronos. |date=January 1987 |volume=12 |issue=1 |pages=43–54 }}</ref> "The brittle zone mentioned above [...] corresponds in Dye 3 1979 with the steady state grain size (crystal size) from ~637 - ~1737 m depth range. This is also the Holocene climatic optimum period."<ref name=Marshallsumter1/> "Nuclear Magnetic Resonance and Neutron Scattering of the dynamics and phase-fractions of water/ice systems in templated porous silicas (SBA-15) indicate that what was believed to be a non-frozen surface water layer is actually plastic ice, the quantity varying (continuously and reversibly) with temperature, and converting to a brittle (mainly cubic) ice at lower temperatures."<ref name=Webber>{{ cite book |author=J. Beau W. Webber |author2=John H. Strange |author3=Philip A. Bland |author4=Ross Anderson |author5=Bahman Tohidi |title=Dynamics at Surfaces : Probing the Dynamics of Polar and A-Polar Liquids at Silica and Vapour Surfaces, In: ''9th International Bologna Conference on Magnetic Resonance in Porous Media'' |publisher= |location=Cambridge, MA, USA |date=13 July 2008 |volume= |issue= |pages= |url=http://kar.kent.ac.uk/13472/1/2008-07-14_MRPM9_dynamics-at-surfaces_ed2.pdf |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-08-26 }}</ref> "The ice loads on marine structures are affected by the failure process of ice. Brittle failure is one of the important failure modes. Ice fails in a brittle manner when the loading rate is high or the temperature is low."<ref name=Tuhkuri>{{ cite book |author=J Tuhkuri |title=Experimental investigations and computational fracture mechanics modelling of brittle ice fragmentation |journal=Acta Polytechnica Scandinavica, Mechanical Engineering Series |date= 1996 |volume= |issue=120 |pages=105 |url=http://trid.trb.org/view.aspx?id=481020 |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-08-16 }}</ref> {{clear}} ==Ice flow divides== [[Image:WAISDivideReg.jpeg|thumb|right|250px|This is a detailed map of the WAIS Divide Region. Credit: [[w:User:Tguinane|Tguinane]].{{tlx|free media}}]] "Among numerous other findings, new insights using markers of biological material have proved particularly exciting. Methane has been found to change in time with many rapid climate changes. Spikes of ammonium and organic acids have been found to be markers for biomass burning, while background concentrations of these species indicate the advances of vegetation in North America."<ref name=Stauffer>{{ cite book |author=B. Stauffer |title=The GRIP Ice Coring Effort |publisher=NOAA |location=Washington, DC USA |date= 1992 |url=http://www.ncdc.noaa.gov/paleo/icecore/greenland/summit/document/gripinfo.htm |accessdate=2014-08-24 }}</ref> {{clear}} ==Glaciers== [[Image:Alfred Ernest Ice Shelf.jpg|thumb|right|200px|This is a radar image of Alfred Ernest Ice Shelf on Ellesmere Island, taken by the ERS-1 satellite. Credit: NASA.{{tlx|free media}}]] On the right is a radar image of Alfred Ernest Ice Shelf on Ellesmere Island, taken by the ERS-1 satellite. '''Def.''' "a mass of ice that originates on land, usually having an area larger than one tenth of a square kilometer"<ref name=Beitler/> is called a '''glacier'''. '''Def.''' "a persistent body of [dense]<ref name=Glacier1>{{ cite book |author=[[wikt:User:141.163.203.132|141.163.203.132]] |title=glacier |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=14 August 2013 |url=https://en.wikipedia.org/wiki/Glacier |accessdate=16 September 2022 }}</ref> ice<ref name=Glacier>{{ cite book |author=[[wikt:User:Wsiegmund|Wsiegmund]] |title=glacier |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=11 September 2010 |url=https://en.wikipedia.org/wiki/Glacier |accessdate=16 September 2022 }}</ref> [that is]<ref name=Glacier2>{{ cite book |author=[[wikt:User:MONGO|MONGO]] |title=glacier |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=4 January 2014 |url=https://en.wikipedia.org/wiki/Glacier |accessdate=16 September 2022 }}</ref> [moving under its own]<ref name=Glacier1/> [weight]"<ref name=Glacier3>{{ cite book |author=[[wikt:User:99.236.178.14|99.236.178.14]] |title=glacier |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=28 February 2014 |url=https://en.wikipedia.org/wiki/Glacier |accessdate=16 September 2022 }}</ref> is called a '''glacier'''. {{clear}} ==Surging glaciers== [[Image:Surging glacier.jpg|thumb|right|200px|In 1941, Hole-in-the-Wall Glacier surged. Credit: W.O. Field, World Data Center for Glaciology, Boulder, CO.{{tlx|fairuse}}]] [[Image:Sermersauq Ice Cap Glacier.jpg|thumb|left|200px|The image shows a glacial surge from the Sermersauq Ice Cap. Credit: Robert Gilbert, Niels Nielsen, Henrik Möller, Joseph R. Desloges, and Morten Rasch.{{tlx|fairuse}}]] '''Def.''' "a glacier that experiences a dramatic increase in flow rate, 10 to 100 times faster than its normal rate; usually surge events last less than one year and occur periodically, between 15 and 100 years"<ref name=Beitler/> is called a '''surging glacier'''. "In 1941, Hole-in-the-Wall Glacier [imaged at the right] surged, also knocking over trees during its advance."<ref name=Beitler/> An "outlet glacier of the Sermersauq Ice Cap [on Disko Island, West Greenland, shown at the left with progressive surges marked] has surged 10.5 km downvalley to within 10 km of the fjord. [...] surging of the glacier, begun in 1995, was undetected until July 1999, when it was discovered during a geomorphic survey of the valley. Mapping from TM, Landsat and SPOT satellite imagery, and subsequent field work have documented the history of the event. On 17 June 1995 the terminus of the glacier was about where it appears in the 1985 air photography [...]. By 24 September 1995 the glacier had advanced 1.25 km and by 12 October another 1.25 km (mean advance during the second period : 70 m day^-1). The advance slowed from 18 m day^-1 in 1996 to 5 m day^-1 in 1997 and <1 m day^-1 between 1997 and 1999. By summer 1999 the advance ceased; the maximum extension of the terminus, about 10.5 km down-valley to about 10 km from the head of the fjord, was mapped from imagery on 9 July 1999 [...]."<ref name=Gilbert>{{ cite journal |author=Robert Gilbert |author2=Niels Nielsen |author3=Henrik Möller |author4=Joseph R. Desloges |author5=Morten Rasch |title=Glacimarine sedimentation in Kangerdluk (Disko Fjord), West Greenland, in response to a surging glacier |journal=Marine Geology |year=2002 |volume=191 |issue= |pages=1-18 |url=http://geog.queensu.ca/gilbert/surge%20paper.PDF |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-09-24 }}</ref> {{clear}} ==Classification of glaciers== [[Image:Glacier mapping.jpg|thumb|center|500px|Glacier mapping is performed with Landsat TM and a GIS. Credit: F. Paul, C. Huggel, A. Kääb, T. Kellenberger, and M. Maisch.{{tlx|fairuse}}]] "The low reflectivity of snow and glacier ice in the middle infrared part of the spectrum allows glacier classification".<ref name=Paul>{{ cite book |author=F. Paul |author2=C. Huggel |author3=A. Kääb |author4=T. Kellenberger |author5=M.Maisch |title=Comparison of TM-derived glacier areas with higher resolution data sets, In: ''Proceedings of EARSeL-LISSIG-Workshop Observing our Cryosphere from Space'' |publisher=EARSeL-LISSIG |location= |date=11 March 2002 |editor= |pages=15 |url=http://eproceedings.org/static/vol02_1/02_1_paul1.pdf |arxiv= |bibcode= |doi= |pmid= |isbn= |accessdate=2014-10-14 }}</ref> In the set of images at the center top of this section, glacier mapping steps are shown from left to right with the Landsat 7 enhanced Thematic Mapper (TM) and a geographic information system (GIS).<ref name=Paul/> The images are part of the "102 glaciers of the Mischabel mountain range."<ref name=Paul/> The first image on the left is a ratio image from TM4 and TM4, specifically (TM4 / TM5).<ref name=Paul/> The second is a "derived glacier map after thresholding (blue) and overlay with digitized basins (red)."<ref name=Paul/> The third image from the left identifies "[i]ndividual glaciers after basin intersection (colour-coded) ready for [digital elevation model] DEM-fusion."<ref name=Paul/> The thermal emission and reflectivity have been measured "using the sensors ASTER (Advanced Spaceborne Thermal Emission and reflection Radiometer) on board [the] Terra [satellite]".<ref name=Paul/> Glaciers may be classified on the basis of areal extent or size. "With [a standard deviation of] σ = 3% the values obtained [...] are (resolution / minimum useful glacier size (in km²)): 5 m / all, 10 m / 0.01, 15 m / 0.03, 20 m / 0.05, 25 m / 0.1, 30 m / 0.2, 60 m / 0.5."<ref name=Paul/> "The comparison with higher-resolution satellite imagery reveals: (a) an overall good corre- spondence of the TM-derived glacier outlines with the manual delineation, (b) mapping of debris-covered glacier ice is not possible with TM data alone, and (c) also manual glacier delineation is problematic in the case of debris cover or snowfields."<ref name=Paul/> {{clear}} ==Alpine glaciers== [[Image:Trips 04 - Mt Wedge - 02 (90961463).jpg|thumb|right|250px|The wedgemount alpine glacier is rapidly receding and used to touch the lake as recently as 1990. Credit: [http://www.flickr.com/people/56796376@N00 McKay Savage from London, UK].{{tlx|free media}}]] '''Def.''' "a glacier that is confined by surrounding mountain terrain; also called a mountain glacier"<ref name=Beitler/> is called an '''alpine glacier'''. For "alpine glaciers the imbalance [the change of mass balance/altitude profiles from years with positive to those with negative mean balance] is nearly independent of altitude, in dry, continental regions the imbalance is largest near the equilibrium line, where albedo changes are most pronounced."<ref name=Kuhn>{{ cite journal |author=Michael Kuhn |title=Mass Budget Imbalances as Criterion for a Climatic Classification of Glaciers |journal=Geografiska Annaler. Series A, Physical Geography |year=1984 |volume=66 |issue=3 |pages=229-38 |url=http://www.jstor.org/discover/10.2307/520696?uid=3739552&uid=2&uid=4&uid=3739256&sid=21104337348461 |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-10-14 }}</ref> {{clear}} ==Maritime glaciers== [[Image:Whaler off of NOAA Ship John N. Cobb-Sawyer Glacier.jpg|thumb|right|250px|Sawyer Glacier is in the background. Credit: Personnel of the NOAA ship John N. Cobb.{{tlx|free media}}]] '''Def.''' a glacier that is # found on or near the sea, # bordering on the sea, # in a moist and temperate climate owing to the influence of the sea, # related to the sea, # near or in the sea is called a '''maritime glacier'''. "Maritime glaciers owe their mass balance variations mainly to changes in the accumulation area".<ref name=Kuhn/> {{clear}} ==Tidewater glaciers== [[Image:2008-05-24 12 Jökulsarlón.jpg|thumb|right|250px|The Jökulsarlón tidewater glacier is in Iceland. Credit: [[c:User:Simisa|Hansueli Krapf]].{{tlx|free media}}]] '''Def.''' a glacier occurring in water affected by the flow of the tide, especially tidal streams is called a '''tidewater glacier'''. {{clear}} ==Polar glaciers== [[Image:Pensacola Glacier.jpg|thumb|right|250px|The Pensacola glacier in the Pensacola Range of Antarctica is a polar glacier. Credit: NASA / James Yungel.{{tlx|free media}}]] '''Def.''' a high-latitude glacier that is covered by ice is called a '''polar glacier''', or '''napajäätikkö'''. Polar "glaciers [owe their mass balance variations] to the varying duration of ablation in their lowest parts."<ref name=Kuhn/> {{clear}} ==Rock glaciers== [[Image:Rock glacier.jpg|thumb|right|250px|Frying Pan Glacier is almost entirely covered by rocks and debris. Credit: George L. Snyder.{{tlx|fairuse}}]] '''Def.''' "looks like a mountain glacier and has active flow; usually includes a poorly sorted mess of rocks and fine material; may include: (1) interstitial ice a meter or so below the surface ("ice-cemented"), (2) a buried core of ice ("ice-cored"), and/or (3) rock debris from avalanching snow and rock"<ref name=Beitler/> is called a '''rock glacier'''. '''Def.''' "a mass of rock fragments and finer material, on a slope, that contains either an ice core or interstitial ice, and shows evidence of past, but not present, movement"<ref name=Beitler/> is called an '''inactive rock glacier'''. At the right, "Frying Pan Glacier, Colorado, is almost entirely covered by rocks and debris in this photograph from 1966."<ref name=Beitler/> {{clear}} ==Tributary glaciers== [[Image:03 susitna surge moraines.jpg|thumb|left|250px|This shows the many tributary glaciers of the Susitna Glacier, including surge effects. Credit: Brian John.{{tlx|fairuse}}]] The photo on the left shows many tributary glaciers coming into the Susitna Glacier, including surge effects. {{clear}} ==Valley glaciers== [[Image:Branched valley glacier.jpg|thumb|right|250px|In this photograph from 1969, small glaciers flow into the larger Columbia Glacier from mountain valleys on both sides. Credit: United States Geological Survey.{{tlx|fairuse}}]] '''Def.''' a "glacier that has one or more tributary glaciers that flow into it"<ref name=Beitler/> is called a '''branched-valley glacier'''. "In this photograph from 1969 [at the right], small glaciers flow into the larger Columbia Glacier from mountain valleys on both sides. Columbia Glacier flows out of the Chugach Mountains into Columbia Bay, Alaska."<ref name=Beitler/> {{clear}} ==Outlet glaciers== [[Image:Greenland-glacier hg.jpg|thumb|right|250px|An outlet glacier flows down the side of Fønfjord (Scoresby Sund), Greenland. Credit: [[c:User:Hgrobe|Hannes Grobe, AWI]].{{tlx|free media}}]] "Close to the edges [of an ice sheet], much of the ice flows in narrow and fast-moving outlet glaciers along bedrock troughs [...] Roughly half of the mass loss occurs by iceberg calving from the fronts of these outlets; the other half, by surface melt around the periphery of the whole ice sheet."<ref name=Cuffey>{{ cite book |author=Kurt M. Cuffey |author2=W. S. B. Paterson |title=The Physics of Glaciers |publisher=Elsevier |location=Burlington, Massachusetts USA |date=2010 |editor= |pages=708 |url=http://books.google.com/books?hl=en&lr=&id=Jca2v1u1EKEC&oi=fnd&pg=PP2&ots=KLFO4-pikc&sig=nrAWChisiE5anhb1wFr23YlogvI#v=onepage&f=false |arxiv= |bibcode= |doi= |pmid= |isbn=978-0-12-369461-4 |accessdate=2014-10-15 }}</ref> {{clear}} ==Isolated glaciers== [[Image:Kilimanjaro Glaciers.jpg|thumb|right|250px|Annotated NASA image of Mount Kilimanjaro indicates its glaciers. Credit: NASA and [[c:User:MONGO|MONGO]].{{tlx|free media}}]] [[Image:Mount Kilimanjaro.jpg|thumb|left|250px|This is a panorama of Mount Kilimanjaro showing Kibo peak. Credit: [[w:User:Muhammad Mahdi Karim|Muhammad Mahdi Karim]].{{tlx|free media}}]] [[Image:Mount Kilimanjaro Dec 2009 edit1.jpg|thumb|right|250px|Mount Kilimanjaro is imaged from the air. Credit: [[w:User:Muhammad Mahdi Karim|Muhammad Mahdi Karim]].{{tlx|free media}}]] [[Image:Kilimanjaro glacier retreat.jpg|thumb|left|250px|The two images show the glacial retreat on Mount Kilimanjaro between February 17, 1993, upper, and February 21, 2000, lower. Credit: NASA and U.S. Government.{{tlx|free media}}]] [[Image:Kilimanjaro-1938-uwm.png|thumb|right|250px|This aerial view is from 1938 and shows much more snow than the one above from 2009. Credit:Mary Meader, American Geographical Society Library, University of Wisconsin-Milwaukee Libraries.{{tlx|free media}}]] [[Image:Kibo ice fields.jpg|thumb|left|250px|Shown are the outlines of the Kibo (Kilimanjaro) ice fields in 1912, 1953, 1976, 1989, and 2000, using the OSU aerial photographs taken on 16 February 2000. Credit: Lonnie G. Thompson, Ellen Mosley-Thompson, Mary E. Davis, Keith A. Henderson, Henry H. Brecher, Victor S. Zagorodnov, Tracy A. Mashiotta, Ping-Nan Lin, Vladimir N. Mikhalenko, Douglas R. Hardy, Jürg Beer.{{tlx|fairuse}}]] "Mount Kilimanjaro is the highest [...] "stand-alone" [...] mountain in the world. [...] Mount Kilimanjaro started to be formed about 750000 years ago being currently constituted by three major volcanic cones, Kibo, Mawenzi, and Shira. The first reaches approximately 5900m."<ref name=Fernandes>{{ cite journal |author=Rui M. S. Fernandes |author2=John Msemwa |author3=Machiel Bos |author4=Joaquim Luís |author5=Jorge Santos |author6=André Sá |author7=Saburi John |author8=Essau Mligo |author9=Goodchance J. Tetti |author10=Hassan M. Ubwa |author11=John R. Sorwa |author12=Maenda Kwimbere |author13=Elifuraha Saria |author14=Paul Emmanuel |author15=Hussein Farah |author16=Charles M. Kamamia |author17=Elsayed Issawi |author18=Anwar Radwan |author19=Rob Painter |author20=Lívia Moreira |author21=João Ferreira |title=Precise Determination of the Orthometric Height of Mt. Kilimanjaro |journal=Surveyors Key Role in Accelerated Development |month=3-8 May |year=2009 |volume=TS 8C |issue=Instruments and Calibration |pages=1-11 |url=http://www.fig.net/pub/fig2009/papers/ts08c/ts08c_fernandes_teamkili2008_3438.pdf |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-10-03 }}</ref> Its "location [is] close to [the] equator associated with the existence of permanent glaciers and its almost perfect volcano shape"<ref name=Fernandes/> For "the Uhuru Peak with respect to the KILI2008 datum ... a final value of 5890.79m was determined for the orthometric height of the highest point in Africa considering the Tanzanian vertical datum."<ref name=Thompson/> Kilimanjaro is located at 3°04.6'S and 37°21.2'E.<ref name=Thompson>{{ cite journal |author=Lonnie G. Thompson |author2=Ellen Mosley-Thompson |author3=Mary E. Davis |author4=Keith A. Henderson |author5=Henry H. Brecher |author6=Victor S. Zagorodnov |author7=Tracy A. Mashiotta |author8=Ping-Nan Lin |author9=Vladimir N. Mikhalenko |author10=Douglas R. Hardy |author11=Jürg Beer |title=Kilimanjaro Ice Core Records: Evidence of Holocene Climate Change in Tropical Africa |journal=Science |month=18 October |year=2002 |volume=298 |issue= |pages=589-93 |url=ftp://ftp.soest.hawaii.edu/engels/Stanley/Textbook_update/Science_298/Thompson-02.pdf |arxiv= |bibcode= |doi=10.1126/science.1073198 |pmid= |accessdate=2014-10-04 }}</ref> "Aerial photographs taken on 16 February 2000 allowed production of a recent detailed map of ice cover extent on the summit plateau [diagram at the lower left]."<ref name=Thompson/> "Total ice area calculated from successive maps (1912, 1953, 1976, 1989, and 2000) reveals [diagram at the lower left, inset] that the areal extent of Kilimanjaro’s ice cover has decreased approximately 80% from ~12 km² in 1912 to ~2.6 km² in 2000 and that since 1989, a hole has developed near the center of the NIF. A nearly linear relationship (R² = 0.98) suggests that if climatological conditions of the past 88 years continue, the ice on Kilimanjaro will likely disappear between 2015 and 2020."<ref name=Thompson/> {{clear}} ==Crater glaciers== [[Image:Nevados de Sollipulli.jpg|thumb|right|250px|The image shows the crater glacier of the volcano Sollipulli. Credit: [[c:User:Roka1953|Roka1953]].{{tlx|free media}}]] [[Image:Iss038e012569.jpg|thumb|left|250px|The summit of Sollipulli is occupied by a four-kilometer wide caldera, currently filled with a snow-covered glacier. Credit: William L. Stefanov.{{tlx|fairuse}}]] "While active volcanoes are obvious targets of interest because they pose natural hazards, there are some dormant volcanoes that also warrant concern because of their geologic history. One such volcano is Sollipulli, located in central Chile near the border with Argentina. The volcano sits in the southern Andes Mountains within Chile’s Parque Nacional Villarica. This photograph by an astronaut on the International Space Station features the summit (2,282 meters, or 7,487 feet, above sea level) and the bare slopes above the tree line. Lower elevations are covered with green forests indicative of Southern Hemisphere summer."<ref name=Stefanov2013>{{ cite book |author=William L. Stefanov |title=Sollipulli Caldera, Chile and Argentina |publisher=NASA |location=Washington, DC USA |date=23 December 2013 |url=http://earthobservatory.nasa.gov/IOTD/view.php?id=82676 |accessdate=2014-10-15 }}</ref> "The summit of Sollipulli is occupied by a four-kilometer wide caldera, currently filled with a snow-covered glacier. While most calderas form after violent, explosive eruptions, the types of rock and other deposits associated with such events have not been found at Sollipulli. Geologic evidence does indicate explosive activity occurred about 2,900 years ago, and lava flows were produced approximately 700 years ago. Together with the craters and scoria cones along the outer flanks of the caldera, this history suggests Sollipulli could erupt violently again, presenting a potential hazard to towns such as Melipeuco and the wider region."<ref name=Stefanov2013/> {{clear}} ==Cirque glaciers== [[Image:Backed up Against the Wall.jpg|thumb|right|250px|A quarter mile of glacial ice is all that remains from the retreat of the glacier of Southwind Fiord, Baffin Island, Nunavut, Canada. Credit: [http://www.flickr.com/people/31856336@N03 Mike Beauregard from Nunavut, Canada].{{tlx|free media}}]] [[Image:Glacial Cirque Formation EN.svg|thumb|left|250px|Schematic profile of a cirque and cirque glacier, shows Bergschrund, randkluft and the headwall gap. Credit: [[c:User:ClemRutter|Clem Rutter]].{{tlx|free media}}]] Cirques, as diagrammed at the left, are formed by a glacier (the cirque glacier) and usually exhibit a Bergschrund, randkluft and the headwall gap. The image at the right shows a glacier on Baffin Island that has retreated back to a cirque glacier. {{clear}} ==Temperate glaciers== [[Image:HafrahvammagljúfurIV 02092006.jpg|thumb|right|250px|The canyons of Hafrahvammar are shown. Credit: [[c:User:Fbd|Friðrik Bragi Dýrfjörð]].{{tlx|free media}}]] At the right is an image of a temperate glacier; i.e., one flowing through a temperate region, as evidenced by the green plants. {{clear}} ==Ice fields== [[Image:Icefield.jpg|thumb|250px|right|This is an aerial image of the Kalstenius Icefield on Ellesmere Island, Canada. Credit: the Royal Canadian Air Force, archived at the World Data Center for Glaciology, Boulder, CO.{{tlx|fairuse}}]] [[Image:Juneau Icefield.jpg|thumb|left|250px|Picture shows the Juneau Icefield just north of Juneau, Alaska. Credit: Mendenhall, U.S. Forest Service.{{tlx|free media}}]] '''Def.''' "a mass of glacier ice; similar to an ice cap, and usually smaller and lacking a dome-like shape; somewhat controlled by terrain"<ref name=Beitler/> is called an '''icefield'''. The image at the right of "Kalstenius Icefield, located on Ellesmere Island, Canada, shows vast stretches of ice. The icefield produces multiple outlet glaciers that flow into a larger valley glacier. The glacier in this photograph is three miles wide."<ref name=Beitler/> The Juneau Icefield is located just north of Juneau, Alaska, continuing north through the border with British Columbia,<ref>{{cite web |url=http://www.juneauicefield.org/ |title = Juneau Icefield Research Program }}</ref> extending through an area of {{convert|3,900|km2|mi2}} in the Coast Range {{convert|140|km|abbr=on}} north to south and {{convert|75|km|abbr=on}} east to west. {{clear}} ==Ice caps== [[Image:Drill sites on Greenland.jpg|thumb|right|250px|The most important drill sites on the inland ice and on two small separate ice caps: Hans Tavsen in Peary Land in the north and Renland in the east are indicated. Credit: Willi Dansgaard.{{tlx|fairuse}}]] [[Image:Aaj-13201212214-1377205687.jpeg|thumb|left|250px|Looking south on Renland is across the Edward Bailey Glacier into the Alpine Bowl. Credit: Silvan Schüpbach.{{tlx|fairuse}}]] [[Image:Ice cap.jpg|thumb|left|250px|This is an aerial image of the ice cap on Ellesmere Island, Canada. Credit: National Snow and Ice Data Center.{{tlx|fairuse}}]] [[Image:Vatnajökull.jpeg|thumb|left|250px|Vatnajökull, Iceland has an ice cap. Credit: NASA.{{tlx|free media}}]] '''Def.''' "a dome-shaped mass of glacier ice that spreads out in all directions"<ref name=Beitler/> is called an '''ice cap'''. In addition to many of the ice core drilling sites on Greenland in the image at the right, there are the separate ice caps on Hans Tavsen in Peary Land way to the north and Renland in the east. In "1985, when [the final version of “the Rolls Royce drill”] penetrated the separate, high-lying Renland ice cap in the Scoresbysund Fiord [...] down to 325 m, world record for this type of drill".<ref name=Dansgaard/> The Renland ice core from East Greenland apparently covers a full glacial cycle from the Holocene into the previous Eemian interglacial. The Renland ice core is 325 m long.<ref name=Hansson1>{{ cite journal |author=Hansson M, Holmén K |title= |journal=Geophy Res Lett. |month=November |year=2001 |volume=28 |issue=22 |pages=4239-42 |doi=10.1029/2000GL012317 }}</ref> "The δ-profile [...] proved that the Renland ice cap has always been separated from the inland ice. Since all of the δ-leaps revealed by the Camp Century core recurred in the small Renland ice cap, the Renland peninsula cannot have been overrun by ice streams from the inland ice, not even during the glaciation.<ref name=Dansgaard/> The Penny Ice Cap is on Baffin Island, Canada, at 67° 15'N, 65° 45'W, 1900 masl. In April 1998 on the Devon Ice Cap filtered lamp oil was used as a drilling fluid. In the Devon core it was observed that below about 150 m the [[stratigraphy]] was obscured by microfractures.<ref name=P1386j>{{ cite book |author=C. SIMON L. OMMANNEY | title=HISTORY OF GLACIER INVESTIGATIONS IN CANADA | url=http://pubs.usgs.gov/prof/p1386j/history/history-lores.pdf | accessdate=October 14, 2005 }}</ref> "Beginning in 1995, a large outlet glacier of the Sermersauq Ice Cap on Disko Island [Greenland] surged 10.5 km downvalley to within 10 km of the head of the fjord, Kuannersuit Sulluat, reaching its maximum extent in summer 1999 before beginning to retreat. Sediment discharge to the fjord increased from 13 x 10³ t day^-1 in 1997 to 38 x 10³ t day^-1 in 1999. CTD results, sediment traps and cores from the 2000 melt season document the impact of the surge on the glacimarine environment of the fjord."<ref name=Gilbert>{{ cite journal |author=Robert Gilbert |author2=Niels Nielsen |author3=Henrik Möller |author4=Joseph R. Desloges |author5=Morten Rasch |title=Glacimarine sedimentation in Kangerdluk (Disko Fjord), West Greenland, in response to a surging glacier |journal=Marine Geology |year=2002 |volume=191 |issue= |pages=1-18 |url=http://geog.queensu.ca/gilbert/surge%20paper.PDF |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-09-24 }}</ref> "Short gravity cores were taken and CTD profiles were recorded at stations throughout Kuannersuit Sulluat [...]. Positions located by GPS are accurate to ±10 m or less. The stream flowing over the sandur to the head of the fjord was gauged and integrated suspended sediment samples were recovered from primary channels."<ref name=Gilbert/> "The cores were photographed, X-rayed and logged. X-radiographs provided measures of the number and size of gravel particles interpreted as ice-rafted debris (IRD) and the grey-scale (GS) of the scanned images was plotted as a measure of the properties of the sand and silt."<ref name=Gilbert/> "The twelve layers in core D4 [imaged at the right] suggest a mean period of about 20 days for these events based on the accumulation rates in the traps [...]. In general, these layers have both higher MS and X-radiographs have lighter toned GS, the former related to lower water content and the latter also related to greater absorption of X-rays by the larger rock and mineral fragments."<ref name=Gilbert/> There "are notable differences in the surge-generated sediments. The proximal sediments [such as in core D4 at the right] are more clearly laminated and layered in visual examination of the cores and as seen in the X-radiographs [compared to distal sediments as imaged on the left for core D20]. These consist both of the subtle differences in the fine-grained sediments on a millimetre scale, and of the sand layers up to 8 cm thick representing more energetic processes (Ó Cofaigh and Dowdeswell, 2001). Both are a response to greater sediment input to the fjord."<ref name=Gilbert/> The ice core drilled in Guliya ice cap in western China in the 1990s reaches back to 760,000 b2k; farther back than any other core at the time, though the EPICA core in Antarctica equalled that extreme in 2003.<ref name=Bowen>{{ cite book |author=Mark Bowen |year=2005 |title=Thin Ice |publisher=Henry Holt Company |isbn=0-8050-6443-5 }}</ref> Ice cores from Sajama in Bolivia span ~25 ka and help present a high resolution temporal picture of the Late Glacial Stage and the Holocene climatic optimum.<ref name=Thompson1>{{ cite journal |author=Thompson LG |author2=Mosley-Thompson EM |author3=Henderson KA |title=Ice-core palaeoclimate records in tropical South America since the Last Glacial Maximum |journal=J Quaternary Sci. |volume=15 |issue=4 |year=2000 |pages=377-94 |doi=10.1002/1099-1417(200005)15:4<377::AID-JQS542>3.0.CO;2-L }}</ref> Although the ice cores from Quelccaya ice cap only go back ~2 ka,<ref name=Thompson1/> others may go back ~5.2 ka. The Quelccaya ice cores correlate with those from the Upper Fremont Glacier. {{clear}} ==Greenland ice sheets== [[Image:Greenland 42.74746W 71.57394N.jpg|thumb|right|250px|Satellite composite image shows the ice sheet of Greenland. Credit: NASA.{{tlx|free media}}]] [[Image:Une partie de l'hémisphère nord de la Terre avec la banquise, nuage, étoile et localisation de la station météo en Alert.jpg|thumb|left|250px|Earth's northern hemisphere polar ice sheet includes sea ice. Credit: NASA/Goddard Space Flight Center.{{tlx|free media}}]] [[Image:Grl18577-fig-0001.png|thumb|right|250px|(a) The probability is for of a pixel melting at least as many times as observed during the 1995, 1998 and 2002 melt seasons given the last 25 years of melt observations. (b) Melt extent is for 2002: Pixels are color coded for number of melt days during the season. (c) Slopes of the trend lines are fit to the areas observed to melt between April and November from 1979 to 2003. Credit: K. Steffen, S. V. Nghiem, R. Huff, and G. Neumann.{{tlx|fairuse}}]] [[Image:Grl18577-fig-0002.png|thumb|left|250px|Half-decade records for ETH/CU Camp station: (a) Top panel is for QSCAT backscatter, (b) middle panel for QSCAT diurnal signature, and (c) bottom panel for air temperature measured at the AWS site. Credit: K. Steffen, S. V. Nghiem, R. Huff, and G. Neumann.{{tlx|fairuse}}]] [[Image:Grl18577-fig-0003.png|thumb|right|250px|QSCAT melt maps are shown on the climatological peak-melt day (1 August). Red color represents current active melt areas, light blue is for areas that have melted but currently refreeze, white is for areas that will melt later, and magenta is for areas that do not experience any melt throughout the melt season. The dark blue color surrounding Greenland is the ocean mask. Credit: K. Steffen, S. V. Nghiem, R. Huff, and G. Neumann.{{tlx|fairuse}}]] [[Image:Grl18577-fig-0004.png|thumb|left|250px|QSCAT maps of number of melt days (violet to red for 1 to 31 days) in 2000–2003 with the overlaid black contours representing melt extent derived from PM data are shown. Credit: K. Steffen, S. V. Nghiem, R. Huff, and G. Neumann.{{tlx|fairuse}}]] '''Def.''' "a dome-shaped mass of glacier ice that covers surrounding terrain and is greater than 50,000 square kilometers (12 million acres)"<ref name=Beitler/> is called an '''ice sheet'''. At the right is a satellite composite image of the ice sheet over Greenland. "Active and passive microwave satellite data are used to map snowmelt extent and duration on the Greenland ice sheet. The passive microwave (PM) data reveal the extreme melt extent of 690,000 km² in 2002 as compared with an average extent of 455,000 km² from 1979–2003."<ref name=Steffen>{{ cite journal |author=K. Steffen |author2=S. V. Nghiem |author3=R. Huff |author4=G. Neumann |title=The melt anomaly of 2002 on the Greenland Ice Sheet from active and passive microwave satellite observations |journal=Geophysical Research Letters |month=21 October |year=2004 |volume=21 |issue=20 |pages= |url= |arxiv= |bibcode= |doi=10.1029/2004GL020444 |pmid= |accessdate=2014-09-28 }}</ref> "Several PM-based melt assessment algorithms [Mote and Anderson, 1995; Abdalati and Steffen, 1995] are applicable to Scanning Multi-channel, Microwave Radiometer (SMMR) and Special Sensor Microwave/Imager (SSM/I) instruments providing near-continuous coverage since 1979. The PM data as gridded brightness temperatures on polar stereographic grids (25 km resolution) [used] are from the National Snow and Ice Data Center [Maslanik and Stroeve, 2003], containing daily data spanning 25 melt seasons from 1979 to 2003."<ref name=Steffen/> In the second image on the right, (a) "shows the probabilities of the observed melt behavior on the Greenland ice sheet for several large melt years and indicates the extreme melt anomaly observed in northeastern Greenland in 2002."<ref name=Steffen/> "Prior to 2002, both 1995 and 1998 were extreme melt years in terms of maximum areal extent and total melt. During 1995 melt was dominated by a high frequency of melt along the western margin of the ice sheet. During 1998 melt was spatially diverse with slightly more melt than usual in the northeast and southwest. However, the high frequency melt in 2002 in the northeast and along the western margin is unprecedented in the PM record with a log likelihood of occurrence that is 35% lower than the previous record melt anomaly in 1991."<ref name=Steffen/> (c) "depicts the magnitude of the increasing trends in melt extent on a daily basis over the last 25 years. Although there is a large amount of inter-annual variability in melt extent on a given day, 56 days show statistically significant (alpha = 0.1) increasing trends in melt area."<ref name=Steffen/> "Melt along the west coast was extensive during 2002 but not atypical for large melt years. However melt in the north and northeast was highly irregular both in terms of extent and frequency. Nearly 3,000 km²[(b)] were classified as melting during 2002 that had not previously melted during any other year between 1979 and 2003."<ref name=Steffen/> The figure at the left "presents QSCAT backscatter and diurnal signatures, and ETH/CU AWS air temperature."<ref name=Steffen/> Half-decade records for ETH/CU Camp station: (a) Top panel is for QSCAT backscatter, (b) middle panel for QSCAT diurnal signature, and (c) bottom panel for air temperature measured at the AWS site.<ref name=Steffen/> At the lower right QSCAT melt maps are shown on the climatological peak-melt day (1 August). Red color represents current active melt areas, light blue is for areas that have melted but currently refreeze, white is for areas that will melt later, and magenta is for areas that do not experience any melt throughout the melt season. The dark blue color surrounding Greenland is the ocean mask. "QSCAT mapping can reveal details of the spatial pattern of surface melt evolution in time. There are large variabilities in melt extent and melt timing over different regions. [The figure at tje lower right] confirms that 2002 has the most extensive areal melt. In 2002, the northeast quadrant of the Greenland ice sheet, extending well into the dry snow zone, experienced at least some melt where melt never happened before (from satellite data records to date). Since the beginning of the QSCAT data record (July 1999), the smallest spatial extent of melt occurred in 2001, and melt extent was similar for years 2000 and 2003."<ref name=Steffen/> "To provide a direct comparison of PM and QSCAT results, we overlay results for PM melt extent and QSCAT number of melt days in [the figure at the lower left] for years 2000–2003. PM XPGR melt extent is approximately confined to QSCAT melt areas experiencing 2 weeks or more of melting time [the figure at the lower left]. QSCAT melt areas outside of the PM melt extent represent the surface that has less melt corresponding to about 15 melt days or less. This is consistent with the relationship of relative melt strength measured by active and passive data as discussed above. Note that such areas can total up to a large region in year 2002. Surface albedo can reduce considerably once the snow melts for a period of 2 weeks. The albedo reduction may significantly impact the surface heat balance and thus change the mass balance. The large number of melt days around the northern perimeter of the ice sheet, which is shown as the narrow dark-red band in north Greenland in the 2003 map was an anomalous feature [the figure at the lower left]. This band was wider as defined by the PM melt extent in 2002 than in 2003. However, there were more QSCAT melt days in the 2003 northern melt band."<ref name=Steffen/> "The comparison reveals that the PM cross-polarized gradient algorithm classifies melt more conservatively than the scatterometer algorithm. The active microwave identifies melt approximately up to two weeks more than the PM at higher elevation in the percolation zone toward the dry snow zone [the figure at the lower left]. Both methods (active and passive microwave) consistently identify melt areas that have a melt duration of at least 10–14 days. The longer snowmelt duration can be sufficient to decrease surface albedo and affect surface heat and mass balance."<ref name=Steffen/> {{clear}} ==Antarctic ice sheets== [[Image:Antarctica 6400px from Blue Marble.jpg|thumb|right|250px|A satellite composite image shows the ice sheet of Antarctica Credit: [[c:User:Dave Pape|Dave Pape]].{{tlx|free media}}]] [[Image:Antarktyda i Antarktyka.jpg|thumb|left|250px|A satellite composite image shows a global view of the sea ice and ice sheet of Antarctica. Credit: NASA Scientific Visualization Studio Collection.{{tlx|free media}}]] [[Image:Antarctic-ice-flow inline.png|right|thumb|300px|Velocity of ice flowing across Antarctica varies by location. Credit: Jeremie Mouginot, University of California Irvine.{{tlx|fairuse}}]] "The only current ice sheets are in Antarctica and [[Greenland]]; during the last glacial period at Last Glacial Maximum (LGM) the Laurentide ice sheet covered much of North America, the Weichselian ice sheet covered northern Europe and the Patagonian Ice Sheet covered southern South America."<ref name=IceSheet>{{ cite book |title=Ice sheet |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=June 6, 2013 |url=http://en.wikipedia.org/wiki/Ice_sheet |accessdate=2013-06-23 }}</ref> At the south pole, Antactica, there is also an extensive ice sheet shown in the second image on the right. Seasonally, when the North polar sea ice and ice sheet has been contracting, the South polar sea ice and ice sheet has been expanding. Apparent global warming that was progressively melting more and more of the north polar ice sheet each year has been countered by progressive expansion of the south polar ice sheet. "Decades of satellite observations have now provided the most detailed view yet [second image down on the right] of how Antarctica continually sheds ice accumulated from snowfall into the ocean."<ref name=Temming>{{ cite book |author=Maria Temming |title=A new map is the best view yet of how fast Antarctica is shedding ice |publisher=Science News |location= |date=9 August 2019 |url=https://www.sciencenews.org/article/new-map-best-view-yet-how-fast-antarctica-shedding-ice?utm_source=Editors_Picks&utm_medium=email&utm_campaign=editorspicks081119 |accessdate=12 August 2019 }}</ref> The "first comprehensive view of how ice moves across all of Antarctica, [includes] slow-moving ice in the middle of the continent rather than just rapidly melting ice at the coasts."<ref name=Temming/> Subtle "movements of Antarctic ice [were detected] with a kind of measurement called synthetic-aperture radar interferometric phase data."<ref name=Temming/> "By using a satellite to bounce radar signals off a patch of ice, [...] how quickly that ice is moving toward or away from the satellite [can be determined]. Combining observations of the same spot from different angles reveals the speed and direction of the ice’s motion along the ground."<ref name=Temming/> "Inland ice moves incredibly slowly — much of it plods along at fewer than 10 meters per year. Closer to the ocean, ice can travel hundreds to thousands of meters per year."<ref name=Temming/> "To get multiple vantage points of the same swathes of ice, [...] data from about half a dozen satellites launched by Canada, Europe and Japan since the early 1990s [was put together]."<ref name=Temming/> "Each brought a little piece of the puzzle."<ref name=Rignot>{{ cite book |author=Eric Rignot |title=A new map is the best view yet of how fast Antarctica is shedding ice |publisher=Science News |location= |date=9 August 2019 |url=https://www.sciencenews.org/article/new-map-best-view-yet-how-fast-antarctica-shedding-ice?utm_source=Editors_Picks&utm_medium=email&utm_campaign=editorspicks081119 |accessdate=12 August 2019 }}</ref> "Surface ice velocity is a fundamental characteristic of glaciers and ice sheets that quantifies the transport of ice. Changes in ice dynamics have a major impact on ice sheet mass balance and its contribution to sea level rise. Prior comprehensive mappings employed speckle and feature tracking techniques, optimized for fast‐flow areas, with precision of 2‐5 m/yr, hence limiting our ability to describe ice flow in the slow interior. We present a vector map of ice velocity using the interferometric phase from multiple satellite synthetic aperture radars resulting in ten‐times higher precision in speed (20 cm/yr) and direction (5o) over 80% of Antarctica. Precision mapping over areas of slow motion (< 1 m/yr) improves from 20% to 93%, which helps better constrain drainage boundaries, improve mass balance assessment, evaluate regional atmospheric climate models, reconstruct ice thickness, and inform ice sheet numerical models."<ref name=Mouginot>{{ cite journal |author=Jeremie Mouginot |author2=E. Rignot |author3=B. Scheuchl |title=Continent‐wide, interferometric SAR phase, mapping of Antarctic ice velocity |journal=Geophysical Research Letters |date=29 July 2019 |volume= |issue= |pages= |url=https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2019GL083826 |arxiv= |bibcode= |doi=10.1029/2019GL083826 |pmid= |accessdate=12 August 2019 }}</ref> {{clear}} ==Himalayas ice sheets== [[Image:Himalayas landsat 7.png|thumb|right|250px|This is a Landsat 7 image of the Himalayas. NASA.{{tlx|fairuse}}]] Often called the third pole, the image on the right shows the rocky ice sheet over the top of the Himalayas. {{clear}} ==Icequakes== [[Image:Stations-1.jpg|thumb|right|250px|This map of Antarctica shows the icequakes triggered by Chile's 2010 earthquake. Credit: Zhigang Peng, Georgia Tech.{{tlx|fairuse}}]] "Only 12 of Antarctica's 42 seismometers picked up icequakes after the Maule earthquake, but the signals seemed to fit a pattern. The pattern suggests that opening or closing of shallow crevasses generated the tiny tremors. For example, seismic stations near Antarctica's mountain ranges and fast-flowing ice rivers known as ice streams were more likely to see icequakes. These are areas with a lot of crevasses. The high-frequency shaking also fits with cracking of '''brittle ice'''."<ref name=Oskin>{{ cite book |author=Becky Oskin |title=Faraway Earthquake Triggered Antarctica Icequakes |publisher=LiveScience.com |location= |date=10 August 2014 |url=http://www.livescience.com/47282-chile-earthquake-caused-antarctica-icequakes.html |accessdate=2014-08-16 }}</ref> Bold added. "Antarctica's ice snapped and popped because of a major earthquake in Maule, Chile, halfway around the world [...] Antarctica has been touched by great earthquakes before. In March 2011, Japan's Tohoku tsunami tore off two Manhattan-size icebergs from the Sulzberger Ice Shelf, more than 8,000 miles (13,000 kilometers) south. Sailors also reported a massive Antarctica iceberg-calving event after Chile's 1868 great earthquake."<ref name=Oskin/> "Icequakes are seismic tremblings caused by sudden movement within a glacier or ice sheet, such as from a fracturing crevasse. (Anyone who has dropped an ice cube into a glass of water knows ice snaps under stress.)"<ref name=Oskin/> "Chile's magnitude-8.8 earthquake on Feb. 27, 2010, set off a flurry of Antarctic icequakes, each lasting from one to 10 seconds, researchers report today (Aug. 10) in the journal Nature Geoscience. The epicenter was 2,900 miles (4,700 km) north of Antarctica."<ref name=Oskin/> "We think the crevasses are being activated by the surface waves from this big earthquake coming through, and that's making the icequake."<ref name=Walter>{{ cite book |author=Jacob Walter |title=Faraway Earthquake Triggered Antarctica Icequakes |publisher=LiveScience.com |location= |date=10 August 2014 |url=http://www.livescience.com/47282-chile-earthquake-caused-antarctica-icequakes.html |accessdate=2014-08-16 }}</ref> "Regular icequakes probably occur all the time in Antarctica and other polar regions."<ref name=Peng>{{ cite book |author=Zhigang Peng |title=Faraway Earthquake Triggered Antarctica Icequakes |publisher=LiveScience.com |location= |date=10 August 2014 |url=http://www.livescience.com/47282-chile-earthquake-caused-antarctica-icequakes.html |accessdate=2014-08-16 }}</ref> "What we found is that they occurred more during the seismic waves of the Maule event."<ref name=Peng/> "Many different kinds of icequakes rumble across Antarctica and Greenland. Known icequake triggers include opening and closing of the fractures called crevasses; glaciers tearing away from sticky bedrock; water runoff; and calving, the breaking off of an iceberg. Spooky underwater sounds from melting, cracking icebergs were once called The Bloop."<ref name=Oskin/> Just "one kind of seismic wave, a surface wave, gets the blame for most of Antarctica's icequakes. [...] a Rayleigh wave [...] travels close to the Earth's surface, rolling along like a wave in a lake or the ocean. [...] At some stations, there was also a short icequake burst from a seismic "P wave," which travel through the Earth's interior."<ref name=Oskin/> {{clear}} ==Ice shelves== [[Image:A-62 iceberg connected to Fimbul Ice Shelf at Queen Maud Land.jpg|thumb|right|250px|Iceberg A 62 was connected to the Fimbul Ice Shelf by a mere 800-metre-wide bridge. Credit: DLR - German Space Agency.{{tlx|free media}}]] [[Image:Antarctic shelf ice hg.png|thumb|left|250px|This is a schematic of glaciological and oceanographic processes along the Antarctic coast. Credit: [[c:User:Hgrobe|Hannes Grobe]], Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany.{{tlx|free media}}]] [[Image:Wardhunt.jpg|thumb|right|250px|Canadian RADARSAT image shows the shelf in August 2002, when a crack made its way down the length of the shelf. Credit: Alaska Satellite Facility, Geophysical Institute, University of Alaska Fairbanks.{{tlx|fairuse}}]] [[Image:Iceberg A-38.jpg|thumb|left|250px|This is an image of iceberg A-38 after it detached from the Ronne Ice Shelf. Credit: National Ice Center/National Oceanic and Atmospheric Administration.{{tlx|free media}}]] "A small island obstructs the constant flow of the ice shelf on Queen Maud Land – it is the lighter area at the bottom left of the image [on the right]. From September 2010 until it broke off, Iceberg A 62 was connected to the Fimbul Ice Shelf by a mere 800-metre-wide bridge. Two fissures in the ice from different sides of the bridge approached one another until the break occurred. The images transmitted by the radar satellite TerraSAR-X over a long period of time should enable researchers to achieve a better understanding of how icebergs calve."<ref name=DLR>{{ cite book |author=DLR |title=The iceberg breaks free |publisher=Deutsches Zentrum für Luft- und Raumfahrt |location=Cologne |date=20 January 2011 |url=http://www.dlr.de/media/en/desktopdefault.aspx/tabid-4986/8423_read-18675/8423_page-2 |accessdate=2016-09-20 }}</ref> '''Def.''' a "thick, floating platform of ice that forms where a glacier or ice sheet flows down to a coastline and onto the ocean surface"<ref name=IceShelfWikt>{{ cite book |title=ice shelf, In: ''Wiktionary'' |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=24 May 2014 |url=https://en.wiktionary.org/wiki/ice_shelf |accessdate=2014-10-30 }}</ref> is called an '''ice shelf'''. "Ice shelves are permanent floating sheets of ice that connect to a landmass."<ref name=Staff>{{ cite book |author=Staff |title=Quick Facts on Ice Shelves |publisher=National Snow & Ice Data Center |location= |date= 2014 |url=https://nsidc.org/cryosphere/quickfacts/iceshelves.html |accessdate=2014-10-31 }}</ref> "Ice shelves fall into three categories: (1) ice shelves fed by glaciers, (2) ice shelves created by sea ice, and (3) composite ice shelves (Jeffries 2002). Most of the world's ice shelves, including the largest, are fed by glaciers and are located in Greenland and Antarctica."<ref name=Staff/> "One example of an ice shelf composed of compacted, thickened sea ice is the Ward Hunt Ice Shelf off the coast of Ellesmere Island in northern Canada. Canadian RADARSAT image shows the shelf in August 2002, when a crack made its way down the length of the shelf."<ref name=Staff/> The Ronne Ice Shelf has a nominal location of 78°30'S 61°W. {{clear}} ==Calving== [[Image:Perito Moreno Glacier Patagonia Argentina Luca Galuzzi 2005.JPG|thumb|250px|right|This shows calving by the Perito Moreno Glacier, in Los Glaciares National Park, southern Argentina. Credit: [[c:User:Lucag|Luca Galuzzi]].{{tlx|free media}}]] [[Image:Loosetooth.jpg|thumb|right|250px|These Multi-angle Imaging SpectroRadiometer (MISR) images show the progression of a "loose tooth"—an iceberg calving from the Amery Ice Shelf. Credit: NASA Earth Observatory, Clare Averill and David J. Diner, Jet Propulsion Laboratory; and Helen A. Fricker, Scripps Institution of Oceanography.{{tlx|fairuse}}]] [[Image:Jakobshavn retreat-1851-2006.jpg|thumb|right|200px|Retreating calving front of the Jacobshavn Isbrae glacier in Greenland from 1851 - 2006. Credit: NASA Earth Observatory, Cindy Starr, based on data from Ole Bennike and Anker Weidick (Geological Survey of Denmark and Greenland) and Landsat data.{{tlx|free media}}]] [[Image:Larsen 2006.jpg|thumb|right|200px|Photos show the A54 iceberg calving from the Scar Inlet Shelf (the remainder of the Larsen Ice Shelf). Credit: Ted Scambos, National Snow and Ice Data Center, University of Colorado, Boulder, and NASA Moderate Resolution Imaging Spectroradiometer images courtesy NASA Earth Observatory.{{tlx|fairuse}}]] '''Def.''' a "process by which ice breaks off a glacier's terminus"<ref name=Beitler/> is called '''calving'''. '''Def.''' the "breaking away of a mass of ice from an iceberg, glacier etc"<ref name=CalvingWikt>{{ cite book |author=[[wikt:User:SemperBlotto|SemperBlotto]] |title=calving |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=11 September 2007 |url=https://en.wiktionary.org/wiki/calving |accessdate=26 September 2022 }}</ref> is called '''calving'''. The image on the right shows calving by the Perito Moreno Glacier, in Los Glaciares National Park, southern Argentina. "Calving of huge, tabular icebergs is unique to Antarctica, and the process can take a decade or longer. Calving results from rifts that reach across the shelf. In the case of Antarctica's Amery Ice Shelf, the calving area resembles a loose tooth [images on the second right]." per Clare Averill and David J. Diner, and Helen A. Fricker, State of the Cryosphere: Ice calves at . On a stable ice shelf, calving is a near-cyclical, repetitive process producing large icebergs every few decades. The icebergs drift generally westward around the continent, and as long as they remain in the cold, near-coastline water, they can survive decades or more. However, they eventually are caught up in north-drifting currents where they melt and break apart. In Greenland, floating ice tongues downstream from large outlet glaciers are more broken up by crevasses. Calving of the ice tongues releases armadas of smaller, steep-sided icebergs that drift south sometimes reaching North Atlantic shipping lanes. Calving of the large glacier, Jacobshavn, on the east coast of Greenland is responsible for the majority of icebergs reaching Atlantic shipping and fishing areas off of Newfoundland and most likely shed the iceberg responsible for the sinking of the Titanic in 1912. The Petermann Glacier in northwestern Greenland also shed a large ice island in August 2010. These denizens of the ocean are now tracked by the National Ice Center in the United States, along with other organizations. By 2006, the Jacobshavn Glacier, third image on the right, had retreated back to where its two main tributaries join, leading to two fast-flowing glaciers where there had previously been just one. The rapidly retreating Jakobshavn Glacier in western Greenland drains the central ice sheet. This image, third one on the right, shows the glacier in 2001, flowing from upper right to lower left. Terminus locations before 2001 were determined by surveys and more recent contours were derived from Landsat data. The recent stages of retreat have widened the ice front, placing more of the glacier in contact with the ocean. In recent years, calving of the largest ice tongues in Greenland (in particular, Jacobshavn, Helheim, and Kangerdlugssuaq) has accelerated probably due to warmer air and/or ocean temperatures. As the ice tongues have retreated, the reduced backpressure against the glacier has allowed these glaciers to accelerate significantly. The images, fourth set of images on the right, show a tabular iceberg calving from an ice shelf. This iceberg happens to be calving from the remnant piece of the Larsen B ice shelf at the southwestern corner of the embayment. While the Larsen B Ice Shelf underwent disintegration [...], this was a normal calving event. Large tabular iceberg calvings are natural events that occur under stable climatic conditions, so they are not a good indicator of warming or changing climate. Over the past several decades, however, meteorological records have revealed atmospheric warming on the Antarctic Peninsula, and the northernmost ice shelves on the peninsula have retreated dramatically (Vaughan and Doake 1996). The most pronounced ice shelf retreat has occurred on the Larsen Ice Shelf, located on the eastern side of the Antarctic Peninsula's northern tip. The shelf is divided into four regions from north to south: A, B, C, and D. {{clear}} ==Icebergs== [[Image:Iceberg in the Arctic with its underside exposed.jpg|right|thumb|250px|When the polar sea is calm, the underside of icebergs can easily be observed in the clear waters of the Arctic Ocean. Credit: [[c:User:AWeith|AWeith]].{{tlx|free media}}]] [[Image:Black ice growler upernavik 2007-07-07a.jpg|right|thumb|250px|Black ice growler from a recently calved iceberg is closing in on the shore at the old heliport in Upernavik, Greenland. Credit: [[c:User:Slaunger|Kim Hansen]].{{tlx|free media}}]] [[Image:Black ice growler texture upernavik 2007-07-07.jpg|left|thumb|250px|Surface texture on a growler of black ice. Credit: [[c:User:Slaunger|Kim Hansen]].{{tlx|free media}}]] '''Def.''' a "huge mass of ocean-floating ice which has broken off a glacier or ice shelf"<ref name=IcebergWikt>{{ cite book |author=[[wikt:User:Interwicket|Interwicket]] |title=iceberg |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=10 September 2010 |url=https://en.wiktionary.org/wiki/iceberg |accessdate=26 September 2022 }}</ref> is called an '''iceberg'''. The first image on the right shows that when the polar sea is calm, the underside of icebergs can easily be observed in the clear waters of the Arctic Ocean. Centered in the image second down on the right is a black ice growler from a recently calved iceberg closing in on the shore at the old heliport in Upernavik, Greenland. Such black ice growlers originate from glacial rifts, or crevasses, filled with melting water, which freezes into transparent ice without air bubbles. On the left is an image of the surface texture on a black ice growler. There are bowl-like depressions in the surface created by the melting process of sea water. {{clear}} ==Sea ices== [[Image:Greenland East Coast 7.jpg|thumb|right|250px|This is an aerial view of the pack ice off the eastcoast of Greenland. Credit: [[c:user:Michael Haferkamp|Michael Haferkamp]].{{tlx|free media}}]] [[Image:Vaxholm ice.jpg|thumb|left|250px|This is pack ice off the coast of Vaxholm, Sweden. Credit: [[c:User:Cyberjunkie|Cyberjunkie]].{{tlx|free media}}]] [[Image:Line3892 - Flickr - NOAA Photo Library.jpg|thumb|right|250px|Pack-ice-covered Auke Bay Harbor, Alaska, in winter. Credit: David Csepp, NOAA/NMFS/AKFSC/ABL.{{tlx|free media}}]] [[Image:Seaice 04.jpg|thumb|left|250px|When waves buffet the freezing ocean surface, characteristic "pancake" sea ice forms. Credit: Ted Scambos, NSIDC.{{tlx|fairuse}}]] A "climate interpretation was supported by very low δ’s in the 1690’es, a period described as extremely cold in the Icelandic annals. In 1695 Iceland was completely surrounded by sea ice, and according to other sources the sea ice reached half way to the Faeroe Islands."<ref name=Dansgaard2005>{{ cite book |author=Willi Dansgaard |title=Frozen Annals Greenland Ice Cap Research |publisher=Niels Bohr Institute |location=Copenhagen, Denmark |year=2005 |editor=The Department of Geophysics of The Niels Bohr Institute for Astronomy Physics and Geophysics at The University of Copenhagen Denmark |pages=123 |url=http://www.iceandclimate.nbi.ku.dk/publications/FrozenAnnals.pdf/ |arxiv= |bibcode= |doi= |pmid= |isbn=87-990078-0-0 |accessdate=2014-10-05 }}</ref> "The correlation is astonishing, because it implies that the dramatic climate changes during the first more than 50 kyrs of the glaciation elapsed nearly in parallel on both sides of the North Atlantic Ocean, presumably controlled by varying sea ice cover. Thus, the Gulf Stream was not just deflected toward North Africa in cold periods, it was rather turned off."<ref name=Dansgaard2005/> '''Def.''' a "large consolidated mass of floating sea ice"<ref name=PackIceWikt>{{ cite book |title=pack ice, In: ''Wiktionary'' |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=29 May 2014 |url=https://en.wiktionary.org/wiki/pack_ice |accessdate=2014-11-01 }}</ref> is called '''pack ice'''. Pack ice in the image on the right is drifting southward in the East Greenland current during July 1996. In the second image on the left, when "waves buffet the freezing ocean surface, characteristic "pancake" sea ice forms."<ref name=Scambos>{{ cite book |author=Ted Scambos |title=Quick Facts on Arctic Sea Ice |publisher=National Snow & Ice Data Center |location= |date= 2004 |url=http://nsidc.org/cryosphere/quickfacts/seaice.html |accessdate=2014-11-03 }}</ref> "Sheets of sea ice form when frazil crystals float to the surface, accummulate and bond together. Depending upon the climatic conditions, sheets can develop from grease and congelation ice, or from pancake ice."<ref name=Scambos1>{{ cite book |author=Ted Scambos |title=Ice formation |publisher=National Snow & Ice Data Center |location= |date= 2004 |url=http://nsidc.org/cryosphere/seaice/characteristics/formation.html |accessdate=2014-11-03 }}</ref> "If the ocean is rough, the frazil crystals accummulate into slushy circular disks, called pancakes or pancake ice, because of their shape. A signature feature of pancake ice is raised edges or ridges on the perimeter, caused by the pancakes bumping into each other from the ocean waves. If the motion is strong enough, rafting occurs. If the ice is thick enough, ridging occurs, where the sea ice bends or fractures and piles on top of itself, forming lines of ridges on the surface. Each ridge has a corresponding structure, called a keel, that forms on the underside of the ice. Particularly in the Arctic, ridges up to 20 meters (60 feet) thick can form when thick ice deforms. Eventually, the pancakes cement together and consolidate into a coherent ice sheet. Unlike the congelation process, sheet ice formed from consolidated pancakes has a rough bottom surface."<ref name=Scambos1/> {{clear}} ==Lahars== [[Image:MSH82 lahar from march 82 eruption 03-21-82.jpg|thumb|right|250px|An explosive eruption of Mount St. Helens on March 19, 1982, sent pumice and ash 9 miles (14 kilometers) into the air, and resulted in a lahar (the dark deposit on the snow) flowing from the crater into the North Fork Toutle River valley. Credit: Tom Casadevall.{{tlx|free media}}]] "Because the volcano itself is covered by 15 square miles of glaciers, the lava that flows down the side and mixes with ice and snow to form lahars — a mudflow slurry that can move extremely quickly and destroy towns in their path. According to the Smithsonian, "lahars have damaged towns on Villarica's flanks." The BBC reports that more than 100 people are believed to have been killed by the volcano's mudflows in the past century."<ref name=Plumer>{{ cite book |author=Brad Plumer |title=Chile's recent volcanic eruption looked absolutely stunning — and terrifying |publisher=Vox |location=Villarrica, Chile |date=4 March 2015 |url=http://www.vox.com/2015/3/4/8147751/chile-volcano-villarrica |accessdate=2015-03-27 }}</ref> '''Def.''' a "volcanic mudflow"<ref name=LaharWikt>{{ cite book |author=[[wikt:User:Emperorbma|Emperorbma]] |title=lahar |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=1 December 2004 |url=https://en.wiktionary.org/wiki/lahar |accessdate=4 May 2019 }}</ref> is called a '''lahar'''. Part of the Mount St. Helens lahar entered Spirit Lake (lower left corner of the image on the right) but most of the flow went west down the Toutle River, eventually reaching the Cowlitz River, 50 miles (80 kilometers) downstream. {{clear}} ==Lightning== {{main|Lightning}} [[Image:Rinjani 1994.jpg|thumb|right|250px|The 1995 eruption of Mount Rinjani in Indonesia exhibits volcanic lightning. Credit: [[commons:User:Spolloman|Oliver Spalt]].{{tlx|free media}}]] [[Image:Galunggung.jpg|thumb|left|250px|The slide depicts a spectacular view of lightning strikes during a third eruption on December 3, 1982. Credit: R. Hadian, U.S. Geological Survey.{{tlx|free media}}]] Many volcanic eruptions put on impressive lightning displays such as during the 1995 eruption of Mount Rinjani in Indonesia shown in the image on the right which exhibits many leaders. The image on the left shows spectacular lightning strikes around Galunggung, including multiple leaders apparently involved in cloud to cloud lightning. "This stratovolcano with a lava dome is located in western Java. Its first eruption in 1822 produced a 22-km-long mudflow that killed 4,000 people. The second eruption in 1894 caused extensive property loss. The photo depicts a spectacular view of lightning strikes during a third eruption on December 3, 1982, which resulted in 68 deaths. A fourth eruption occurred in 1984."<ref name=Hadian>{{ cite book |author=R. Hadian |title=Galunggung, Indonesia |publisher=NOAA National Geophysical Data Center |location= |date=3 December 1982 |url=http://www.ngdc.noaa.gov/nndc/servlet/ShowDatasets?EQ_0=603&bt_0=&st_0=&EQ_1=&bt_1=&st_1=&query=&dataset=101634&search_look=2&group_id=null&display_look=4,44&submit_all=Select+Data |accessdate=2015-03-24 }}</ref> Volcanic lightning arises from colliding, fragmenting particles of volcanic ash (and sometimes ice),<ref>{{Cite news |url=https://www.washingtonpost.com/news/capital-weather-gang/wp/2016/04/13/scientists-think-theyve-solved-the-mystery-of-how-volcanic-lightning-forms/?utm_term=.80462ad8051d |title=Scientists think they've solved the mystery of how volcanic lightning forms |last=Fritz |first=Angela |date=2016 |work=The Washington Post |accessdate= }}</ref><ref>{{Cite news |url=https://www.seeker.com/mystery-of-volcano-lightning-explained-1771209774.html |title=Mystery of Volcano Lightning Explained |last=Mulvaney |first=Kieran |date=2016 |work=Seeker |accessdate= }}</ref> which generate [[static electricity]] within the volcanic plume.<ref>{{Cite news |url=https://blogs.agu.org/geospace/2016/04/12/new-studies-uncover-mysterious-processes-generate-volcanic-lightning/ |title=New studies uncover mysterious processes that generate volcanic lightning |last=Lipuma |first=Lauren |date=2016 |work=American Geophysical Union GeoSpace Blog |accessdate= }}</ref> Volcanic eruptions have been referred to as '''dirty thunderstorms'''<ref>{{Cite journal |last=Hoblitt |first=Richard P. |date=2000 |title=Was the 18 May 1980 lateral blast at Mt St Helens the product of two explosions? |journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences |volume=358 |issue=1770 |pages=1639–1661 |doi=10.1098/rsta.2000.0608 |doi-access=free }}</ref><ref name=":6">{{Cite journal |last=Bennett |first=A J |last2=Odams |first2=P |last3=Edwards |first3=D |last4=Arason |first4=Þ |date=2010-10-01 |title=Monitoring of lightning from the April–May 2010 Eyjafjallajökull volcanic eruption using a very low frequency lightning location network |journal=Environmental Research Letters |volume=5 |issue=4 |pages=044013 |doi=10.1088/1748-9326/5/4/044013 |doi-access=free |issn=1748-9326 }}</ref> due to [http://glossary.ametsoc.org/wiki/Moist_convection moist convection] and ice formation that drive the eruption plume dynamics<ref name=":7">{{Cite journal |last=Woods |first=Andrew W. |date=1993 |title=Moist convection and the injection of volcanic ash into the atmosphere |journal=Journal of Geophysical Research: Solid Earth |volume=98 |pages=17627–17636 |doi=10.1029/93JB00718 }}</ref><ref name=":8">{{Cite journal |last=Van Eaton |first=Alexa R. |last2=Mastin |first2=Larry G. |last3=Herzog |first3=Michael |last4=Schwaiger |first4=Hans F. |last5=Schneider |first5=David J. |last6=Wallace |first6=Kristi L. |last7=Clarke |first7=Amanda B. |date=2015-08-03 |title=Hail formation triggers rapid ash aggregation in volcanic plumes |journal=Nature Communications |volume=6 |issue=1 |doi=10.1038/ncomms8860 |doi-access=free |issn=2041-1723 }}</ref> and can trigger volcanic lightning.<ref>{{Cite journal |last=Williams |first=Earl R. |last2=McNutt |first2=Stephen R. |date=2005 |title=Total water contents in volcanic eruption clouds and implications for electrification and lightning |url=http://www.giseis.alaska.edu/Input/steve/PUBS/williams-mcn-signpost.PDF |journal=Proceedings of the 2nd International Conference on Volcanic Ash and Aviation Safety |volume= |pages=67–71 }}</ref><ref name=":9">{{Cite journal |last=Van Eaton |first=Alexa R. |last2=Amigo |first2=Álvaro |last3=Bertin |first3=Daniel |last4=Mastin |first4=Larry G. |last5=Giacosa |first5=Raúl E. |last6=González |first6=Jerónimo |last7=Valderrama |first7=Oscar |last8=Fontijn |first8=Karen |last9=Behnke |first9=Sonja A. |date=2016-04-12 |title=Volcanic lightning and plume behavior reveal evolving hazards during the April 2015 eruption of Calbuco volcano, Chile |journal=Geophysical Research Letters |volume=43 |issue=7 |pages=3563–3571 |doi=10.1002/2016gl068076 |doi-access=free |issn=0094-8276 |via= }}</ref> But unlike ordinary thunderstorms, volcanic lightning can also occur before any ice crystals have formed in the ash cloud.<ref>{{Cite journal |last=Cimarelli |first=C. |last2=Alatorre-Ibargüengoitia |first2=M.A. |last3=Kueppers |first3=U. |last4=Scheu |first4=B. |last5=Dingwell |first5=D.B. |date=2014 |title=Experimental generation of volcanic lightning |journal=Geology |volume=42 |issue=1 |pages=79–82 |doi=10.1130/g34802.1 |doi-access=free |issn=1943-2682 }}</ref><ref>{{Cite journal |last=Cimarelli |first=C. |last2=Alatorre-Ibargüengoitia |first2=M. A. |last3=Aizawa |first3=K. |last4=Yokoo |first4=A. |last5=Díaz-Marina |first5=A. |last6=Iguchi |first6=M. |last7=Dingwell |first7=D. B. |date=2016-05-06 |title=Multiparametric observation of volcanic lightning: Sakurajima Volcano, Japan |journal=Geophysical Research Letters |volume=43 |issue=9 |pages=4221–4228 |doi=10.1002/2015gl067445 |doi-access=free |issn=0094-8276 }}</ref> {{clear}} ==Blues== {{main|Radiation astronomy/Blues}} [[Image:Argentina - Bariloche trekking 013 - Glacier Castaño Overo spilling water and ice over the cliff on Cerro Tronador (6797419529).jpg|thumb|right|250px|This image shows the Glacier Castaño Overo spilling blue water ice, or blue ice. Credit: [https://www.flickr.com/people/56796376@N00 McKay Savage from London, UK].{{tlx|free media}}]] '''Blue ice''' occurs when snow falls on a glacier, is compressed, and becomes part of a [[w:glacier|glacier]] ... blue ice was observed in [[w:Tasman Glacier|Tasman Glacier]], New Zealand in January 2011.<ref name="NZ_Herald_10699700">{{ cite book |url=http://www.nzherald.co.nz/travel/news/article.cfm?c_id=7&objectid=10699700 |title=NZ blue ice sighting an unexpected treat for tourists, In: ''The New Zealand Herald'' |author=Harvey, Eveline |date=14 January 2011 |accessdate=21 September 2011 }}</ref> Ice is blue for the same reason water is blue: it is a result of an [[w:overtone|overtone]] of an oxygen-hydrogen (O-H) bond stretch in water which absorbs light at the red end of the visible spectrum.<ref name=Dartmouth>[http://www.dartmouth.edu/~etrnsfer/water.htm Why Is Water Blue]</ref> {{clear}} ==Glaciations== [[Image:GlaciationsinEarthExistancelicenced annotated.jpg|thumb|center|450px|Geologic time is annotated with glacial or ice age periods. Credit: [[c:User:William M. Connolley|William M. Connolley]].{{tlx|free media}}]] [[Image:IceAgeEarth.jpg|thumb|right|200px|Earth at the last glacial maximum of the current ice age. Credit: [[c:User:Ittiz|Ittiz]], based on: "Ice age terrestrial carbon changes revisited" by Thomas J. Crowley (Global Biogeochemical Cycles, Vol. 9, 1995, pp. 377-389.{{tlx|free media}}]] [[Image:Iceage north-intergl glac hg.png|thumb|left|200px|Recent (black) and maximum (grey) glaciation of the northern hemisphere are during the Quaternary climatic cycles. Credit: [[c:User:Hgrobe|Hannes Grobe]]/AWI.{{tlx|free media}}]] [[Image:Iceage south-intergl glac hg.png|thumb|right|200px|Recent (black) and maximum (grey) glaciation of the southern hemisphere are during the Quaternary climatic cycles. Credit: [[c:User:Hgrobe|Hannes Grobe]]/AWI.{{tlx|free media}}]] '''Def.''' the "process of covering with a glacier,<ref name=GlaciationWikt1>{{ cite book |author=[[wikt:User:66.32.178.103|66.32.178.103]] |title=glaciation |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=22 April 2010 |url=https://en.wiktionary.org/wiki/glaciation |accessdate=4 July 2019 }}</ref> or the state of being glaciated;<ref name=GlaciationWikt/> the production of glacial phenomena;<ref name=GlaciationWikt>{{ cite book |author=[[wikt:User:Poccil|Poccil]] |title=glaciation |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=20 October 2004 |url=https://en.wiktionary.org/wiki/glaciation |accessdate=4 July 2019 }}</ref> an ice age<ref name=GlaciationWikt2>{{ cite book |author=[[wikt:User:SemperBlotto|SemperBlotto]] |title=glaciation |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=1 November 2018 |url=https://en.wiktionary.org/wiki/glaciation |accessdate=4 July 2019 }}</ref>" is called a '''glaciation'''. The ice ages or periods of glaciations on Earth have occurred apparently from the early Proterozoic (Huronian), during the late Proterozoic (Cryogenian), into the early Paleozoic (Andean-Saharan) during the Ordovician and Silurian periods, the late Paleozoic (Karoo Ice Age) during the Carboniferous and early Permian periods, and most recently the Late Cenozoic Ice Age (Holarctic-Antarctic). Although these ice ages are widely separated in geological time, there appear to be types of astronomical or orbital forcings: "in most parts of the Earth major climatic and palaeoenvironmental units typically have a duration of the order of half a precession cycle (around 10 ka) rather than half an eccentricity cycle (around 50 ka) so that the level of stratigraphic resolution provided by the Middle Pleistocene [Marine Isotope Stage] MIS (typical duration 50 ka) is not sufficiently fine to constitute a universal stratigraphic template."<ref name=Shackleton>{{ cite journal |author=Nicholas J. Shackleton |author2=Maria Fernanda Sánchez-Goñi |author3=Delphine Pailler |author4=Yves Lancelot |title=Marine Isotope Substage 5e and the Eemian Interglacial |journal=Global and Planetary Change |year=2003 |volume=36 |issue= |pages=151-5 |url=http://www.colorado.edu/geography/class_homepages/geog_5241_f09/media/Readings/shackletonetal.pdf |arxiv= |bibcode= |doi=10.1016/S0921-8181(02)00181-9 |pmid= |accessdate=2014-10-11 }}</ref> {{clear}} ==Ice ages== [[Image:Northern_icesheet_hg.png|thumb|upright=1.3|right|300px|Map is of the Northern Hemisphere ice during the last glacial maximum. Credit: [[c:user:Hgrobe|Hannes Grobe/AWI]].{{tlx|free media}}]] [[Image:Karoo Glaciation.png|thumb|left|300px|Approximate extent of the Karoo Glaciation is shown in blue, over the Gondwana supercontinent during the Carboniferous and Permian periods. Credit: [[c:user:GeoPotinga|GeoPotinga]].{{tlx|free media}}]] Throughout Earth's climate history or paleoclimate there have been fluctuations between two primary states: greenhouse and icehouse Earth.<ref name=Auteur>{{Cite book|last=Auteur|first=Summerhayes, Colin P.|title=Palaeoclimatology : from snowball earth to the anthropocene|date=8 September 2020|{{isbn|978-1-119-59138-2}}|oclc=1236201953|accessdate=17 April 2021|url=https://web.archive.org/web/20210418104911/https://www.worldcat.org/oclc/1236201953 }}</ref> [[Image:4600 Myr climate change.svg|thumb|right|400px|Number 4 is the Andean-Saharan glaciation. Credit: [[c:user:Pedros.lol|Pedros.lol]].{{tlx|free media}}]] [[Image:Snowball Huronian.jpg|thumb|left|350px|The Earth is depicted during Huronian Glaciation. Credit: Oleg Kuznetsov.{{tlx|free media}}]] [[Image:Snowball.gif|thumb|right|200px|Earth is depicted during the Cryogenian as a snowball. Credit: [[c:user:たけまる|たけまる]].{{tlx|free media}}]] Both climate states last for millions of years and should not be confused with glacial and interglacial periods, which occur as alternate phases within an icehouse period and tend to last less than 1 million years.<ref name=Paillard>{{Cite journal|last=Paillard|first=D.|date=2006-07-28|title=ATMOSPHERE: What Drives the Ice Age Cycle?|journal=Science|volume=313|issue=5786|pages=455–456|doi=10.1126/science.1131297|pmid=16873636|s2cid=128379788|issn=0036-8075|accessdate=2021-04-17|url=https://web.archive.org/web/20211121183234/https://www.science.org/doi/10.1126/science.1131297}}</ref> '''Def.''' a "period of long-term reduction in the temperature of Earth's surface and atmosphere, resulting in the presence of major polar ice sheets that reach the ocean and calve icebergs"<ref name=Tweeden>UMass Lowell - A human-induced hothouse climate? https://web.archive.org/web/20190510115634/http://faculty.uml.edu/lweeden/documents/HumaninducedhothouseGSAToday.pdf</ref> is called an '''ice age'''. '''Def.''' "a period during which no continental glaciers exist anywhere on the planet"<ref name=NationalResearchCouncil>{{ cite book|date=2011-08-02|title=Understanding Earth's Deep Past|doi=10.17226/13111|{{isbn|978-0-309-20915-1}}|accessdate=2021-04-17|url=https://web.archive.org/web/20211121183307/https://www.nap.edu/catalog/13111/understanding-earths-deep-past-lessons-for-our-climate-future|author=National Research Council }}</ref> is called a '''greenhouse Earth'''. Earth has been in a greenhouse state for about 85% of its history.<ref name=NationalResearchCouncil/> There are five known Icehouse periods in Earth's climate history: the Huronian glaciation, Cryogenian, Andean-Saharan glaciation, Late Paleozoic Ice Age (Karoo Ice Age), and Late Cenozoic Ice Age (Holarctic-Antarctic Ice Age).<ref name=Auteur/> The Sturtian glaciation and Marinoan glaciation occurred during the Cryogenian Period. These events were formerly considered together as the Varanger glaciations, from their first detection in Norway's Varanger Peninsula. These are the greatest ice ages known to have occurred on Earth. The Cryogenian Period was ratified in 1990 by the International Commission on Stratigraphy.<ref name=Plumb>{{ cite journal |last=Plumb |first=Kenneth A. |title=New Precambrian time scale |journal=Episodes |year=1991 |volume=14 |series=2 |issue=2 |pages=134–140 |doi=10.18814/epiiugs/1991/v14i2/005 |accessdate=7 September 2013 |url=http://www.stratigraphy.org/bak/Precambrian.pdf }}</ref> In glaciology, ''ice age'' implies the presence of extensive ice sheets in both northern and southern hemispheres.<ref name=Imbrie>{{cite book |author1=Imbrie, J. |author2=Imbrie, K. P. |title=Ice ages: solving the mystery |url=https://archive.org/details/iceagessolvingmy0000imbr |date=1979 |publisher=Enslow Publishers |{{isbn|978-0-89490-015-0}} |location=Short Hills NJ }}</ref> By this definition, Earth is currently in an interglacial period—the Holocene. {{clear}} ==Little Ice Ages== [[Image:Carbon14 with activity labels.svg|thumb|center|500px|Changes in the ¹⁴C record, which are primarily (but not exclusively) caused by changes in solar activity, are graphed over time. Credit: [[w:User:Leland McInnes|Leland McInnes]].{{tlx|free media}}]] The Little Ice Age (LIA) appears to have lasted from about 1218 (782 b2k) to about 1878 (122 b2k). A "climate interpretation was supported by very low δ’s in the 1690’es, a period described as extremely cold in the Icelandic annals. In 1695 Iceland was completely surrounded by sea ice, and according to other sources the sea ice reached half way to the Faeroe Islands."<ref name=Dansgaard2005>{{ cite book |author=Willi Dansgaard |title=Frozen Annals Greenland Ice Cap Research |publisher=Niels Bohr Institute |location=Copenhagen, Denmark |year=2005 |editor=The Department of Geophysics of The Niels Bohr Institute for Astronomy Physics and Geophysics at The University of Copenhagen Denmark |pages=123 |url=http://www.iceandclimate.nbi.ku.dk/publications/FrozenAnnals.pdf/ |arxiv= |bibcode= |doi= |pmid= |isbn=87-990078-0-0 |accessdate=2014-10-05 }}</ref> In the image at the top, "before present" is used in the context of radiocarbon dating, where the "present" has been fixed at 1950. The apparent decreases in solar activity are called the "Maunder Minimum", "Spörer Minimum", "Wolf Minimum", and "Oort Minimum". "Northern Hemisphere summer temperatures over the past 8000 years have been paced by the slow decrease in summer insolation resulting from the precession of the equinoxes."<ref name=Miller>{{ cite journal |author=Gifford H Miller |author2=Aslaug Geirsdottir |author3=Yafang Zhong |author4=Darren J Larsen |author5=Bette L Otto-Bliesner |author6=Marika M Holland |author7=David Anthony Bailey |author8=Kurt A. Refsnider |author9=Scott J. Lehman |author10=John R. Southon |author11=Chance Anderson |author12=Helgi Björnsson |author13=Thorvaldur Thordarson |title=Abrupt onset of the Little Ice Age triggered by volcanism and sustained by sea-ice/ocean feedbacks |journal=Geophysical Research Letters |month=January |year=2012 |volume=39 |issue=2 |pages=L02708 |url=http://adsabs.harvard.edu/abs/2012GeoRL..39.2708M |arxiv= |bibcode=2012GeoRL..39.2708M |doi=10.1029/2011GL050168 |pmid= |accessdate=2014-10-09 }}</ref> Precisely "dated records of ice-cap growth from Arctic Canada and Iceland [show] that LIA summer cold and ice growth began abruptly between 1275 and 1300 AD, followed by a substantial intensification 1430-1455 AD. Intervals of sudden ice growth coincide with two of the most volcanically perturbed half centuries of the past millennium. [Explosive] volcanism produces abrupt summer cooling at these times, and that cold summers can be maintained by sea-ice/ocean feedbacks long after volcanic aerosols are removed. [The] onset of the LIA can be linked to an unusual 50-year-long episode with four large sulfur-rich explosive eruptions, each with global sulfate loading >60 Tg. The persistence of cold summers is best explained by consequent sea-ice/ocean feedbacks during a hemispheric summer insolation minimum; large changes in solar irradiance are not required."<ref name=Miller/> {{clear}} ==Venus== [[Image:Maxwell_Montes_of_planet_Venus.jpg|thumb|right|250px|Brightening of the radar reflection from the surface of Venus at high elevations such as Maxwell Montes. Credit: NASA/JPL.{{tlx|free media}}]] While there is little or no water on Venus, there is a phenomenon which is quite similar to snow. The Magellan probe imaged a highly reflective substance at the tops of Venus's highest mountain peaks which bore a strong resemblance to terrestrial snow. This substance arguably formed from a similar process to snow, albeit at a far higher temperature. Too volatile to condense on the surface, it rose in gas form to cooler higher elevations, where it then fell as precipitation. The identity of this substance is not known with certainty, but speculation has ranged from elemental tellurium to lead sulfide (galena).<ref name=Otten>{{ cite book |title='Heavy metal' snow on Venus is lead sulfide |author=Carolyn Jones Otten |publisher=Washington University in St Louis |url=http://news-info.wustl.edu/news/page/normal/633.html |date=2004 |accessdate=2007-08-21}}</ref> {{clear}} ==Moon== [[Image:Chandrayaan1 Spacecraft Discovery Moon Water.jpg|thumb|left|350px|These images show a very young lunar crater on the far side, as imaged by the Moon Mineralogy Mapper aboard Chandrayaan-1. Credit: ISRO/NASA/JPL-Caltech/USGS/Brown University.{{tlx|free media}}]] [[Image:The image shows the distribution of surface ice at the Moon's south pole (left) and north pole (right).webp|thumb|left|350px|The image shows the distribution of surface ice at the Moon's south pole (left) and north pole (right) as viewed by NASA's Moon Mineralogy Mapper (M³) spectrometer onboard India's Chandrayaan-1 orbiter. Credit: NASA.{{tlx|free media}}]] "The comet hypothesis of the origin of lunar ice, which was recently discovered in the polar regions of the moon by Lunar Prospector, is [...] that a comet impact produces a temporary atmosphere whose volatile component accumulates essentially completely in cold traps - the permanently shadowed regions of the Moon."<ref name=Klumov>{{ cite journal |author=B.A. Klumov and A.A. Berezhnoi |title=Possible origin of lunar ice |journal=Advances in Space Research |date=October 2002 |volume=30 |issue=8 |pages=1875-1881 |url=https://www.sciencedirect.com/science/article/abs/pii/S0273117702004891 |arxiv= |bibcode= |doi=10.1016/S0273-1177(02)00489-1 |pmid= |accessdate=19 September 2022 }}</ref> "Due to small oblique angle of the Moon׳s spin axis with respect to ecliptic (1.54°), the plausibility of existence of water ice in cold traps was initially discussed by Watson ''et al''. (1961). Cold traps favorably harbor water ice that originates from occasional comets, water-containing meteorites, and solar-wind-induced iron reduction of regolith; yet ice is lost due to solar wind sputter erosion (Arnold, 1979; Crider and Vondrak, 2002, 2003; Klumov and Berezhnoi, 2002). The processes of deposition and sublimation in these regions have been sustained for nearly 2 Gyr, since the Moon׳s orbital evolution became stable (Arnold, 1979; Bills and Ray, 1999)."<ref name=Wei>{{ cite journal |author=Guangfei Wei, Xiongyao Li and Shijie Wang |title=Thermal behavior of regolith at cold traps on the moon's south pole: Revealed by Chang'E-2 microwave radiometer data |journal=Planetary and Space Science |date=2 March 2016 |volume=122 |issue= |pages=101-10 |url=https://www.sciencedirect.com/science/article/abs/pii/S0032063315300635 |arxiv= |bibcode= |doi=10.1016/j.pss.2016.01.013 |pmid= |accessdate=19 September 2022 }}</ref> "The 31 km diameter and 7.5 km deep de Gerlache crater, located 30 km from the southern pole of the Moon was surveyed. At its bottom a 15 km diameter younger crater can be also found beside many smaller overprinting craters."<ref name=Kereszturi>{{ cite journal |author=A. Kereszturi, R. Tomka, P.A. Gläser, B.D. Pal, V. Steinmann and T. Warren |title=Characteristics of de Gerlache crater, site of girlands and slope exposed ice in a lunar polar depression |journal=Icarus |date=December 2022 |volume=388 |issue= |pages=115231 |url=https://www.sciencedirect.com/science/article/pii/S0019103522003256 |arxiv= |bibcode= |doi=10.1016/j.icarus.2022.115231 |pmid= |accessdate=19 September 2022 }}</ref> "At all locations [these “girland like features” ... which seem to be produced by mass movements on slopes] are superposed by recently formed 10–50 m diameter craters".<ref name=Kereszturi/> "In de Gerlache crater ice occurrences have previously been located on moderately steep slopes, indicating they might be exposed by mass movement processes, where active movements might have happened in the last some 10 Ma using crater statistics based age of the shallow regolith layer."<ref name=Kereszturi/> Impacts on the Moon could send ice chunks toward the Earth. {{clear}} ==Mars== {{main|Liquids/Liquid objects/Mars}} [[Image:2005-1103mars-full.jpg|thumb|right|250px|This Hubble Space Telescope image shows a dust storm, just above center and lighter in contrast than the surface of Mars. Credit: NASA, ESA, The Hubble Heritage Team (STScI/AURA), J. Bell (Cornell University) and M. Wolff (Space Science Institute).{{tlx|free media}}]] [[Image:Icy Crater on Mars ESP 016954 2245.jpg|thumb|left|250px|A newly formed impact crater is observed by HiRISE on Mars Reconnaissance Orbiter. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]] [[Image:Icy Crater on Mars ESP 016954 2245 subimage 2.jpg|thumb|right|250px|Another newly formed impact crater is observed by HiRISE on Mars Reconnaissance Orbiter. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]] [[Image:Small Crater on Planum Boreum PSP 009942 2645 subimage 1.jpg|thumb|left|250px|An impact crater on Planum Boreum, or the North Polar Cap, of Mars, is observed by HiRISE on the Mars Reconnaissance Orbiter. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]] [[Image:ESP 011425 1775.jpg|thumb|right|250px|This freshly formed impact crater occurred on Mars between February 2005 and July 2005. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]] Martian meteors are thought to be from Mars because they have elemental and isotopic compositions that are similar to rocks and atmosphere gases analyzed by spacecraft on Mars.<ref name=Treiman>{{ cite journal |author=A. H. Treiman, coauthors=''et al.'' |title=The SNC meteorites are from Mars |journal=Planetary and Space Science |volume=48 |issue=12–14 |month=October |year=2000 |pages=1213–30 |bibcode=2000P&SS...48.1213T |doi=10.1016/S0032-0633(00)00105-7 }}</ref> Figure 109 is a Hubble Space Telescope image of a dust storm on Mars. The picture was snapped on October 28, 2005. The regional dust storm on Mars had "been growing and evolving over the past few weeks. The dust storm, which is nearly in the middle of the planet in this Hubble view is about 930 miles (1500 km) long measured diagonally, which is about the size of the states of Texas, Oklahoma, and New Mexico combined. No wonder amateur astronomers with even modest-sized telescopes have been able to keep an eye on this storm. The smallest resolvable features in the image (small craters and wind streaks) are the size of a large city, about 12 miles (20 km) across. The occurrence of the dust storm is in close proximity to the NASA Mars Exploration Rover Opportunity's landing site in Sinus Meridiani. Dust in the atmosphere could block some of the sunlight needed to keep the rover operating at full power. ... The large regional dust storm appears as the brighter, redder cloudy region in the middle of the planet's disk. This storm has been churning in the planet's equatorial regions for several weeks now, and it is likely responsible for the reddish, dusty haze and other dust clouds seen across this hemisphere of the planet in views from Hubble, ground based telescopes, and the NASA and ESA spacecraft studying Mars from orbit. Bluish water-ice clouds can also be seen along the limbs and in the north (winter) polar region at the top of the image."<ref name=Bell>{{ cite book |author=Jim Bell |author2=Mike Wolff |author3=Keith Noll |title=Mars Kicks Up the Dust as it Makes Closest Approach to Earth |publisher=HubbleSite NewsCenter |location= |date=November 3, 2005 |url=http://hubblesite.org/newscenter/archive/releases/2005/34/image/a/ |accessdate=2013-02-24 }}</ref> Figure 110 is an image of a "newly formed impact crater, observed by HiRISE on Mars Reconnaissance Orbiter. The impact that formed the crater exposed the water ice beneath the surface. Some of the ice can be seen scattered at the adjascent area in the subimages. The blast zone (excavated dark material) is almost 800 meters (half a mile) across. The crater itself is just over 20 meters (66 feet) across".<ref name=Byrne>{{ cite book |author=Shane Byrne |title=Icy Craters on Mars |publisher=NASA/JPL/University of Arizona |location=Tucson, Arizona USA |date=April 21, 2010 |url=http://www.uahirise.org/ESP_016954_2245 |accessdate=2013-05-25 }}</ref> "This crater is one of a special group that have excavated down to buried ice. This ice gets thrown out of the crater onto the surrounding terrain. Although buried ice is common over about half the Martian surface, we can only easily discover craters in dusty regions. The overlap between areas that both have buried ice and surface dust is unfortunately small. So even though we have discovered over 100 new impact craters we have only discovered 7 new craters that expose buried ice."<ref name=Byrne/> "When craters excavate this buried ice it tells us something about the extent and depth of buried ice on Mars (controlled by climate); this information is used by planetary scientists to figure out what the recent climate of Mars was like. It has also been a surprise that this ice is so clean. Scientists expected this buried ice to be a mixture of ice and dirt; instead this ice seems to have formed in pure lenses. Yet another surprise that Mars had in store for us!"<ref name=Byrne/> The ice (presumably water ice) is white in the image, but take note of the blue dust or regolith also exposed. Figure 111 is a subimage of Figure 110. It is natural color and shows in better detail both the ice (white) and the blue material. Figure 112 is an image showing an impact crater on Planum Boreum, or the North Polar Cap, of Mars, as observed by HiRISE on Mars Reconnaissance Orbiter in natural color. "Impact craters on the surface of Planum Boreum, popularly known as the north polar cap, are rare. This dearth of craters has lead scientists to suggest that these deposits may be geologically young (a few million years old), not having had much time to accumulate impact craters throughout their lifetime."<ref name=Fishbaugh>{{ cite book |author=Kate Fishbaugh |title=Small Crater on Planum Boreum |publisher=NASA/JPL/University of Arizona |location=Tucson, Arizona USA |date=October 15, 2008 |url=http://www.uahirise.org/ESP_016954_2245 |accessdate=2013-05-25 }}</ref> "It is also possible that impacts into ice do not retain their shape indefinitely, but instead that the ice relaxes (similar to glass in an old window), and the crater begins to disappear. This subimage shows an example of a rare, small crater ( approximately 115 meters, or 125 yards, in diameter). Scientists can count these shallow craters to attain an estimate of the age of the upper few meters of the Planum Boreum surface."<ref name=Fishbaugh/> "The color in the enhanced-color example comes from the presence of dust and of ice of differing grain sizes. The blueish ice has a larger grain size than the ice that has collected in the crater. The reddish material is dust. The smooth area stretching to the upper right, away from the crater may be due to winds being channeled around the crater or to fine-grained ice and frost blowing out of the crater."<ref name=Fishbaugh/> Figure 113 shows a freshly formed impact crater that occurred on Mars between February 2005 and July 2005.<ref name=Team1>{{ cite book |author=HiRISE Team1 |title=Fresh Impact Crater Formed between February 2005 and July 2005 |publisher=NASA/JPL/University of Arizona |location=Tucson, Arizona USA |date=January 2, 2009 |url=http://hirise.lpl.arizona.edu/ESP_011425_1775 |accessdate=2013-05-25 }}</ref> Note the blue material expelled from the crater rock onto the nearby Martian landscape. Very light snow is known to occur at high latitudes on Mars.<ref name=Minard>{{ cite book |url=http://news.nationalgeographic.com/news/2009/07/090702-snow-mars-phoenix.html |title="Diamond Dust" Snow Falls Nightly on Mars |author=Anne Minard |date=2009-07-02 |publisher=National Geographic News }}</ref> {{clear}} ==1 Ceres== The detection of water vapor on Ceres, the largest object in the asteroid belt,<ref name=Kuppers>{{cite journal |last1=Küppers |first1=Michael |last2=O'Rourke |first2=Laurence |last3=Bockelée-Morvan |first3=Dominique |last4=Zakharov |first4=Vladimir |last5=Lee |first5=Seungwon |last6=von Allmen |first6=Paul |last7=Carry |first7=Benoît |last8=Teyssier |first8=David |last9=Marston |first9=Anthony |last10=Müller |first10=Thomas |last11=Crovisier |first11=Jacques |last12=Barucci |first12=M. Antonietta |last13=Moreno |first13=Raphael |date=2014 |title=Localized sources of water vapour on the dwarf planet (1) Ceres |journal=Nature |volume=505 |issue=7484 |pages=525–527 |doi=10.1038/nature12918 |bibcode=2014Natur.505..525K |pmid=24451541 |s2cid=4448395 }}</ref> was made by using the far-infrared of the Herschel Space Observatory.<ref name=Harrington>{{cite book |last=Harrington |first=J.D. |date=22 January 2014 |title=Herschel Telescope Detects Water on Dwarf Planet |id=Release 14-021 |publisher=NASA |url=http://www.nasa.gov/press/2014/january/herschel-telescope-detects-water-on-dwarf-planet |accessdate=22 January 2014 }}</ref> ==24 Themis== The surface of 24 Themis appears completely covered in ice as detected using NASA's Infrared Telescope Facility, as this ice layer is sublimating, it may be getting replenished by a reservoir of ice under the surface.<ref name=Cowen>{{cite journal | first=Ron | last=Cowen | date=8 October 2009 | title=Ice confirmed on an asteroid | journal=Science News | accessdate=9 October 2009 |url= https://web.archive.org/web/20091012075224/http://www.sciencenews.org/view/generic/id/48174/title/Ice_confirmed_on_an_asteroid }}</ref><ref name=Atkinson>{{cite book |last=Atkinson |first=Nancy |date=8 October 2009 |title=More water out there, ice found on an asteroid |accessdate=11 October 2009 |url=https://web.archive.org/web/20091011051040/http://spacefellowship.com/2009/10/08/more-water-out-there-ice-found-on-an-asteroid/ }}</ref><ref name=Campins>{{cite journal |last1=Campins |first1= H. |last2=Hargrove |first2= K |last3=Pinilla-Alonso |first3= N. |last4=Howell |first4= E.S. |last5=Kelley |first5= M.S. |last6=Licandro |first6= J. |last7=Mothé-Diniz |first7= T. |last8=Fernández |first8= Y. |last9=Ziffer |first9= J. |year=2010 |title=Water ice and organics on the surface of the asteroid 24 Themis |journal=Nature |volume=464 |issue=7293 |pages=1320–132 |doi=10.1038/nature09029 |pmid=20428164 |bibcode=2010Natur.464.1320C |s2cid=4334032 }}</ref><ref name=Rivkin>{{ cite journal |last1=Rivkin |first1=Andrew S. |last2=Emery |first2=Joshua P. |year=2010 |title=Detection of ice and organics on an asteroidal surface |journal=Nature |volume=464 |issue=7293 |pages=1322–1323 |doi=10.1038/nature09028 |pmid=20428165 |bibcode=2010Natur.464.1322R |s2cid=4368093 }}</ref> ==Europa== {{main|Rocks/Ice sheets/Europa}} [[Image:PIA02500.jpg|thumb|right|250px|Frozen sulfuric acid on Jupiter's moon Europa is depicted in this image produced from data gathered by NASA's Galileo spacecraft. Credit: NASA/JPL.{{tlx|free media}}]] [[Image:Europa densely packed plates.jpg|thumb|right|250px|This chaotic terrain on Europa has areas consisting of densely packed blocks with fractures and narrow lanes of matrix between them. Credit: G. C. Collins, J. W. Head III, R. T. Pappalardo, and N. A. Spaun.{{tlx|fairuse}}]] [[Image:Europa mostly matrix.jpg|thumb|left|250px|The image shows areas on Europa consisting of almost all matrix and no blocks. Credit: G. C. Collins, J. W. Head III, R. T. Pappalardo, and N. A. Spaun.{{tlx|fairuse}}]] [[Image:Conamara Chaos.jpg|thumb|right|250px|Conamara Chaos, the most intensely studied chaos area, lies near the middle of this continuum. Credit: G. C. Collins, J. W. Head III, R. T. Pappalardo, and N. A. Spaun.{{tlx|fairuse}}]] [[Image:High resolution Conamara Chaos.jpg|thumb|left|250px|High-resolution (10 m/pixel) image shows a plate surrounded by matrix material within Conamara Chaos. Credit: G. C. Collins, J. W. Head III, R. T. Pappalardo, and N. A. Spaun.{{tlx|fairuse}}]] [[Image:Europa Chaos.jpg|thumb|right|250px|This view from the Galileo spacecraft of a small region of the thin, disrupted, ice crust in the Conamara region of Jupiter's moon Europa shows the interplay of surface color with ice structures. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]] [[Image:PIA01296 Conomara Chaos regional view.jpg|thumb|left|250px|This Galileo spacecraft image of Jupiter's icy satellite Europa shows surface features such as domes and ridges. Credit: NASA/Jet Propulsion Laboratory/University of Arizona.{{tlx|free media}}]] [[Image:Europa chaotic terrain.jpg|thumb|right|260px|Craggy, 250&nbsp;m high peaks and smooth plates are jumbled together in a close-up of Conamara Chaos. Credit: NASA/JPL.{{tlx|free media}}]] [[Image:PIA01125 Europa chaos and gray band.jpg|thumb|left|250px|Chaotic terrain is typified by the area in the upper right-hand part of the image. Credit: NASA / JPL.{{tlx|free media}}]] "Frozen sulfuric acid on Jupiter's moon Europa is depicted in [Figure 114] produced from data gathered by NASA's Galileo spacecraft. The brightest areas, where the yellow is most intense, represent regions of high frozen sulfuric acid concentration. Sulfuric acid is found in battery acid and in Earth's acid rain."<ref name=Lavoie09301999>{{ cite book |author=Sue Lavoie |title=PIA02500: Sulfuric Acid on Europa |publisher=NASA's Office of Space Science |location=Washington DC USA |date=September 30, 1999 |url=http://photojournal.jpl.nasa.gov/catalog/PIA02500 |accessdate=2013-06-24 }}</ref> "The morphology of chaotic terrain forms a continuum from areas consisting of densely packed blocks with fractures and narrow lanes of matrix between them ([Figure 115]), to areas consisting of almost all matrix and no blocks ([Figure 116]). Conamara Chaos, the most intensely studied chaos area ([Figure 117]), lies near the middle of this continuum, with -60% of its area consisting of matrix and the remainder consisting of blocks [Spaun ''et al''., 1998]. In addition to these large chaos areas, chaotic terrain also occurs in the interiors of some small (-10 km diameter) features [Spaun ''et al''., 1999] known as "lenticulae"."<ref name=Collins/> "In Conamara Chaos, where data with spatial resolution of up to ten meters per pixel were obtained, the hummocky matrix appears to be a jumbled collection of ice chunks of all sizes, from a kilometer to tens of meters across ([Figure 118])."<ref name=Collins/> "Galileo spacecraft observations of Europa suggest the existence of a brittle ice crust (or lithosphere) at most -2 km thick, and maybe thinner locally, overlying a liquid water or ductile ice layer [Carr ''et al''., 1998; Pappalardo ''et al''., 1998, 1999]. Elastic and viscous models of buckling based on the spacing between possible folds in the Astypalaea Linea region give a thickness for the buckling layer of -2 km [Prockter and Pappalardo, 2000]. Evidence derived from the width troughs (interpreted as possible grabens) in the surroundings of Callanish, a possible impact structure, might denote a brittle-ductile transition locally as shallow as 0.5 km [Moore ''et al''., 1998]. Besides this, study of ice flexion induced by a dome-type structure located close to Conamara Chaos suggests an elastic lithosphere thickness of only -0.1-0.5 km [Williams and Greeley, 1998]."<ref name=Ruiz>{{ cite journal |author=Javier Ruiz |author2=Rosa Tejero |title=Heat flows through the ice lithosphere of Europa |journal=Journal of Geophysical Research |date=25 December 2000 |volume=105 |issue=E12 |pages=29,283-9 |url=http://onlinelibrary.wiley.com/store/10.1029/1999JE001228/asset/jgre1197.pdf;jsessionid=8B4256297C7AAD49444947999112809F.f02t01?v=1&t=hzbw8b6c&s=8cf7d27f3f3cf8ddd191c3dcfb213c138e6b08ea |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-08-26 }}</ref> The "odd surface terrain patterns [of Europa] likely come about due to convection. [...] The ice shell of Jupiter’s moon Europa is marked by regions of disrupted ice known as chaos terrains that cover up to 40% of the satellite’s surface, most commonly occurring within 40° of the equator. Concurrence with salt deposits implies a coupling between the geologically active ice shell and the underlying liquid water ocean at lower latitudes. Europa’s ocean dynamics have been assumed to adopt a two-dimensional pattern, which channels the moon’s internal heat to higher latitudes. [...] heterogeneous heating promotes the formation of chaos features through increased melting of the ice shell and subsequent deposition of marine ice at low latitudes."<ref name=Goodman>{{ cite book |author=Jason Goodman |title=Scientists Detect Hidden Ocean on Jupiter’s Moon |publisher=Astro Watch |location= |date=December 2, 2013 |url=http://www.astrowatch.net/2013/12/scientists-detect-hidden-ocean-on.html |accessdate=2014-06-11 }}</ref> Figure 119 is a view "of a small region of the thin, disrupted, ice crust in the Conamara region of Jupiter's moon Europa showing the interplay of surface color with ice structures. The white and blue colors outline areas that have been blanketed by a fine dust of ice particles ejected at the time of formation of the large (26 kilometer in diameter) crater Pwyll some 1000 kilometers to the south. A few small craters of less than 500 meters or 547 yards in diameter can be seen associated with these regions. These were probably formed, at the same time as the blanketing occurred, by large, intact, blocks of ice thrown up in the impact explosion that formed Pwyll. The unblanketed surface has a reddish brown color that has been painted by mineral contaminants carried and spread by water vapor released from below the crust when it was disrupted. The original color of the icy surface was probably a deep blue color seen in large areas elsewhere on the moon. The colors in this picture have been enhanced for visibility. North is to the top of the picture and the sun illuminates the surface from the right. The image, centered at 9 degrees north latitude and 274 degrees west longitude, covers an area approximately 70 by 30 kilometers (44 by 19 miles), and combines data taken by the Solid State Imaging (CCD) system on NASA's Galileo spacecraft during three of its orbits through the Jovian system. Low resolution color (violet, green, and infrared) data acquired in September 1996, were combined with medium resolution images from December 1996, to produce synthetic color images. These were then combined with a high resolution mosaic of images acquired on February 20th, 1997 at a resolution of 54 meters (59 yards) per picture element and at a range of 5340 kilometers (3320 miles)."<ref name=PIA01127>{{ cite book |author=NASA / JPL / University of Arizona |title=NASA planetary photojournal, PIA01127 |publisher=NASA/JPL |location=Pasadena, California USA |date=February 4, 1998 |url=http://photojournal.jpl.nasa.gov/catalog/PIA01127 |accessdate=2013-04-01 }}</ref> Figure 120 is another "image of Jupiter's icy satellite Europa shows surface features such as domes and ridges, as well as a region of disrupted terrain including crustal plates which are thought to have broken apart and "rafted" into new positions. The image covers an area of Europa's surface about 250 by 200 kilometer (km) and is centered at 10 degrees latitude, 271 degrees longitude. The color information allows the surface to be divided into three distinct spectral units. The bright white areas are ejecta rays from the relatively young crater Pwyll, which is located about 1000 km to the south (bottom) of this image. These patchy deposits appear to be superposed on other areas of the surface, and thus are thought to be the youngest features present. Also visible are reddish areas which correspond to locations where non-ice components are present. This coloring can be seen along the ridges, in the region of disrupted terrain in the center of the image, and near the dome-like features where the surface may have been thermally altered. Thus, areas associated with internal geologic activity appear reddish. The third distinct color unit is bright blue, and corresponds to the relatively old icy plains."<ref name=SolidStateImaging/> "This product combines data taken by the Solid State Imaging (SSI) system on NASA's Galileo spacecraft during three separate flybys of Europa. Low resolution color data (violet, green, and 1 micron) acquired in September 1996 were combined with medium resolution images from December 1996, to produce synthetic color images. These were then combined with a high resolution mosaic of images acquired in February 1997."<ref name=SolidStateImaging>{{ cite book |author=Solid-State Imaging |title=PIA01296: Europa "Ice Rafts" in Local and Color Context |publisher=NASA/JPL |location=Pasadena, California USA |date=May 8, 1998 |url=http://photojournal.jpl.nasa.gov/catalog/PIA01296 |accessdate=2013-04-01 }}</ref> Figure 121 is a "view of the Conamara Chaos region on Jupiter's moon Europa taken by NASA's Galileo spacecraft shows an area where the icy surface has been broken into many separate plates that have moved laterally and rotated. These plates are surrounded by a topographically lower matrix. This matrix material may have been emplaced as water, slush, or warm flowing ice, which rose up from below the surface. One of the plates is seen as a flat, lineated area in the upper portion of the image. Below this plate, a tall twin-peaked mountain of ice rises from the matrix to a height of more than 250 meters (800 feet). The matrix in this area appears to consist of a jumble of many different sized chunks of ice. Though the matrix may have consisted of a loose jumble of ice blocks while it was forming, the large fracture running vertically along the left side of the image shows that the matrix later became a hardened crust, and is frozen today. The Brooklyn Bridge in New York City would be just large enough to span this fracture."<ref name=Lavoie1998>{{ cite book |author=Sue Lavoie |title=PIA01177: Chaotic Terrain on Europa in Very High Resolution |publisher=NASA's Office of Space Science |location=Washington, DC USA |date=March 2, 1998 |url=http://photojournal.jpl.nasa.gov/catalog/PIA01177 |accessdate=2013-06-24 }}</ref> "North is to the top right of the picture, and the sun illuminates the surface from the east. This image, centered at approximately 8 degrees north latitude and 274 degrees west longitude, covers an area approximately 4 kilometers by 7 kilometers (2.5 miles by 4 miles). The resolution is 9 meters (30 feet) per picture element. This image was taken on December 16, 1997 at a range of 900 kilometers (540 miles) by Galileo's solid state imaging system."<ref name=Lavoie1998/> "Chaotic terrain on Europa is interpreted to be the result of the breakup of brittle surface materials over a mobile substrate."<ref name=Collins>{{ cite journal |author=G. C. Collins |author2=J. W. Head III |author3=R. T. Pappalardo |author4=N. A. Spaun |title=Evaluation of models for the formation of chaotic terrain on Europa |journal=Journal of Geophysical Research |date=25 January 2000 |volume=105 |issue=E1 |pages=1709-16 |url=http://onlinelibrary.wiley.com/store/10.1029/1999JE001143/asset/jgre1144.pdf?v=1&t=hzbx3jkf&s=502393cfea3bb6d9420615af0ca826e8ea8a6a57 |arxiv= |bibcode= |doi= |pmid= |accessdate=2014-08-26 }}</ref> At Figure 122, "the mottled appearance results from areas of the bright, icy crust that have been broken apart (known as "chaos" terrain), exposing a darker underlying material. This terrain is typified by the area in the upper right-hand part of the image. The mottled terrain represents some of the most recent geologic activity on Europa. Also shown in this image is a smooth, gray band (lower part of image) representing a zone where the Europan crust has been fractured, separated, and filled in with material derived from the interior. The chaos terrain and the gray band show that this satellite has been subjected to intense geological deformation."<ref name=Ciclops>{{ cite book |author=Ciclops |title=Regional Mosaic of Chaos and Gray Band on Europa |publisher=NASA/JPL |location=Pasadena, California USA |date=6 November 1997 |url=http://ciclops.org/view.php?id=4352&js=1 |accessdate=2014-08-26 }}</ref> {{clear}} ==Technology== [[Image:250mm Rain Gauge.jpg|thumb|upright|right|125px|The image shows a standard rain gauge. Credit: [[c:User:Bidgee|Bidgee]].{{tlx|free media}}]] The standard way of measuring rainfall or snowfall is the standard rain gauge, which can be found in 100-mm (4-in) plastic and 200-mm (8-in) metal varieties.<ref name=NationalWeatherService>{{ cite book |author=National Weather Service Office, Northern Indiana 2009 |url=http://www.crh.noaa.gov/iwx/program_areas/coop/8inch.php |title=8 Inch Non-Recording Standard Rain Gauge |accessdate=2009-01-02 }}</ref> The inner cylinder is filled by {{convert|25|mm|in|abbr=on}} of rain, with overflow flowing into the outer cylinder. Plastic gauges have markings on the inner cylinder down to {{convert|0.25|mm|in|abbr=on}} resolution, while metal gauges require use of a stick designed with the appropriate {{convert|0.25|mm|in|abbr=on}} markings. After the inner cylinder is filled, the amount inside it is discarded, then filled with the remaining rainfall in the outer cylinder until all the fluid in the outer cylinder is gone, adding to the overall total until the outer cylinder is empty.<ref name=Lehmann>{{ cite book |author=Chris Lehmann 2009 |url=http://nadp.sws.uiuc.edu/CAL/2000_reminders-4thQ.htm |title=10/00 |publisher=Central Analytical Laboratory |accessdate=2009-01-02 }}</ref> {{clear}} ==Cryosats== [[Image:CryoSat.jpg|thumb|right|250px|Image shows an artist's impression of CryoSat-2 in orbit. Credit: P. Carril, ESA.{{tlx|fairuse}}]] CryoSat-2 is a European Space Agency (ESA) Earth Explorer Mission that launched on 8 April 2010,<ref>{{Cite web |title=Earth Explorers: ESA’s world-class research missions |url=https://www.esa.int/Applications/Observing_the_Earth/FutureEO/Earth_Explorers_ESA_s_world-class_research_missions |accessdate=2022-08-09 |website=www.esa.int }}</ref> dedicated to measuring land and polar sea ice thickness and monitoring changes in ice sheets.<ref>{{Cite web |title=CryoSat |url=https://www.esa.int/Applications/Observing_the_Earth/FutureEO/CryoSat |accessdate=2022-08-09 |website=www.esa.int }}</ref><ref name=Cryosat2>{{Cite web |title=CryoSat-2 Product Handbook |url=https://earth.esa.int/eogateway/documents/20142/37627/CryoSat-Baseline-D-Product-Handbook.pdf |accessdate=8 August 2022 |publisher=The European Space Agency }}</ref> The primary payload of the mission is a synthetic aperture radar (SAR) Interferometric Radar Altimeter (SIRAL), which measures surface elevation.<ref name=Cryosat2/> By subtracting the difference between the surface height of the ocean and the surface height of sea ice, the sea ice freeboard (the portion of ice floating above the sea surface) can be calculated. Freeboard can be converted to sea ice thickness by assuming the sea ice is floating in hydrostatic equilibrium.<ref name=Laxon>{{Cite journal |last=Laxon |first=Seymour W. |last2=Giles |first2=Katharine A. |last3=Ridout |first3=Andy L. |last4=Wingham |first4=Duncan J. |last5=Willatt |first5=Rosemary |last6=Cullen |first6=Robert |last7=Kwok |first7=Ron |last8=Schweiger |first8=Axel |last9=Zhang |first9=Jinlun |last10=Haas |first10=Christian |last11=Hendricks |first11=Stefan |date=2013-02-28 |title=CryoSat-2 estimates of Arctic sea ice thickness and volume |url=http://doi.wiley.com/10.1002/grl.50193 |journal=Geophysical Research Letters |volume=40 |issue=4 |pages=732–737 |doi=10.1002/grl.50193}}</ref> {{clear}} ==Global Precipitation Measurement== [[Image:Visualization of the GPM Core Observatory and Partner Satellites.jpg|thumb|right|250px|This image depicts the GPM Core Observatory satellite orbiting Earth, with several other satellites from the GPM Constellation in the background. Credit: NASA.{{tlx|free media}}]] "The Global Precipitation Measurement (GPM) mission is an international network of satellites [shown in the image at right] that provide the next-generation global observations of rain and snow. Building upon the success of the Tropical Rainfall Measuring Mission (TRMM), the GPM concept centers on the deployment of a “Core” satellite carrying an advanced radar / radiometer system to measure precipitation from space and serve as a reference standard to unify precipitation measurements from a constellation of research and operational satellites. Through improved measurements of precipitation globally, the GPM mission will help to advance our understanding of Earth's water and energy cycle, improve forecasting of extreme events that cause natural hazards and disasters, and extend current capabilities in using accurate and timely information of precipitation to directly benefit society. GPM, initiated by NASA and the Japan Aerospace Exploration Agency (JAXA) as a global successor to TRMM, comprises a consortium of international space agencies, including the Centre National d’Études Spatiales (CNES), the Indian Space Research Organization (ISRO), the National Oceanic and Atmospheric Administration (NOAA), the European Organization for the Exploitation of Meteorological Satellites (EUMETSAT), and others."<ref name=Hou>{{ cite book |author=Arthur Hou |title=Precipitation Measurement Missions |publisher=Goddard Space Flight Center |location=Greenbelt, Maryland USA |date=July 26, 2013 |url=http://pmm.nasa.gov/ |accessdate=2013-08-03 }}</ref> The launch occurred on February 28, 2014 at 3:37am JST on the first attempt.<ref>{{cite book|title=GPM Launch Information|date=22 January 2014|url=http://www.nasa.gov/mission_pages/GPM/launch/index.html|publisher=NASA|accessdate=2014-02-19}}</ref> {{clear}} == Additional information == === Acknowledgements === Any people, organisations, or funding sources that you would like to thank. === Competing interests === The author has no competing interest. === Ethics statement === An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section. == References == {{reflist|35em}} iudb0hdgh0u8la0b2jor7b1oh9xrgxm 0 293672 2690264 2680068 2024-12-04T13:36:04Z Bocardodarapti 289675 2690264 wikitext text/x-wiki {{ Mathematical text/Method |Text= Let {{mat|term=M|pm=}} denote a {{ Definitionlink |square matrix| |pm=. }} How can we decide whether the matrix is {{ Definitionlink |invertible| |Context=matrix| |pm=, }} and how can we find the {{ Definitionlink |inverse matrix| |pm= }} {{mathl|term= M^{-1} |pm=?}} For this we write down a table, on the left-hand side we write down the matrix {{mat|term= M |pm=,}} and on the right-hand side we write down the identity matrix {{ Extra/Bracket |text=of the right size| |Ipm=|Epm=. }} Now we apply on both sides step by step the same elementary row manipulations. The goal is to produce in the left-hand column, starting with the matrix, in the end the identity matrix. This is possible if and only if the matrix is invertible. We claim that we produce, by this method, in the right column the matrix {{mat|term= M^{-1} |pm=}} in the end. This rests on the following {{Keyword|invariance principle|pm=.}} Every elementary row manipulation can be realized, according to {{ Factlink |Factname= Matrix/Elementary row operations/Elementary matrix from left/Fact |Nr= |pm=, }} as a matrix multiplication with some {{ Definitionlink |elementary matrix| |pm= }} {{mat|term= E |pm=}} from the left. If in the table we have somewhere the pair {{ Math/display|term= (M_1, M_2) |pm=, }} after the next step {{ Extra/Bracket |text=in the next line| |Ipm=|Epm= }} we have {{ Math/display|term= (EM_1,EM_2) |pm=. }} If we multiply the inverse of the second matrix {{ Extra/Bracket |text=which we do not know yet; however, we do know its existence, in case the matrix is invertible| |Ipm=|Epm= }} with the first matrix, then we get {{ Relationchain/display | (EM_1)^{-1} EM_2 || M_1^{-1} E^{-1} E M_2 || M_1^{-1} M_2 || || |pm=. }} This means that this expression is not changed in each single step. In the beginning, this expression equals {{mathl|term= M^{-1} {{Identity matrix/abr|n}} |pm=,}} hence in the end, the pair {{mathl|term= ( {{Identity matrix/abr|n}} , N) |pm=}} must fulfil {{ Relationchain/display | N || {{Identity matrix/abr|n}}^{-1} N || M^{-1} {{Identity matrix/abr|n}} || M^{-1} |pm=. }} |Textform=Method |Category= }} 5vcxewiaph8lk17bu3gcn021zefp9v5 0 293675 2690249 2545775 2024-12-04T12:17:15Z Bocardodarapti 289675 2690249 wikitext text/x-wiki {{ Mathematical text/Proof{{{opt|}}} |Text= {{ Proofstructure |Strategy= |Notation= |Proof= The mapping {{mat|term= \varphi |pm=}} has the property {{ Relationchain/display | \varphi(v_j) || {{sumi1m|s_{ij} w_i }} || || || |pm=, }} where {{mathl|term= s_{ij} |pm=}} is the {{mat|term= i |pm=-}}th entry of the {{mat|term= j |pm=-}}th column vector {{mat|term= s_j |pm=.}} Therefore, {{ Relationchain/display | \varphi {{mabr| {{sumj1n| a_j v_j }} |}} || {{sumj1n|a_j {{mabr| {{sumi1m|s_{ij } w_i }} |}} }} || {{sumi1m| {{mabr| {{sumj1n| a_j s_{ij} }} |}} w_i}} || || |pm=. }} This is {{mat|term= 0 |pm=}} if and only if {{ Relationchain | {{sumj1n| a_j s_{ij} }} || 0 || || || |pm= }} for all {{mat|term= i |pm=,}} and this is equivalent with {{ Relationchain/display | {{sumj1n|a_js_j}} || 0 || || || |pm=. }} For this vector equation, there exists a nontrivial tuple {{mathl|term= {{tuple1n|a}} |pm=,}} if and only if the columns are linearly dependent, and this holds if and only if the {{ Definitionlink |kernel| }} of {{mat|term= \varphi |}} is not trivial. Due to {{ Factlink |Factname= Linear mapping/Kernel/Injectivity/Fact |Nr= |pm=, }} this is equivalent with {{mat|term= \varphi |SZ=}} not being injective. |Closure= }} |Textform=Proof |Category=See }} h0i1io7slvlp32vynrgm48lb5o72wi7 0 302470 2690351 2690233 2024-12-05T10:50:43Z ThaniosAkro 2805358 /* Implementation */ 2690351 wikitext text/x-wiki Pi is a mathematical concept related to circles, geometry, and probably more. ----- ;Subpages {{Subpages/List}} ----- '''Notice:''' [[Pi/Real cosine function/Definition]] and [[Pi/Zero of cosine/Introduction/Section]] are two subpages to [[Pi]] that should not be deleted because they are transclusions to another learning resource. '''Please do not delete this page because it might help "protect" the subpages.''' ==Learning resources== ===Wikipedia=== * [[Wikipedia:Pi|Pi]] * [[Wikipedia:Category:Pi|Pi]] (Category) * [[Wikipedia:Irrational number|Irrational number]] * [[Wikipedia:Approximations of π|Approximations of π]] * [[Wikipedia:List of topics related to π|List of topics related to π]] * [[Wikipedia:List of formulae involving π|List of formulae involving π]] * [[Wikipedia:A History of Pi|A History of Pi]] * [[Wikipedia:Turn_(angle)#Tau_proposals|Turn_(angle)#Tau_proposals]] * [[Wikipedia:Six nines in pi|Six nines in pi]] * [[Wikipedia:Proof that π is irrational|Proof that π is irrational]] ===External=== * [https://www.exploratorium.edu/pi/history-of-pi A Brief History of Pi (π)] * [https://press.princeton.edu/ideas/pi-is-magic Pi is magic] * [https://sites.math.rutgers.edu/~cherlin/History/Papers2000/wilson.html The History of Pi] * [https://www.math.utah.edu/~alfeld/math/pi.html pi to 10,000 digits] * [https://news.web.baylor.edu/news/story/2023/baylor-math-chair-explains-magic-mystery-p-pi Baylor Math Chair Explains Magic, Mystery of π (Pi)] * [https://curtiscenter.math.ucla.edu/are-there-any-practical-applications-of-pi-aside-from-calculating-the-area-or-circumference-of-a-circle/ Are there any practical applications of pi, aside from calculating the area or circumference of a circle?] * [https://content.byui.edu/file/b8b83119-9acc-4a7b-bc84-efacf9043998/1/Math-2-5-1.html Pi and the Circumference of a Circle] * [https://news.harvard.edu/gazette/story/2019/03/harvard-physics-lecturer-on-why-pi-endlessly-fascinates-us/ Our endless fascination with pi] ==Calculation of π== ===Radians, the natural angle=== [[File:1129radian00.png|thumb|400px|'''Diagram illustrating one radian of angular measurement.''' </br> Arc of circle (red curved line with arrows) with length equal to radius of circle subtends one radian at center. </br> In diagram above, length of radius = length of arc = 1. </br> One radian <math>= 57.29577951308232\dots^\circ.</math> ]] If you were a mathematician among the ancient Sumerians of the 3rd millennium BC and you were determined to define the angle that could be adopted as a standard to be used by all users of trigonometry, you would probably suggest the angle in an equilateral triangle. This angle is easily defined, easily constructed, easily understood and easily reproduced. It would be easy to call this angle the "natural" angle. The numeral system used by the ancient Sumerians was Sexagesimal, also known as base 60, a numeral system with sixty as its base. In practice the natural angle could be divided into 60 parts, now called degrees, and each degree could be divided into 60 parts, now called minutes, and so on. Three equilateral triangles fit neatly into a semi-circle, hence 180 degrees in a semi-circle. We know that <math>\tan 30^\circ = \frac{\sqrt{3}}{3}.</math> Therefore, <math>\arctan (\frac{\sqrt{3}}{3})</math> should be <math>0.5,</math> or one half of our concept of the natural angle. Whatever the natural angle might be, it has existed for billions of years, but it has come to light only in recent times with invention of the calculus. In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function: <math>\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} + \cdots.</math> The following python code calculates <math>\arctan (\frac{\sqrt{3}}{3})</math> using Gregory's series: <math></math> <syntaxhighlight lang=python> # python code r3 = 3 ** .5 x = r3/3 arctan_x = ( x - x**3/3 + x**5/5 - x**7/7 + x**9/9 - x**11/11 + x**13/13 - x**15/15 + x**17/17 - x**19/19 + x**21/21 - x**23/23 + x**25/25 - x**27/27 + x**29/29 - x**31/31 + x**33/33 - x**35/35 + x**37/37 - x**39/39 + x**41/41 - x**43/43 + x**45/45 - x**47/47 + x**49/49 - x**51/51 + x**53/53 - x**55/55 + x**57/57 - x**59/59 + x**61/61 - x**63/63 + x**65/65 - x**67/67 + x**69/69 ) sx = 'arctan_x' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> arctan_x = 0.5235987755982988 </syntaxhighlight> Our assessment of the natural angle as the angle in an equilateral triangle was a very reasonable guess. However, the natural angle is the radian, the angle subtended at center of circle by an arc on the circumference equal to the radius. Six times arctan_x <math>= 180^\circ</math> or the number of radians in a semi-circle: <syntaxhighlight lang=python> # python code sx = 'arctan_x * 6' ; print (sx, '=', eval(sx)) sx = '180/(arctan_x * 6)' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> arctan_x * 6 = 3.141592653589793 180/(arctan_x * 6) = 57.29577951308232 </syntaxhighlight> <math>\pi = 3.141592653589793\dots,</math> number of radians in semi-circle. One radian <math>= 57.29577951308232\dots^\circ,</math> slightly less than <math>60^\circ.</math> Because the value <math>\frac\sqrt{3}{3}</math> is fairly large, calculation of <code>arctan_x</code> above required 34 operations to produce result accurate to 16 places of decimals. The calculation did not converge quickly. Python code below uses much smaller values of <math>x</math>, and calculation of <code>arctan_x</code> for precision of 1001 is quite fast. ===tan(A/2)=== {{RoundBoxTop|theme=2}} [[File:1122tanA_200.png|thumb|400px|'''Graphical calculation of <math>\tan \frac{A}{2}</math>.''' </br> <math>OQ = 1;\ QP = t.</math> </br> <math>\tan(A) = \frac{QP}{OQ} = \frac{t}{1} = t.</math> </br> <math>OP = OR = \sqrt{1 + t^2}</math> ]] In diagram: Point <math>P</math> has coordinates <math>(1,t).</math> Point <math>R</math> has coordinates <math>(\sqrt{1 + t^2},0).</math> Mid point of <math>PR,\ M</math> has coordinates <math>( \frac{ 1 + \sqrt{1 + t^2} }{2}, \frac{t}{2} ).</math> <math>\tan \frac{A}{2} = \frac{t}{2} / \frac{ 1 + \sqrt{1 + t^2} }{2} = \frac{t}{1 + \sqrt{1 + t^2} }</math> <math>= \frac{t}{1 + \sqrt{1 + t^2} } \cdot \frac{1 - \sqrt{1 + t^2}}{1 - \sqrt{1 + t^2} }</math> <math>= \frac{t( 1 - \sqrt{1 + t^2} )}{1-(1+t^2)}</math> <math>= \frac{t( 1 - \sqrt{1 + t^2} )}{-t^2}</math> <math>= \frac{-1 + \sqrt{1 + t^2} }{t}</math> * <math>\tan \frac{A}{2} = \frac{\tan(A)}{1 + \sqrt{1 + \tan^2(A)}} = \frac{-1 + \sqrt{1 + \tan^2 (A)} }{\tan (A)}</math> * <math>\tan (2A) = \frac{2\tan (A)}{ 1 - \tan^2 (A) }</math> {{RoundBoxBottom}} ===Implementation=== {{RoundBoxTop|theme=2}} This section calculates five values of <math>\pi</math> using the following known values of <math>\tan(A):</math> {| class="wikitable" |- ! Angle <math>A</math> || <math>\tan(A)</math> |- | <math>45^\circ</math> | <math>1</math> |- | <math>36^\circ</math> | <math>\sqrt{ 5 - 2\sqrt{5} }</math> |- | <math>30^\circ</math> | <math>\frac{\sqrt{3}}{3}</math> |- | <math>27^\circ</math> | <math>\sqrt{ 11 - 4\sqrt{5} + (\sqrt{5} - 3) \sqrt{ 10 - 2\sqrt{5} } }</math> |- | <math>24^\circ</math> | <math>\frac{ (3\sqrt{5} + 7) \sqrt{5 - 2\sqrt{5}} - (\sqrt{5} + 3)\sqrt{3} }{2}</math> |} Values of <math>x</math> in table below are derived from the above values by using identity <math>\tan(\frac{A}{2}) = \frac{-1 + \sqrt{1 + \tan^2(A)}}{\tan(A)}</math>: {| class="wikitable" |- ! Angle <math>\theta</math> || <math>x = \tan(\theta)</math> |- | <math>\frac{45^\circ}{2^{33}}</math> | <code>0.00000_00000_91432_37995_4197.....089_03901_63759_3912</code> |- | <math>\frac{36^\circ}{2^{33}}</math> | <code>0.00000_00000_73145_90396_3357.....211_97500_56173_0713</code> |- | <math>\frac{30^\circ}{2^{33}}</math> | <code>0.00000_00000_60954_91996_9464.....024_32806_94580_0689</code> |- | <math>\frac{27^\circ}{2^{33}}</math> | <code>0.00000_00000_54859_42797_2518.....791_30634_03540_9738</code> |- | <math>\frac{24^\circ}{2^{32}}</math> | <code>0.00000_00000_97527_87195_1143.....736_60376_04724_6778</code> |} <syntaxhighlight lang=python> # python code desired_precision = 1001 number_of_leading_zeroes = 10 # See below. import decimal dD = decimal.Decimal # Decimal object is like float with (almost) infinite precision. dgt = decimal.getcontext() Precision = dgt.prec = desired_precision + 3 # Adjust as necessary. Tolerance = dD("1e-" + str(Precision-2)) # Adjust as necessary. adjustment_to_precision = number_of_leading_zeroes * 2 + 3 def tan_halfA(tan_A) : dgt.prec += adjustment_to_precision top = -1 + (1+tan_A**2).sqrt() dgt.prec -= adjustment_to_precision tan_A_2 = top/tan_A return tan_A_2 def tan_2A (tanA) : ''' 2 * tanA tan(2A) = ----------- 1 - tanA**2 ''' if tanA in (1,-1) : return '1/0' dgt.prec += adjustment_to_precision bottom = (1 - tanA**2) output = 2*tanA/bottom dgt.prec -= adjustment_to_precision return output+0 def θ_tanθ_from_A_tanA (angleA, tanA) : ''' if input == 45,1 output is: "dD(45) / (2 ** (33))", "0.00000_00000_91432_37995_....._63759_3912" ^^^^^^^^^^^ number_of_leading_zeroes refers to these zeroes. θ,tanθ = θ_tanθ_from_A_tanA (angleA, tanA) ''' θ, tanθ = angleA, tanA for p in range (1,100) : θ /= 2 tanθ = tan_halfA(tanθ) if tanθ >= dD('1e-' + str(number_of_leading_zeroes)) : continue str1 = str(tanθ) # str1 = "n.nnnnnnnnnnnnn ..... nnnnnnnnnnnnE-11" str1a = str1[0] + str1[2:-4] list1 = [ str1a[q:q+5] for q in range (0, len(str1a), 5) ] str2 = '0.00000_00000_' + ('_'.join(list1)) dD2 = dD(str2) (dD2 == tanθ) or ({}[2]) ((θ * (2**p)) == angleA ) or ({}[3]) str3 = 'dD({}) / (2 ** ({}))'.format(angleA,p) (θ == eval(str3)) or ({}[4]) return str3, str2 ({}[5]) r3 = dD(3).sqrt() r5 = dD(5).sqrt() tan36 = (5 - 2*r5).sqrt() tan45 = dD(1) tan30 = r3/3 v1 = 3*r5+7 v2 = (5 - 2*r5).sqrt() v3 = (r5+3)*r3 tan24 = ( v1*v2 - v3 )/2 v1 = r5 - 3 ; v2 = (10 - 2*r5).sqrt() tan27 = ( 11 - 4*r5 + v1*v2 ).sqrt() values_of_A_tanA = ( (dD(45), tan45), (dD(36), tan36), (dD(30), tan30), (dD(27), tan27), (dD(24), tan24), ) values_of_θ_tanθ = [] for (A, tanA) in values_of_A_tanA : θ, tanθ = θ_tanθ_from_A_tanA (A, tanA) print() sx = 'θ' ; print (sx, '=', eval(sx)) # sx = 'tanθ' ; print (sx, '=', eval(sx)) print ('tanθ =', '{}.....{}'.format(tanθ[:30], tanθ[-20:])) values_of_θ_tanθ += [ (θ, tanθ) ] # Check for (v1,v2),(v3,v4) in zip (values_of_A_tanA, values_of_θ_tanθ) : A, tanA = v1,v2 θ = eval(v3) tanθ = dD(v4) status = 0 for p in range (1,100) : θ *= 2 tanθ = tan_2A (tanθ) if θ == A : dgt.prec = desired_precision (+tanθ == +tanA) or ({}[10]) dgt.prec = Precision status = 1 break status or ({}[11]) </syntaxhighlight> <syntaxhighlight> θ = dD(45) / (2 ** (33)) tanθ = 0.00000_00000_91432_37995_4197.....089_03901_63759_3912 θ = dD(36) / (2 ** (33)) tanθ = 0.00000_00000_73145_90396_3357.....211_97500_56173_0713 θ = dD(30) / (2 ** (33)) tanθ = 0.00000_00000_60954_91996_9464.....024_32806_94580_0689 θ = dD(27) / (2 ** (33)) tanθ = 0.00000_00000_54859_42797_2518.....791_30634_03540_9738 θ = dD(24) / (2 ** (32)) tanθ = 0.00000_00000_97527_87195_1143.....736_60376_04724_6778 </syntaxhighlight> <syntaxhighlight lang=python> # python code def calculate_π (angleθ, tanθ) : ''' angleθ may be: "dD(27) / (2 ** (33))" tanθ may be: "0.00000_00000_54859_42797_ ..... _03540_9738" π = calculate_π (angleθ, tanθ) ''' thisName = 'calculate_π (angleθ, tanθ) :' if isinstance(angleθ, dD) : pass elif isinstance(angleθ, str) : angleθ = eval(angleθ) else : ({}[21]) if isinstance(tanθ, dD) : pass elif isinstance(tanθ, str) : tanθ = dD(tanθ) else : ({}[22]) x = tanθ ; multiplier = -1 ; sum = x ; count = 0; status = 0 # x**3 x**5 x**7 x**9 # y = x - ---- + ---- - ---- + ---- # 3 5 7 9 # # Each term in the sequence is roughly the previous term multiplied by x**2. # Each value of x contains 10 leading zeroes after decimal point. # Therefore, each term in the sequence is roughly the previous term with 20 more leading zeroes. # Each pass through main loop adds about 20 digits to current value of sum # and θ is calculated to precision of 1004 digits with about 50 passes through main loop. # for p in range (3,200,2) : # This is main loop. count += 1 addendum = (multiplier * (x**p)) / p sum += addendum if abs(addendum) < Tolerance : status = 1; break multiplier = -multiplier status or ({}[23]) print(thisName, 'count =',count) π = sum * 180 / angleθ dgt.prec = desired_precision π += 0 # This forces π to adopt precision of desired_precision. dgt.prec = Precision return π # Calculate five values of π: values_of_π = [] for θ,tanθ in values_of_θ_tanθ : π = calculate_π (θ,tanθ) values_of_π += [ π ] </syntaxhighlight> Each calculation of π required about 50 passes through main loop: <syntaxhighlight> calculate_π (angleθ, tanθ) : count = 50 calculate_π (angleθ, tanθ) : count = 49 calculate_π (angleθ, tanθ) : count = 49 calculate_π (angleθ, tanθ) : count = 49 calculate_π (angleθ, tanθ) : count = 50 </syntaxhighlight> Check that all 5 values of π are equal: <syntaxhighlight lang=python> # python code set1 = set(values_of_π) sx = 'len(values_of_π)' ; print (sx, '=', eval(sx)) sx = 'len(set1)' ; print (sx, '=', eval(sx)) sx = 'set1' ; print (sx, '=', eval(sx)) π, = set1 # Note the syntax. If length of set1 is not 1, this statement fails. </syntaxhighlight> <syntaxhighlight> len(values_of_π) = 5 len(set1) = 1 set1 = {Decimal('3.141592653589793238462643383279.....12268066130019278766111959092164201989')} </syntaxhighlight> Because all five calculated values of π are equal, there is very high probability that this value of π is accurate. Print value of π as python command formatted: <syntaxhighlight lang=python> # python code newLine = ''' '''[-1:] def print_π (π) : ''' Input π is : Decimal('3.141592653589793238 ..... 66111959092164201989') This function prints: π = ( "3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510" + "58209_74944_59230_78164_06286_20899_86280_34825_34211_70679" ..... + "18577_80532_17122_68066_13001_92787_66111_95909_21642_01989" ) ''' πstr = str(π) (len(πstr) == (desired_precision + 1)) or ({}[31]) (πstr[:2] == '3.') or ({}[32]) twenty_rows = [] for p in range (2, len(πstr), 50) : str1a = πstr[p:p+50] list1a = [ str1a[q:q+5] for q in range(0, len(str1a), 5) ] str1b = '_'.join(list1a) twenty_rows += [str1b] twenty_rows[0] = '3.' + twenty_rows[0] joiner = '"{} + "'.format(newLine) str3 = '( "{}" )'.format(joiner.join(twenty_rows)) str4 = eval(str3) (dD(str4) == π) or ({}[33]) lines = str3.split(newLine) paragraphs = [ newLine.join(lines[p:p+4]) for p in range(0,len(lines),4) ] str5 = (newLine*2).join(paragraphs) str6 = eval(str5) (dD(str6) == π) or ({}[34]) print ('π =', str5) return str5 π1 = print_π (π) </syntaxhighlight> <syntaxhighlight> π = ( "3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510" + "58209_74944_59230_78164_06286_20899_86280_34825_34211_70679" + "82148_08651_32823_06647_09384_46095_50582_23172_53594_08128" + "48111_74502_84102_70193_85211_05559_64462_29489_54930_38196" + "44288_10975_66593_34461_28475_64823_37867_83165_27120_19091" + "45648_56692_34603_48610_45432_66482_13393_60726_02491_41273" + "72458_70066_06315_58817_48815_20920_96282_92540_91715_36436" + "78925_90360_01133_05305_48820_46652_13841_46951_94151_16094" + "33057_27036_57595_91953_09218_61173_81932_61179_31051_18548" + "07446_23799_62749_56735_18857_52724_89122_79381_83011_94912" + "98336_73362_44065_66430_86021_39494_63952_24737_19070_21798" + "60943_70277_05392_17176_29317_67523_84674_81846_76694_05132" + "00056_81271_45263_56082_77857_71342_75778_96091_73637_17872" + "14684_40901_22495_34301_46549_58537_10507_92279_68925_89235" + "42019_95611_21290_21960_86403_44181_59813_62977_47713_09960" + "51870_72113_49999_99837_29780_49951_05973_17328_16096_31859" + "50244_59455_34690_83026_42522_30825_33446_85035_26193_11881" + "71010_00313_78387_52886_58753_32083_81420_61717_76691_47303" + "59825_34904_28755_46873_11595_62863_88235_37875_93751_95778" + "18577_80532_17122_68066_13001_92787_66111_95909_21642_01989" ) </syntaxhighlight> {{RoundBoxTop|theme=3}} [[File:1202pi_highlighted.png|thumb|400px|'''Value of <math>\pi</math> highlighted.''']] If you highlight the above expression for <math>\pi</math> as shown in diagram, you can copy and paste it into your python source file as valid python code. {{RoundBoxBottom}} {{RoundBoxBottom}} ==Discussion questions== * What are useful applications of pi in engineering or science? * How is pi useful in computer science? ==See also== * [[Math]] * [[Geometry]] * [[Exact_Trigonometric_Values]] jhf4ur7ac7b8w7hsooeg9bfecv4w115 0 305362 2690335 2679936 2024-12-05T03:50:30Z Dc.samizdat 2856930 /* 6-cell rings */ fix incorrect description of fibrations, including which cell rings contain which great hexagons 2690335 wikitext text/x-wiki {{Short description|Regular object in four dimensional geometry}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Omnitruncated tesseract|21]] | Index=22 | Next=[[W:Rectified 24-cell|23]] | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}} | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]] | Face_List=96 [[W:Triangle|{3}]] | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=[[W:Dodecagon|{12}]] | Coxeter_Group=[[W:F4 (mathematics)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], order 192 | Vertex_Figure=[[W:Cube|cube]] | Dual=[[W:Polytope#Self-dual polytopes|self-dual]] | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math display="block">\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math> Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a closed spiral not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object in the same orientation, indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration]|Hopf fibration] of four interlocking rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two interlocking great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1968 | title=The Beauty of Geometry: Twelve Essays | publisher=Dover | place=New York | edition=2nd }} * {{Cite journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. 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It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math display="block">\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math> Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a closed spiral not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object in the same orientation, indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration]|Hopf fibration] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite web|last=Egan|first=Greg|date=23 December 2021|title=Symmetries and the 24-cell|url=https://www.gregegan.net/SCIENCE/24-cell/24-cell.html|author-link=W:Greg Egan|website=gregegan.net|access-date=10 October 2022}} * {{Cite journal | last1=Mamone|first1=Salvatore | last2=Pileio|first2=Giuseppe | last3=Levitt|first3=Malcolm H. | year=2010 | title=Orientational Sampling Schemes Based on Four Dimensional Polytopes | journal=Symmetry | volume=2 |issue=3 | pages=1423–1449 | doi=10.3390/sym2031423 |bibcode=2010Symm....2.1423M |doi-access=free }} * {{Cite thesis|title=Applications of Quaternions to Dynamical Simulation, Computer Graphics and Biomechanics|last=Mebius|first=Johan|date=July 2015|publisher=[[W:Delft University of Technology|Delft University of Technology]]|orig-date=11 Jan 1994|doi=10.13140/RG.2.1.3310.3205}} * {{Cite book|title=Elementary particles and the laws of physics|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987}} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|doi=10.1007/s00006-019-0960-5 |s2cid=253592159 |doi-access=free}} * {{Cite journal|last1=Koca|first1=Mehmet|last2=Al-Ajmi|first2=Mudhahir|last3=Koc|first3=Ramazan|date=November 2007|title=Polyhedra obtained from Coxeter groups and quaternions|journal=Journal of Mathematical Physics|volume=48|issue=11|pages=113514|doi=10.1063/1.2809467|bibcode=2007JMP....48k3514K |url=https://www.researchgate.net/publication/234907424}} {{Refend}} ==External links== * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] iib3ualrmgdruk6saqthztxhrrvo4d4 0 305986 2690321 2690096 2024-12-05T00:33:33Z Jtneill 10242 Copyediting 2690321 wikitext text/x-wiki {{title|Comprehensive action determination model:<br>What is the CADM and how can it be applied to understanding human motivation?}} {{MECR3|1=https://youtu.be/Fde1L08wMl4}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}}{{Image|float=right|pad=20px|name=Footprint water.png|width=300px|caption=Figure 1: Water footprint}} '''Scenario''' A psychology researcher is exploring how university students’ water use habits relate to their broader attitudes towards sustainability. They are curious about the internal and external factors that influence water use habits, like whether students are unaware of the environmental impact of their actions, or if they just don’t prioritize sustainability in their daily routines. Additionally, They want to investigate how their broader attitudes toward the environment influence their actions. Coming across an “integrated model” of behaviour they chose to utilize it. They think this model could be useful for investigating water usage behaviours in university students. {{RoundBoxBottom}} The Comprehensive action determination model is a theoretical framework designed to explain human behaviour. It proposes that behaviour is directly predicted by three processes: habitual, situational, and intentional, and indirectly influenced by normative processes. This complex model can aid research in understanding human motivation by breaking down the processes, evaluating the outcomes, and examining the underlying context behind behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is motivation? * What is the Comprehensive Action Determination Model (CADM)? * What are the underpinning theories behind the CADM? * How can the CADM be applied to understanding human motivation? {{RoundBoxBottom}} == What is motivation? == [[File:Maslow's Hierarchy of Needs.svg|thumb|442x442px|Figure 2: Pyramid depicting the structure of Maslow's hierarchy of needs. ]] [[Motivation]] is the "why" behind behaviour {{g}} the reasons underlying why people act in certain ways. Theorists explain motivation as the attribute that moves us to do or not to do something (Lai, 2011). The word ''motivation'' comes from the Latin verb ''movere'', meaning "to move" (Dörnyei & Ushioda, 2021){{ic|not in the References}}. Motivation has three important components that drive behaviour direction, intensity, and persistence. Direction refers to how motivation shapes peoples choices and goals. Intensity is how motivation affects the effort people put in. Persistence is how motivation influences how long people keep working toward our goals. (Dörnyei & Ushioda, 2021). Many theories have been developed to understand motivation, these include Herzberg's [[wikipedia:Two-factor_theory|Two-factor theory]] (Herzberg et al., 1959; Herzberg et al., 2011), [[Self-determination theory]] (Deci, 1972, 2000){{ic|not in the References}}, and [[Maslow's hierarchy of needs]] (Figure 2) (Maslow, 1943). Theorists argue that it is unrealistic to create an elaborate super-theory to understand motivation due to its complexity and the countless variables that affect it. They argue that the complexity of motivation cannot be understood through a single theory or model (Dörnyei & Ushioda, 2021). == Comprehensive action determination model == The Comprehensive action determination model (CADM) was first proposed by Klöckner and Blöbaum in 2010 (Klöckner & Blöbaum, 2010){{ic|not in the References}}. They argue that one of the main goals of [[environmental psychology]] is to understand what determines people's actions, {{g}} in relation to environmental influences. Klöckner and Blöbaum also note that several well-established models have been proposed, however none of these models provide an adequate representation of the multiple factors that determine behaviour (Klöckner & Blöbaum, 2010). They further argue that integrated approaches combining two models, the Theory of planned behaviour (Ajzen, 1991) and the Norm-activation model (Schwartz, 1977), have shown promise (Liu et al., 2017, Onwezen et al., 2013), especially when the concept of [[wikipedia:Habit|habit]] is also incorporated (Klöckner et al., 2003). Additionally, Klöckner and Blöbaum highlight that the Ipsative theory of behaviour offers a valuable perspective on the situational determination of behaviour (Klöckner & Blöbaum, 2010). However, these theories (the Theory of planned behaviour and the Norm-activation model) have proven successful in specific domains, they exhibit limitations in other areas (Klöckner & Blöbaum, 2010). The CADM unifies the Theory of planned behaviour, the Norm-activation model, the Ipsative theory of behaviour, and the concept of habit into one integrated framework. Klöckner and Blöbaum (2010) propose that combining these existing theories into a single model may result in a more universally applicable framework that accounts for all relevant factors and influences affecting behaviour. (Klöckner & Blöbaum, 2010). === Underpinning theories === To understand the CADM, an understanding of the underpinning models, theories and theoretical concepts is required: the theory of planned behaviour, the Norm-activation model, the Ipsative theory of behaviour and the theoretical concept of habit. ==== Theory of planned behaviour ==== [[File:Theory of planned behaviour.png|thumb|405x405px|Figure 3: Flow diagram of the Theory of Planned Behaviour ]] The Theory of planned behaviour was first proposed by Ajzen in 1991 (Ajzen, 1991). The theory consists of three constructs: behavioural intention, attitude toward behaviour, and subjective norms. Behavioural intention refers to the motivation behind a behaviour, the stronger the intention, the more likely the behaviour is to be performed. Attitudes refers to how positively or negatively a person evaluates a specific behaviour. Subjective norms refer to the social pressures to perform or not perform a given behaviour. Perceived behavioural control is also a key construct in the Theory of Planned Behaviour, and it refers to how a person perceives the difficulty of performing a specific behaviour (Asare, 2015). The theory suggests that attitudes toward the behaviour, subjective norms, and perceived behavioural control all influence behavioural intention, which in turn leads to behaviour, as depicted in Figure 3. Klockner and Blobaum (2010) argue that the Theory of planned behaviour focuses too much on intention while neglecting the role of objective situational constraints, habits, and personal norms. ==== Norm-activation model ==== The Norm-activation model was first proposed by Schwartz in 1977. The model suggests that behaviour is predicted by personal norms. Schwartz (1977) defines personal norms as “feelings of moral obligation, not as intentions” (Onwezen et al., 2013). The model posits that personal norms are influenced by two factors: awareness of consequences and awareness of needs (sometimes referred to as the ascription of responsibility). Awareness of consequences refers to the understanding that performing or not performing a certain behaviour leads to specific outcomes. Awareness of needs involves the feeling of responsibility to perform a particular behaviour (Onwezen et al., 2013; Klöckner & Blöbaum, 2010). As depicted in Figure 4, both awareness of consequences and awareness of needs predict personal norms, which in turn predict behaviour. Klockner and Blobaum (2010) identify limitations of the model, stating that “The Norm-activation model focuses on personal norms but underestimates the roles of habits, intentions, attitudes, and the situational context.”{{example}} [[File:Norm activation model.png|center|thumb|610x610px|Figure 4: Flow diagram of the Norm-Activation Model ]] ==== Ipsative theory of behaviour ==== The Ipsative theory of behaviour was proposed by Frey (1988). The theory suggests that a person's behaviour can be limited or obstructed by the absence of genuine or perceived opportunities, influenced by both internal and external circumstances (Tanner, 1999). The theory consists of three presumptions about human behaviour. Firstly, objective constraints are assumed to influence behaviour. These constraints determine what a person can do, what they ought to do, or what they are permitted to do within a specific society. These variables make up the "objective possibility set," which limits or hinders people's ability to engage in certain activities (Tanner, 1999). Secondly, ipsative constraints prevent the activation of alternative behaviours. Ipsative constraints form the "ipsative possibility set," which individuals regard as relevant to their behavioural decisions (Klockner & Blobaum, 2010, Tanner, 1999){{example}}. Finally, subjective constraints are believed to directly affect preferences rather than determining participation in specific actions, {{g}} they influence a person's willingness to act (Tanner, 1999). Klockner and Blobaum (2010) argue that while the Ipsative theory of behaviour effectively outlines the objective and subjective aspects of situations as predictors of behaviour, it overlooks intentional, habitual, and normative processes.{{example}} ==== Habits ==== The theoretical concept of habit was incorporated into the CADM to address limitations found in the Theory of planned behaviour and the Norm-activation model in predicting repetitive behaviours (Klockner & Blobaum, 2010). Habits can be defined as learned tendencies to repeat previous behaviours. They are activated by contextual elements that are often linked to past performances, such as specific locations, preceding actions in a sequence, and particular individuals (Woods & Neal, 2007). The concept of habit was added to account for structural differences between actions that are frequent and those that are rare or performed for the first time. When decisions are frequently made with satisfying outcomes, the influence of decision-making in given situations decreases, resulting in more automated behavioural patterns (Klockner & Blobaum, 2010, Triandis, 1979). Klockner and Blobaum (2010) note that although the concept of habit recognises the interaction between intentions and habits, it does not fully account for non-automatic situational facilitation, constraints on behaviour, or normative processes.{{example}} === How does the CADM work? === The Comprehensive action determination model proposes that behaviour is determined by three possible direct sources or processes: habitual, intentional, and situational as depicted in figure 5. Habitual processes include schemata, heuristics, and associations of behaviour. Intentional processes include attitudes and intentions behind behaviour and situational processes include both objective and subjective constraints on behaviour. A fourth process, the normative process, is also present, but it does not directly affect behaviour. Instead, it influences intentional and habitual processes. The normative process includes social norms, personal norms, and awareness of needs and consequences. Klockner and Blobaum (2010) explain that attitudes, subjective constraints (e.g., [[wikipedia:Perceived_control|perceived behavioural control]]), and personal and social norms are used to generate intentions. They further explain that attitudes reflect cognitive and emotional beliefs about behaviour, while perceived behavioural control represents beliefs about the degree of control or determination one has over their actions. Additionally, personal and social norms shape the moral framework that guides the decision-making process leading to behaviour. Personal norms are rooted in an individual’s value system and can be seen as the motivations behind decision-making (Klockner & Blobaum, 2010). However, the normative process also influences habits, as it has higher temporal stability compared to attitudes and perceived behavioural control. Situational processes and perceived behavioural control are also thought to activate personal norms by creating awareness of needs and consequences, which then generate intentions. Habitual and situational processes are believed to interfere with intentional processes and can moderate the influence of intentions on behaviour. Perceived behavioural control is essential for activating both normative and intentional processes. As a result, situational influences affect both normative and intentional processes. Furthermore, habits are said to form through the successful execution of behaviour in specific situations, meaning situational processes also influence habitual behaviours. Finally, behaviour influences changes in personal norms, and habits, in turn, affect future behaviour (Klockner & Blobaum, 2010).{{example}} [[File:Comprehensive action determination model.png|center|thumb|508x508px|Figure 5: Simplified flow diagram of the Comprehensive Action Determination Model ]] The CADM in Figure 5 is a simplified version of the CADM, {{g}} Klockner an Bloblaum delve into further detail on the individual aspects apart of {{g}} different processes and how they affect one another. Figure 6 depicts the detailed CADM utilizing the example of water conservation behaviours for the scenario at the start of the chapter. According to Klockner and Blobaum (2010), behaviour is primarily predicted by intentions and perceived behavioural control. Intentions, in turn, are generated from perceived behavioural control, social norms, and attitudes. In this scenario, habitual processes, such as water usage habits, directly predict the likelihood of engaging in water conservation efforts and moderate the relationship between intention and conservation behaviour. Personal norms are identified as predictors of intention, {{g}} they do not directly predict conservation behaviour. Personal norms are shaped by awareness of environmental needs and consequences and are activated through perceived behavioural control, which creates a sense of moral obligation. Furthermore, social norms influence personal norms, as they are internalized and are a part of the individual’s value system (Klockner & Blobaum, 2010). Personal norms also influence the formation of habits. Klockner and Blobaum (2010) indicate that both subjective constraints (perceived behavioural control) and objective constraints (access to water) are direct predictors of conservation behaviour. Since habits tend to demonstrate long-term stability, water usage habits should be influenced by perceived behavioural control and water access. Both perceived behavioural control and water access also act as mediators in the relationship between intention and water conservation behaviour. Finally, conservation behaviour feeds back onto personal norms and habits. [[File:Comprehensive Action Determination Model water usage behaviour.png|center|thumb|445x445px|Figure 6: Detailed flow diagram of Comprehensive action determination model In context to water usage behaviours]] === Current research on the CADM === Since its proposal in 2010, the CADM has gained some traction in recent literature especially in [[wikipedia:Environmental_psychology|environmental behaviour]] (Klockner, 2013). Environmental behaviour refers to actions, attitudes, and practices related to the environment. Current research applying the CADM include sustainable farming (Tan, 2024), food waste behaviour (Cheng et al., 2024), clothing consumption (Joanes et al., 2020), and recycling behaviours (Fang et al., 2021., Klockner & Oppedal, 2011., Ofstad et al., 2017). CADM has proven effective in predicting agricultural practices (Tan, 2024), food waste behaviours (Cheng et al., 2024), and recycling and consumption patterns (Joanes et al., 2020). Research conducted by Tan (2024) on sustainable farming found that the CADM was the most effective model for analysing and predicting persistent agricultural practices, contributing to a deeper understanding of behavioural determinants in sustainable agriculture. Although the CADM has shown notable success in recent applications and studies, it is important to recognise that the existing body of literature remains limited. There is need for further research to enhance the generalisability of these findings and ensure their applicability across a broader range of contexts === Limitations to the CADM === The CADM has two notable limitations, complexity and being data intensive. The model’s complex nature can be difficult to interpret initially, as it combines elements from the Theory of planned Behaviour, the Norm-activation model, the Ipsative theory of behaviour, and the concept of habit. This complexity is further compounded by the presence of numerous variables that not only influence each other but also interact, with some variables acting as mediators, moderators, or both. These interrelationships make it challenging to understand how each factor contributes to behaviour. Furthermore, the complex nature of the CADM may not be practical in situations where quick decisions are needed, as the analysis of CADM is typically too thorough for quick decision-making. Secondly, the model is limited by its data-intensive nature. The data analysis process can be viewed as time-consuming, labour-intensive and could be difficult to replicate. == CADM and understanding human motivation == {{expand}} === Examining context === The CADM allows researchers to examine the context behind human behaviour. Examining the context behind behaviour helps researchers gain deeper insights into the motivations that drive specific behaviours. Furthermore, it allows researchers to assess the strength of the motivation in relation to the context. Additionally, Normative process such as personal norms and social norms are known to influence intentions which lead to behaviour. Based on the behaviour, researchers can identify if social norms have a stronger influence on intentions more than personal norms, even though it is said by Klockner and Blobaum (2010) that personal norms are affected by social norms. The use of the CADM in behavioural research provides a framework for breaking down the motivations underlying behaviour. === Evaluating outcomes === Using the CADM in longitudinal studies could help researchers explore how past behaviours influence future actions. By following participants over time, studies can offer insights into how behaviours, habits, and perceived control in specific situations shape future behaviour. The CADM allows researchers to track patterns and identify the motivations driving behaviour change. The CADM emphasises the role of feedback loops in behaviour change. Positive or negative outcomes from past behaviours can strengthen or weaken future actions. These feedback loops can either reinforce existing behaviours or lead to changes. The model allows researchers to track these feedback mechanisms and how they impact future behaviour. By using CADM in longitudinal studies, researchers can better understand how past behaviour’s{{g}} motivate future actions. Evaluating outcomes can help develop more effective strategies for behaviour change and intervention, providing a deeper understanding of the factors that motivate long-term behaviour.{{example}} === Understanding behavioural processes === The CADM offers a valuable framework for understanding human motivation by systematically breaking down the various behavioural processes that drive actions. This model allows researchers to analyse the factors influencing motivation at different stages, such as habitual and situational processes. By applying the CADM, studies can identify specific points in the model where motivation may excel such as during moments of goal accomplishment as well as areas where motivation may falter, such as in the face of adversity or lack of support. The CADM provides a deeper understanding of how different variables interact to either enhance or undermine motivation. == Conclusion == Motivation is the fundamental "why" behind Behaviour; it is the core driver of behaviour, fuelling the pursuit of goals, overcoming challenges, and sustaining effort. Understanding motivation is important because it helps us identify the driving forces behind behaviour. The CADM explains how individuals make decisions in relation to habitual, intentional, situational and normative processes, all of which affect each other in complex ways. The CADM integrates pre-existing "action determination models," such as the Theory of Planned Behaviour, the Norm-Activation Model, the Ipsative Theory of Behaviour, and the theoretical concept of habit, into a comprehensive model. The CADM helps understand human motivation by analysing the context and underlying motives behind actions. It can also evaluate how past behaviour shapes future actions and where motivation may succeed or falter. The Comprehensive action determination model helps explain how motivation interacts with various factors like habits, intentions, and situational influences. By understanding this model, Researcher {{g}} can gain insight into the underlying motives behind actions and improve decision-making, ultimately shaping future behaviour. {{RoundBoxTop|theme=14}} '''Quiz''' <quiz display="simple"> {'''What are the underlying theories of the CADM?''': |type="()"} - Norm-activation model, Ipsative theory of behaviour and habit. + Theory of planned behaviour, Norm-activation model, Ipsative theory of behaviour and habit. - Theory of planned behaviour, Norm-activation model, Maslow's hierarchy of needs. - Theory of planned behaviour, Ipsative theory of behaviour and habit. {'''In The CADM, normative process predict {{g}} which of the following processes:''' |type="()"} - Habitual, situational and intentional. - situational and intentional. - Habitual and situational. + Intentional and habitual. {'''What are two limitations of the CADM?''' |type="()"} - Data availability, complexity. - Data availability and generalisation - Data intensive and generalisation. + Complexity and data intensive. {'''In the literature, what field of psychology is the CADM most researched?:''' |type="()"} - Clinical psychology in mental health issues, antisocial behaviour, attitudes and mood. + Environmental psychology, in recycling behaviours, sustainable farming, food wastage and clothing consumption. - Social psychology in social identity, interpersonal relationships, discrimination and social influence. - Positive psychology in goal setting, positive emotions and optimism. </quiz> {{RoundBoxBottom}} ==See also== * [[Motivation and emotion/Book/2013/Environmental behaviour|Environmental behaviour]] (Book chapter, 2013) * [[Motivation and emotion/Book/2024/Environmental cues and habits|Environmental cues and habits]] (Book chapter, 2024) * [[Motivation and emotion/Book/2018/Impulse buying motivation|Impulse buying motivation]] (Book chapter, 2018) ==References== {{Hanging indent|1= Ajzen, I. (1991). The theory of planned behavior. Organizational Behavior and Human Decision Processes, 50(2), 179–211. https://doi.org/10.1016/0749-5978(91)90020-T Asare, M. (2015). Using the theory of planned behavior to determine the condom use behavior among college students. American Journal of Health Studies, 30(1), 43. Cheng, X., Zhang, J., & Li, W. (2024). What shapes food waste behaviors? New insights from a comprehensive action determination model. Waste Management, 181, 188–198. https://doi.org/10.1016/j.wasman.2024.04.017 Deci, E. L. (1971). Effects of externally mediated rewards on intrinsic motivation. Journal of Personality and Social Psychology, 18(1), 105–115. https://doi.org/10.1037/h0030644 Fang, W.-T., Huang, M.-H., Cheng, B.-Y., Chiu, R.-J., Chiang, Y.-T., Hsu, C.-W., & Ng, E. (2021). Applying a comprehensive action determination model to examine the recycling behavior of Taipei City residents. Sustainability, 13(2), 490. https://doi.org/10.3390/su13020490 Frey, B. S. (1988). Ipsative and objective limits to human behavior. Journal of Behavioral Economics, 17(4), 229–248. https://doi.org/10.1016/0090-5720(88)90012-5 Frey, B. S. (1992). An ipsative theory of human behaviour. In Economics as a science of human behaviour (pp. 179–191). Springer. https://doi.org/10.1007/978-94-017-1374-0_12 Herzberg, F., Mausner, B., & Snyderman, B. B. (2011). The motivation to work (2nd ed.). Transaction Publishers. Herzberg, F., Mausner, B., & Snyderman, B. (1959). The motivation to work. John Wiley & Sons. Joanes, T., Gwozdz, W., & Klöckner, C. A. (2020). Reducing personal clothing consumption: A cross-cultural validation of the comprehensive action determination model. Journal of Environmental Psychology, 71, 101396. https://doi.org/10.1016/j.jenvp.2020.101396 Klöckner, C. A. (2013). A comprehensive model of the psychology of environmental behaviour—a meta-analysis. Global Environmental Change, 23(5), 1028–1038. https://doi.org/10.1016/j.gloenvcha.2013.05.014 Klöckner, C. A., Matthies, E., & Hunecke, M. (2003). Problems of operationalizing habits and integrating habits in normative decision-making models. Journal of Applied Social Psychology, 33(2), 396–417. https://doi.org/10.1111/j.1559-1816.2003.tb01902.x Klöckner, C. A., & Oppedal, I. O. (2011). General vs. domain-specific recycling behaviour: Applying a multilevel comprehensive action determination model to recycling in Norwegian student homes. Resources, Conservation and Recycling, 55(4), 463–471. https://doi.org/10.1016/j.resconrec.2010.12.009 Lai, E. R. (2011). Motivation: A literature review. Pearson Research Report. Liu, Y., Sheng, H., Mundorf, N., Redding, C., & Ye, Y. (2017). Integrating the norm activation model and theory of planned behavior to understand sustainable transport behavior: Evidence from China. International Journal of Environmental Research and Public Health, 14(12), 1593. https://doi.org/10.3390/ijerph14121593 Maslow, A. H. (1943). A theory of human motivation. Psychological Review, 50(4), 370–396. https://doi.org/10.1037/h0054346 Ofstad, S., Tobolova, M., Alim Nayum, & Klöckner, C. A. (2017). Understanding the mechanisms behind changing people’s recycling behavior at work by applying a comprehensive action determination model. Sustainability, 9(2), 204. https://doi.org/10.3390/su9020204 Onwezen, M. C., Antonides, G., & Bartels, J. (2013). The norm activation model: An exploration of the functions of anticipated pride and guilt in pro-environmental behavior. Journal of Economic Psychology, 39, 141–153. https://doi.org/10.1016/j.joep.2013.07.005 Ryan, R. M., & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68 Schwartz, S. H. (1977). Normative influences on altruism. In L. Berkowitz (Ed.), Advances in experimental social psychology (Vol. 10, pp. 221–279). Academic Press. https://doi.org/10.1016/S0065-2601(08)60358-5 Tan, J. J. H. (2024). Sustaining sustainable farming: An evaluation of the reasoned action and comprehensive action determination frameworks for persistence (Master's thesis, California Polytechnic State University). Available from ProQuest Dissertations & Theses database. Tanner, C. (1999). Constraints on environmental behavior. Journal of Environmental Psychology, 19(2), 145–157. https://doi.org/10.1006/jevp.1999.0121 Triandis, H. C. (1979). Values, attitudes, and interpersonal behavior. Nebraska Symposium on Motivation, 27, 195–259. Wood, W., & Neal, D. T. (2007). A new look at habits and the habit-goal interface. Psychological Review, 114(4), 843–863. https://doi.org/10.1037/0033-295X.114.4.843 Dörnyei, Z., & Ushioda, E. (2021). Teaching and researching motivation. Routledge. https://doi.org/10.4324/9781351006743 }} ==External links== * [https://www.healthdirect.gov.au/motivation-how-to-get-started-and-staying-motivated Motivation] (Healthdirect) * [https://www.sciencedirect.com/topics/medicine-and-dentistry/theory-of-planned-behavior Theory of planned behaviour] (ScienceDirect) * [https://www.sciencedirect.com/science/article/abs/pii/S0167487013000950 Norm-activation model] (ScienceDirect) * [https://water.dpie.nsw.gov.au/our-work/projects-and-programs/water-efficiency/water-saving-tips Water saving behaviours] (NSW Government) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Behaviour]] [[Category:Motivation and emotion/Book/Environment]] qlx5ueck3r730dbpnj9cv5cg9412tvv 1 307040 2690322 2684060 2024-12-05T00:34:10Z Jtneill 10242 Book chapter feedback 2690322 wikitext text/x-wiki ==Initial suggestions== {{ping|HassanAlsamara}} Thanks for tackling this topic. Some initial suggestions: * Check out other related chapters and see how you can build on, link to, and integrate with that work: ** [[:Category:Motivation and emotion/Book/Behaviour]] ** [[:Category:Motivation and emotion/Book/Environment]] * Also [[Motivation and emotion/Book|search past book chapters for related topics]] * For the [[Motivation and emotion/Assessment/Topic|topic development]], consider: ** What psychological theory(ies) can help to understand and explain this topic? ** What is the main research in this area? * Let me know if I can do anything else to support the development of this chapter. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:16, 11 August 2024 (UTC) <!-- Official topic development feedback --> {{METF/2024 |1= <!-- Title --> # The title and/or sub-title were not correctly worded and/or formatted (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|Heading casing]] # Insufficient development of headings to unpack the topic <!-- Alignment with focus questions --> # Reasonably good alignment between focus questions and heading structure, but consider closer alignment <!-- Other ---> # Check and correct capitalisation # Background info about motivation is not needed; instead, unpack the CADM and its application |3= <!-- Overview--> <!-- Scenario --> # Add a scenario or case study in a feature box (with an image) at the start of this section to help catch reader interest <!-- Description --> # Add a brief, evocative description of the problem/topic <!-- Focus questions --> # Reasonably good alignment between focus questions and heading structure, but consider closer alignment # Tailor the focus questions away from general questions about motivation to more specific unpacking of the sub-title |4= <!-- Key points--> # Promising development of key points for each section, with relevant citations # For sections which include sub-sections, include the key points for an overview paragraph prior to branching into the sub-headings # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Then focus most of the content on ''directly answering the core question(s)'' posed by the chapter sub-title. # Use APA style for citations (e.g., alphabetical order) # Use Australian spelling <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples <!-- Conclusion --> # Conclusion (the most important section): ## Underway ## What might the take-home, practical messages be? (What are the answer(s) to the question(s) in the sub-title and/or focus questions?) |5= <!-- Figure --> # One or more figure(s) is/are presented and captioned # The relevance of one or more figures isn't clear <!-- Caption --> # The figure caption(s) could better explain how the image connects to key points being made in the main text <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Include in-text [[m:Help:Interwiki linking|interwiki links]] for the first mention of key terms to relevant Wikipedia articles and/or to other relevant book chapters (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Examples/case studies --> # Consider use of one or more scenarios/examples/case studies <!-- Quiz --> # Focus the quiz question(s) on the take-home messages for each focus question <!-- Tables --> # Also consider using one or more tables to summarise key information |7= <!-- References --> <!-- Overall --> # OK # Messy # Reeve is overused as a citation <!-- Systematic reviews --> # Are there any systematic reviews about this topic? <!-- Suggestions --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links |8= <!-- Resources --> <!-- See also --> # See also ## Very good ## Link to the most relevant Wikipedia page(s) - emotion is too general ## Use alphabetical order <!-- External links --> # External links ## OK ## Move Wikipedia link to see also (much better than the emotion link) ## Other link is too general ## Only include links directly related to the sub-title |9= <!-- User page --> # Very good <!-- Description about self --> # Excellent description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Good – two out of three types of contributions made with with direct link(s) to evidence. The other type of contribution is making: ## posts about the unit or project on other platforms such as the {{Motivation and emotion/Canvas}} discussion forum or on [https://x.com X] using the {{Motivation and emotion/Hashtag}} }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:40, 26 August 2024 (UTC) {{MEMF/2024 |1= <!-- Overall comments ... --> <!-- Overall - Overall --> # Overall, this is a basic presentation |2= <!-- Overview comments ... --> <!-- Overview - Opening --> # The opening conveys the purpose of the presentation in a basic way # Incorrect date on opening slide <!-- Overview - Introduction --> # Create an engaging introduction to hook audience interest (e.g., through an example) <!-- Overview - Context --> # A basic context for the presentation is established <!-- Overview - Focus --> # Consider asking focus questions to help focus and discipline the presentation |3= <!-- Content comments ... --> # Comments about the book chapter may also apply to this section <!-- Content - Addresses topic --> # The presentation addresses the topic <!-- Content - Amount --> # An appropriate amount of content is presented — not too much or too little, but the balance could be improved (it is very theoretically dry) <!-- Content - Theory --> # The presentation makes good use of relevant psychological theory <!-- Content - Research --> # The presentation makes insufficient/no use of relevant psychological research <!-- Content - Citations --> # The presentation makes basic use of citations to support claims <!-- Content - Examples --> # The presentation makes insufficient use of examples <!-- Content - Practical advice --> # The presentation could be improved by providing practical advice |4= <!-- Conclusion comments ... --> <!-- Conclusion - Slide --> # Provide a conclusion which summarises the most relevant psychological theory and research about this topic, with take-home messages for each focus question |5= <!-- Audio comments ... --> <!-- Audio - Narration --> # The presentation makes very basic use of narrated audio <!-- Audio - Pacing --> # Audio communication is reasonably well-paced <!-- Audio - Voice --> # Basic [[w:Intonation (linguistics)|intonation]] <!-- Audio - Practice --> # The narration could benefit from further scripting and/or practice <!-- Audio - Recording quality --> # Audio recording quality was excellent <!-- Audio - Topic --> # The narrated [[#Content|content]] is reasonably well matched to the target topic |6= <!-- Video comments ... --> <!-- Video - Overall --> # Overall, visual display quality is basic <!-- Video - Video, Image, Text --> # The presentation makes basic use of text and image based slides # The presentation makes basic use of text-based slides <!-- Video - Text - Font --> # The font size is sufficiently large to make it easy to read <!-- Video - Text - Amount --> # The amount of text presented on one or more slides could be reduced to make it easier to read and listen at the same time <!-- Video - Images --> # The visual communication is supplemented in a basic way by relevant images and/or diagrams <!-- Video - Production --> # The presentation is basically produced using simple tools <!-- Video - Topic --> # The visual [[#Content|content]] is reasonably well matched to the target topic |7= <!-- Meta-data comments ... --> <!-- Meta-data - Title/sub-title --> # The (almost) correct title is used, but the sub-title (or a shortened version of it) is not used, as the name of the presentation. This would help to convey the purpose of the presentation and be consistent. <!-- Meta-data - Description --> # A brief written description of the presentation is provided. Expand. <!-- Meta-data - Links --> # Links to and from the book chapter are provided |8= <!-- Licensing comments ... --> <!-- Licensing - Images --> # Image sources and their copyright status are communicated # Provide clickable links to the image sources and license details (e.g., in the description) <!-- Licensing - Presentation --> # A copyright license for the presentation is clearly indicated }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:40, 10 November 2024 (UTC) == Heading casing == {| style="float: center; background:transparent;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:HassanAlsamara|Hi [[User:HassanAlsamara|HassanAlsamara]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:16, 12 November 2024 (UTC) |} ==Draft feedback== {{ping|HassanAlsamara}} From a very quick look at current draft: * Looks comprehensive (but this isn't based on reading, only a quick scan) * See comment above about heading capitalisation * Remove trailing colons from headings * Follow [https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style for capitalisation] * Use ''focused'' focus questions (e.g., avoid general questions such as why is understanding motivation important) * Use APA style for citations (e.g., alphabetical order for multiple citations) * Use Australian spelling (internalized -> internalised) * The draft is over the maximum word count (e.g., reduce the section "What is motivation?") * Use APA style for references * Use 3rd person perspective rather than first person perspective (e.g., remove "we") * Some paragraphs overly long (aim for 3 to 5 sentences) Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:16, 12 November 2024 (UTC) <!-- Official book chapter feedback --> {{MEBF/2024 |1= <!-- Overall comments... --> # Overall, this is a basic chapter <!-- Overall – Citations --> # Very good use of academic, peer-reviewed citations to support claims # In some places, citations are used which are not in the References <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FComprehensive_action_determination_model&diff=2690321&oldid=2690096 these copyedits] |2= <!-- Overview comments... --> # Well developed <!-- Overview – Case study --> # Somewhat engaging case study or scenario in a feature box with a relevant image <!-- Overview – Explains problem --> # Explains the psychological problem or phenomenon reasonably well <!-- Overview – Focus questions --> # The focus questions are clear and relevant |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # A solid good range of relevant theories are selected, described, and explained <!-- Theory – Builds on --> # Builds somewhat on other [[Motivation and emotion/Book|chapters]] and/or [[w:|Wikipedia]] articles <!-- Theory – Depth --> # Reasonably good depth is provided about relevant theory(ies) <!-- Theory – Tables/Figures/Lists --> # Very good use of tables, figures, and/or lists to help clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used <!-- Theory – Examples --> # Basic use of examples to illustrate theoretical concepts # Consider using more tangible, practical, simple examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Insufficient review of relevant research # More detail about key studies would be ideal # Any systematic reviews or meta-analyses in this area? <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Insufficient integration between theory and research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Insufficient integration with [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Reasonably good summary and conclusion # Key points are well summarised # Address the focus questions # Add practical, take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## Overall, the quality of written expression is reasonably good <!-- Written expression – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid directional referencing (e.g., "as previously mentioned") because it's usually unnecessary. If needed, use [[w:Help#Section linking|section linking]]. <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## Include an introductory paragraph before branching into the sub-sections (see {{expand}} tags) <!-- Written expression – Grammar --> # Grammar ## The grammar for some sentences could be improved (e.g., see {{g}} tags). Consider using a grammar checking tool, Studiosity, and/or peer feedback on draft work. <!-- Written expression – Abbreviations --> ## Abbreviations ### Once an abbreviation has been established (e.g., PTSD), use it consistently aftwarwards <!-- Written expression – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) <!-- Written expression – Proofreading --> # Proofreading ## More proofreading is needed (e.g., fix punctuation and typographical errors) to bring the quality of written expression closer to a professional standard ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] (e.g., remove first letter capitalisation of models/theories) <!-- Written expression – APA style --> # APA style ## [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies Use sentence casing for the names of disorders, therapies, theories, etc.] ## Direct quotes need page numbers – even better, communicate about concepts in your own words <!-- Written expression – Figures --> ## Figures ### Reasonably well captioned ### Use this format for captions: ''Figure X''. Descriptive caption goes here in sentence casing. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). <!-- Written expression – Citations --> ## Citations use very good [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]: ### List multiple citations in alphabetical order by first author surname <!-- Written expression – References --> ## References use very good APA style: ### Check and correct use of italicisation |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Basic use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # No use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Very good use of figure(s) # No use of table(s) # Basic use of feature box(es) # The chapter is conceptually abstract. Insufficient use of scenarios, case studies, or practical examples <!-- Learning features – Quizzes --> # Very good use of quiz(zes) and/or reflection question(s) # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end <!-- Learning features – See also --> # Basic use of the "See also" section ## Also include links to related Wikipedia articles <!-- Learning features – External links --> # Very good use of the "External links" section |9= <!-- Social contribution comments... --> # ~3 logged, useful, mostly moderate contributions with direct links to evidence # Some images uploaded to Wikimedia Commons, but not logged as contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:33, 5 December 2024 (UTC) c2fp2r7asz8ir7ub68vv5tmjn0omvlp 0 307962 2690312 2690176 2024-12-04T22:48:12Z Scogdill 1331941 2690312 wikitext text/x-wiki =Event= On Monday 28 June 1897, Queen Victoria hosted a garden party at Buckingham Palace, inviting between 5,000 and 6,000 people. This party was the final official event of the London Diamond Jubilee celebrations. The Queen released to the press the names of people invited, which means the newspapers could print some or all of this list. The very long article in the London ''Morning Post'', for example, prints what may be the comprehensive list of those invited, although two columns are illegible in some places. The original newspaper account seems to have been published by the ''Court Circular'', and then the popular newspapers reprinted pieces of that story, many adding contextualizing paragraphs of their own. Some of these later reports are quite long, perhaps 5 or more full columns. Sometimes the newspapers included short descriptions of the women's dresses, suggesting that for the list of people invited, the source was the ''Court Circular'', but the parts of the stories devoted to context, history or fashion might have been written by a reporter present at the event. ==Logistics== * 28 June 1897, Monday, in the gardens at Buckingham Palace, hosted by Queen Victoria. * Between 5,000 and 6,000 guests were invited. * Many visitors from the empire who were in town for the Jubilee celebrations were invited to this garden party. * The weather was fine, having improved since the day before. * The garden party was held in the grounds around Buckingham Palace, and the Palace itself was open and available for guests to visit:<blockquote>Great preparations had been made in the splendid grounds adjoining the Royal Palace for the party, the whole scene presenting a fascinating appearance. The beautifully-kept grounds were partially covered with tents and marquees for the convenience of the many guests, and the lovely lake was really in the hands of the Queen’s bargemen, who had charge of the many boats which had been placed on the extensive ornamental waters for the use of guests. There was also plenty of music, several regimental bands being in attendance, while for those who wished to become acquainted with the valuable pictures and works art which are to be found at the Royal residence, all the State and reception rooms of the Palace were thrown open.<ref name=":2">“The Queen’s Garden Party. Brilliant Scene at Buckingham Palace.” ''Globe'' 29 June 1897, Tuesday: 6 [of 8], Col. 3a–c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18970629/050/0006. Print p. 6.</ref> (6, Col. 3a)</blockquote> *The streets around the entrances to Buckingham Palace were lined with spectators beginning hours before the Queen was to arrive:<blockquote>Although the Garden Party was not timed to commence until after five o’clock, the Mall from Marlborough House to Buckingham Palace was well lined by two o’clock, and an hour afterwards large crowds, for the most part composed of ladies, had taken up their positions. This was also the case along Constitution-hill, where the assembly which had gathered to witness the Queen’s arrival at the Palace from Windsor nad [sic] to a large extent remained. The heat was somewhat oppressive, but the trees along the Mall and the Green Park afforded welcome shelter. Many ladies had evidently come prepared for a long wait, as they had provided themselves with the now familiar camp stool, which is always prominent on these occasions. On the other hand, the police were waging war against the men who frequent such places with stools and forms, and as soon as any of them put in an appearance they were quickly pounced upon by the officers, who at once proceeded to destroy the intended stands before the eyes of the helpless owners. Among the sightseers were several of the Indian visitors in gorgeous coloured coats, tight-fitting trousers, and turbans, as well as some of the Australian and New Zealand troops.<ref name=":2" /> (6, Col. 3a)</blockquote> ==Related Events== This garden party was the culminating event of the official celebrations for Queen Victoria's Diamond Jubilee, and more specific events led up to it: # Trip from Windsor to Paddington Station Queen Victoria and a large retinue traveled by train from Windsor to Paddington Station the day before, preceded on an earlier train by "the royal equipages sent from Buckingham Palace for the use of the Queen and her suite," which were<blockquote>First came the splendied semi-state landau in which the Queen made her now famous journey on June 22d. It was preceded by scarlet-coated outriders, and horsed by four magnificent bays driven by postilions in navy blue and white uniforms. Two similar carriages followed, and these were in turn succeeded by a number of pair-horse clarences for the conveyance of the household and suite, and several breaks and ‘buses for luggage. A captain's escort, furnished by the 2d Life Guards, and commanded by Captain Ellison, clattered along in rear of the carriages, and took up a position opposite the spot where, by prior arrangement, Her Majesty’s saloon was to be brought to a standstill. These magnificent troops, riding their great black horses, and with the sunlight dancing upon their nodding plumes, and reflected by their burnished helmets, cuirasses, and trappings, made a very fine show indeed. The escort did not carry the colour, as it did on the 21st, nor was it accompanied by the regimental trumpets.<ref name=":0">"Jubilee Festivities. The Queen Again in London. Interesting Functions. A Visit to Kensington. The Garden Party." ''North British Daily Mail'' 29 June 1897, Tuesday: 5 [of 8], Col. 3a–7b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002683/18970629/083/0005. Print p. 5.</ref>{{rp|5, Col. 3b}}</blockquote> # Reception at Paddington # Visit to Kensington # Kensington to Buckingham Palace # The Garden Party # Return to Windsor by Way of Paddington === Foreign Admirals === On 29 June 1897, the day after the garden party, the ''North British Daily Mail'' reports that, after the Queen's garden party, the foreign admirals would return to Spithead for a tour around the dockyard and luncheon:<blockquote>THE FLEET AT SPITHEAD<p> The fleet at Spithead was again illuminated last night, the railway companies having duplicated the ordinary train service to bring visitors down. The Koenig Wilhelm was to have sailed on Sunday evening, but her departure has been deferred, and last night her officers gave a private dinner party aboard for the anniversary of the Queen’s Coronation. All the commissioned ships in the harbour were dressed at noon. A royal salute was fired. The [Col. 6c–7a] foreign admirals will return from their visit to London on the occasion of the Queen’s garden party to be conducted round the dockyard to-day, and they will be entertained to luncheon.<ref name=":0" />{{rp|5, Col. 6c–7a}}</blockquote> === Colonial Premiers === The day of the garden party the colonial premiers attended a meeting with Secretary of State for the Colonies, [[Social Victorians/People/Chamberlain|Joseph Chamberlain]]:<blockquote>THE COLONIAL PREMIERS The whole of the Colonial Premiers went to the Colonial Office yesterday for further conference with Mr Chamberlain, who received them in his private room, attended by Mr F. H. Wilson, legal assistant, Mr Reid and the Hon. T. Cochrane, M.P., assistant private secretaries. The conference lasted hours, and was of a strictly private and confidential character, the matters discussed involving several points of high State policy. Premiers will be entertained at Warwick Castle by the Earl and Countess of Warwick on July 15th. On the same occasion the Attorney General of Queensland will present a loving cup from Warwick, in Queensland, to the old county town of Warwick, from which it takes its name. He will be accompanied by the Colonial troopers.<ref name=":0" />{{rp|5, Col. 7a}}</blockquote> For these visitors to London during the Diamond Jubilee, the next major social event was on 15 July, at Warwick Castle, hosted by [[Social Victorians/People/Warwick|Daisy, Countess of Warwick and Francis, 5th Earl of Warwick]], although perhaps some attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]]. == Who Was Present == In the absence of a copy of the report about the garden party in the ''Court Circular'', the newspaper account with the fullest list of names is from the ''Morning Post'', although people further down the list can be impossible to identify, and two full columns are damaged (Col. 7 on p. 4 and Col. 1 on p. 5).<ref name=":1">“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004 and https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005.</ref> Whenever possible, then, what is here has been amended with other newspaper reports that have names to help decipher the illegible ones in the ''Morning Post'' account. The names in the Morning Post are grouped, mostly by rank and name. === People of Color at This Event === One purpose of a closer look at this event is to get a more precise list of names of people of color from the various countries in the empire, who were not recognized and thus not named in newspaper descriptions of other events. For example, the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]] was said to include a number of South Asian dignitaries, but because the Duchess did not release to the newspapers the names of those who were invited, those dignitaries went mostly unnamed in the newspaper reports, if their presence was noted at all. Besides the South Asian guests invited to this garden party, some South Asian visitors to London were spectators as well:<blockquote>Among the sightseers were several of the Indian visitors in gorgeous coloured coats, tight-fitting trousers, and turbans, as well as some of the Australian and New Zealand troops.<ref name=":2" /> (6, Col. 3a)</blockquote>In a section on what people — mostly women — wore, the reporter for the ''Daily News'' said,<blockquote>Suffice to say, the modistes had done their best, and that their achievements excited general admiration. Here and there, however, was an Eastern beauty whose golden lace drapery, loosely enveloping a figure that owed nothing to the corset, challenged comparison, we will not say with what success, with the European model. In the almost entire absence of uniforms or Court dress, the costumes of the East Indian notables lent colour to the assemblage, while their pearls and diamonds, the wealth of Ormuz and of Ind, were not allowed to pass unobserved.<ref name=":3" /> (5, Col. 6b)</blockquote> === People Invited === # Queen Victoria, with escort and attendants ## Captain's Escort of the 2nd Life Guards ## The Duchess of Buccleuch, Mistress of the Robes ## The Dowager Lady Churchill, Lady in Waiting ## The Hon. Harriet Phipps, Woman of the Bedchamber ## Maids of Honour in Waiting ### The Hon. Mary Hughes ### The Hon. Aline Majendie ## the Earl of Kintore, Lord in Waiting ## Captain Drummond, Groom in Waiting ## Equerries in Waiting ### Major-General Sir John M'Neill, V.C. ### Lieutenant Colonel Davidson, M.V O. [sic] #Grand Duke and Grand Duchess Serge of Russia #Princess Henry of Battenberg, with attendants ##Miss Minnie Cochrane ##Colonel John Clerk, C.S.I., C.V.O. #Her Imperial Majesty the Empress Frederic, attended by ##the Dowager Lady Ampthill ##Lord Harris ##Colonel S. Waller ##Princess Hatzfeldt Trachenberg ##Count Seckendorff ##Baron and Baroness Reischach #Their Royal Highnesses the Prince and Princess of Wales, with attendants ##Lady Suffield, Lady in Waiting ##Miss Knollys, Woman of the Bedchamber ##Lord Colville of Culross, K.T., G.C.V.O., Chamberlain to the Princess of Wales ##The Earl of Gosford, K.P., Lord in Waiting ##General Sir D. Probyn, G.C.V.O., K.C.B., K.C.S.I., V.C, Comptroller ##Sir Francis Knollys, K.C.M.G., C.B., Groom in Waiting ##Major-General Stanley Clarke, C.M.G., Equerry in Waiting #Princess Victoria of Wales #Their Royal Highnesses Prince and Princess Charles of Denmark #Their Royal Highnesses the Grand Duke and Grand Duchess of Mecklenburg-Strelitz, attended by ##Lady Caroline Cust ##Mr. Hugo Erskine Wemyss ##Count Reventlow Criminil ##Baron von der Wense #Their Royal Highnesses Prince and Princess Christian, attended by ##Baroness von und zu Egloffstein ##Colonel the Hon. Charles Eliot #Her Highness Princess Victoria #His Highness Prince Christian Victor #His Highness Prince Albert of Schleswig-Holstein #Her Royal Highness Princess Louise Marchioness of Lorne and the Marquis of Lorne, attended by ##Lady Sophia Macnamara ##[[Social Victorians/People/Arthur Collins|Colonel Arthur Collins]], M.V.O. #Their Royal Highnesses Prince and Princess Henry of Prussia, attended by ##Admiral of the Fleet Sir Edmund Commerell ##Baron and Baroness Seckendorff ##Count Hahn ##Captain Muller #Their Royal Highnesses the Duke and Duchess of Saxe-Coburg and Gotha, attended by ##The Hon. Mrs. Monson ##His Excellency Herr von Schön ##Captain the Hon. D. J. Mouson [sic, s/b Monson?], M.V.O. ##Mr. A. D. J. Monson ##Captain von Ruxleben #Princess Beatrice of Saxe-Coburg and Gotha #The Hereditary Prince of Saxe-Coburg and Gotha #Their Royal Highnesses the Duke and Duchess of Connaught and Strathearn, attended by ##Colonel and the Hon. Mrs. A. Egerton #Her Royal Highness the Duchess of Albany, attended by ##Sir Robert and Lady Collins ##Miss Potts #Her Royal Highness Princess Frederica of Hanover and Baron von Pawel Raminingen, attended by ##Mr. and Mrs. Charles Wood #His Royal Highness the Duke of Cambridge, attended by ##Colonel A. C. FitzGeorge, C.B. #Her Royal Highness the Duchess of Teck and his Highness the Duke of Teck, attended by ##Lady Katherine Coke ##The Hon. A. Nelson Hood #Her Royal Highness Princess Louise Duchess of Fife and the Duke of Fife #His Highness the Prince and her Royal Highness Princess Frederic Charles of Hesse, attended by ##The Hon. A. Hay ##Fraulein von Tasmund ##Baron von Kotwitz #Their Highnesses Prince and Princess Aribert of Anhalt, attended by ##Miss Deverell ##Major Evan Martin #Her Royal Highness the Hereditary Princess of Saxe-Meiningen and her Serene Highness Princess Feodore of Saxe-Meiningen, attended by ##The Hon. Aubrey FitzClarence ##Miss von Dreskan ##Baron von Roeder #His Serene Highness the Prince of Schaumburg-Lippe #Their Highnesses Prince and Princess Edward of Saxe-Weimar #Her Serene Highness Princess Victor of Hohenlohe #Countess Gleichen (x2) #Their Serene Highnesses Prince and Princess Adolphus of Teck #The Prince Francis and Prince Alexander of Teck #His Highness Prince Augustus Leopold of Saxe-Coburg #Their Serene Highnesses Prince and Princess Blucher von Wahlstatt #Their Serene Highnesses Prince and Princess Joachim Murat #Their Serene Highnesses [[Social Victorians/People/Pless|Prince and Princess Hans Henry Pless]] #Prince and Princess Loewenstein #Their Serene Highnesses the Duke and Duchess of Arenberg #Prince Victor Duleep Singh #Prince Frederick Duleep Singh #Princess Duleep Singh (x2) #ARGENTINE REPUBLIC — M. Florencio Dominguez and M. Carlos Dominguez #BADEN — Herr yon Brauer, Mr. Brook Taylor, and Baron Bohlen Halbach #BAVARIA — His Royal Highness the Prince Rupert, General Sir L. Gardiner, K.C.V.O., C.B., Major Fairholme, Lieutenant-Colonel Emile von le Bret Nucourt, and Captain Othon von Stettin #BELGIUM — His Serene Highness the Prince Charles de Ligne, Princess de Ligne, Madlle. de Ligne, Mr. C. lnnes Ker, Count de Jonghe d'Ardoye, and the Marquis d’Asshe #BOLIVIA — M. Caso, Mr. Conway Seymour, M. Pedro Suarez, Madame Suarez, and M. Adolfo Bolivian #BRAZIL — M. [[Social Victorians/People/Souza Correa|de Souza Correa]] [Corréa?] #BULGARIA — Their Royal Highnesses the Prince and Princess of Bulgaria, Colonel J. R. Slade, C.B., Madame Petrow Tchomakoff, Count Robert de Bourboulon, Lieutenant-Colonel Marcoff, Major Petrew, Captain Stoïanow, and Mr. Martin Furth #CENTRAL AMERICA (Greater Republic) — M. Medina and Miss Medina #CHILI — M. Ramon Subercasseaux and Mr. Raglan Somerset #CHINA — His Excellency Chang Yen Hoon, Colonel Mark Bell, V.C,. Mr. Liang, Mr. Jui, and Mr. Koo #COREA — His Excellency Min Young Hwan, Major A. Cavendish, Mr. Min Young Chan, Mr. Min Shangho, and Mr. von Rautenfeld #COSTA RICA — Senor Don Demetrio Iglesias, Mr. C. Alban Young, Dona Eudoxia Castro, Señorita Maria Iglesias, Don Ricardo Fernandez Guardia, and Dona Christina Castro Keith #DENMARK — His Royal Highness the Prince Waldemar, Major-General Arthur Ellis, C.S.I., M. Charles Rothe, and Captain Evers #EGYPT — Prince Mohammed Ali Pasha, Colonel Larking, Tigrane Pasha, Colonel Aziz Bey, Mr. George Smart, Said Zoulfikar Bey #ECUADOR — M. Navares, Colonel Concha #FRANCE — General Davoust, Duc d'Auerstadt, Duchesse d'Auerstadt, and Madlle. Davoust, Colonel Brabazon, Colonel Dawson, General Hagron, M. Crozier, Colonel Humbert, and Captain Riviers de Mauny #GERMANY — His Royal Highness the Prince Albert of Prussia, Prince Regent of Brunswick, Major-General Sir C. du Piat, K.C.B., Colonel Grierson, Lieutenant-General von Plessen, Colonel von Arnim, Captain Fischel, Count von der Schulenberg (Hofmarschall), Major Freiherr von Stein, Dr. Schreibe, Captain von Unzer #GREECE — M. Rangabi, Mr. R. D. Norton #GUATEMALA — Dr. Cruz, Madlles. Cruz (2), Señor Estrada #HAWAIIAN ISLANDS — Mr. S. M. Damon, Captain the Hon. H. Napier, Major Curtis P. Jaukea #HESSE — Their Royal Highnesses the Grand Duke and Grand Duchess of Hesse, Colonel the Hon. H. Byng, C.B., Baroness de Grancy, Baron Riedesel zu Eisenbach, Baron de Genadius Grancy #ITALY — Their Royal Highnesses the Crown Prince and Princess, the Earl of Clarendon, Colonel Needham, Countess Giulia Trigona, Lieutenant-General Terzaghi, Major Cavaliere Viganoni, Captain Cavaliere Merli Miglietti, Count Romnaldo Trigona, Cavaliere F. Comotto #JAPAN — His Imperial Highness the Prince Arisugawa, Mr. R. F. Synge, Captain Beaumont, R.N., Marquis Ito, Mr. S. Saito, Marquis Kido, Captain Funaki, Lieutenant-Colonel Murata, Lieutenant Kato, Mr. Nabeshima #LIBERIA — Mr. H. Hayman #LUXEMBURG — His Royal Highness the Hereditary Grand Duke of Luxemburg, Colonel H. D. Browne, Baron Ritter yon Grünstein #MECKLENBURG-SCHWERIN — His Excellency Herr D. yon Vietinghoff, Mr. Eyre A. Crowe #MEXICO — Don Antonio Mier y Celis, Mr. Arnold Royle, C.B., Don Francisco R. Gallardo, Don Eustagino dc Escaudon, and Captain Don Ponfirio Diaz #MONTENEGRO — His Highness the Prince Danilo, Major the Hon. C Harbord, Colonel Djurcovitch, and Captain Pejanovitch #NETHERLANDS — Count van Lynden, Countess van Lynden, Mr. Horace West, and Count W. de Bylandt #PARAGUAY — M. E. Machain and Madame Machain #PERSIA — His Imperial Highness the Prince Amir Khan, General Sir Thomas Gordon, K.C.I.E., C.B., C.S.I.[,] Mr. Harry Churchill, General Karim Khan, Mirza Ahmad Khan, Mirza Ohaness Khan, Mirza Mohamad Ali Khan #PERU — Senor Canevaro, Duchesse de Zoagli Canevaro, Dr. Don A. N. Puente, Don Alfredo Elster, and Don Carlos von der Heyde #PORTUGAL — His Royal Highness the Duke of Oporto, Major the Hon. H. C Legge, M.V.O., Colonel Duval Telles, Captain Moreira de Sà, Major d'Albuquerque, and Lieutenant Jose de Melie[?] #ROME — Right Rev. Monsignore Sambucetti, [[Social Victorians/People/Stonor|Hon. Harry Stonor]], Right Rev. Monsignore Belmont, the Right Rev. Monsignore de Vaz, Marchesi and Marchesa Muccioli, of the Noble Guard #ROUMANIA — General Pancovici, Colonel G. P. Georgescu #RUSSIA — Their Imperial Highnesses the Grand Duke Serge and Grand Duchess Feodrowna, the Grand Duke Cyril, Lord Churchill, Lieutenant-Colonel Waters, Countess Olsouffiew, Princess Youssoupoff, Princess Lobanoff de Rostow, General Stépanoff, Colonel Gadon, and Prince Youssoupoff, Colonel Clements, Mr. Alexander Gordon Ross, and Sub-Lieutenant N. Coubé (A.D.C. to Grand Duke Cyril) #SAXE-COBURG — His Royal Highness the Prince Philip of Saxe-Coburg, Captain Walter Campbell, and Herr von Schön #SAXE-WEIMAR — His Highness the Prince Hermann of Saxe-Weimar, Mr. Frederick Campbell, and Count Zeppelin #SAXONY — His Royal Highness the Prince Frederick Augustus, Duke of Saxony, Colonel Howard, Freiherr yon Reitzenstern, First Lieutenant von Metzsch, and Baron von Oppell #SERVIA — M. Mijatovich and Madame Mijatovich #SIAM — His Royal Highness the Crown Prince and the Prince Mahit of Siam, Colonel E. H. Sartorius, V.C., Lieutenant-Colonel Rajavallabha, Lieutenant-Colonel C. Vernon Hume, Colonel Indaraty, Surgeon-Major Yarr #SPAIN — Duke of Sotomayor, Captain the Hon. A. Greville, Señor José Caro, Señor Alfonso Merry del Val, and Señor Benitez al Villar #SWEDEN AND NORWAY — His Royal Highness the Prince Eugène of Sweden and Norway, Captain G. L. Holford, Count G. Gyldenstolpe, Captain Roeder, Captain Baron Cederstrom #TURKEY — Munir Pasha, Major Surtees, Brigadier-General Nassir Pasha, Captain Enver Bey, Colonel Gordon Ponsonby #UNITED STATES — His Excellency the Hon. Whitelaw Reid, Mrs. Whitelaw Reid, Colonel Hallam Parr, Major-General Nelson A. Miles, Mrs. Nelson Miles, Rear-Admiral Joseph N. Miller, Captain M. P. Maus, Mr. Ogden Mills, Mrs. Ogden Mills, Mr. G. Creighton Webb, Mr. Erskine Hewett, Commander W. H. Emory, Lieutenant Philip Andrews, Lieutenant T. S. Rogers #URUGUAY — Dr. Alberto Nin, Madlle. Nin, Don Alfonso Saenz de Zumaran, Don Luis Posadas, Colonel C. Robido #WURTEMBURG— His Royal Highness the Duke Albert of Wurtemburg, Colonel C. Swaine, Lieutenant-General von Bilfinger, First Lieutenant Count von Degenfeld- Schonburg; five officers of the Queen's German Regiment: Major C. R. Burn (in attendance), Lieutenant-Colonel von Falkenhayn, Major von Arnim, First Lieutenant Baron von Moeller-Lilienstern, First Lieutenant von Gerlach, Second Lieutenant von Studnitz #"Native Princes, and gentlemen and ladies accompanying them"<ref name=":1" /> (4, Col. 2b) ##His Highness the Raja of Kaparthala ##His Highness the Thakur Sahib of Morvi, K.C.I.E. ##His Highness the Thakur Sahib of Gondal, C.I., and her Highness the Maharani of Gondal, C.I. ##Colonel Maharaj Dhiraz ##Sir Pratab Singh, K.C.S.I. ##Thakur Hari Singh[,?] ##Kunwar Dhokal Singh ##Rajah Ajit Singh of Khetri, attended by ##Rajkumar Unmaid Singh of Shahpura, attended by ###Colonel Trevor (in attendance upon the Rajah Ajit Singh of Khetri and the Rajkumar Unmaid Singh of Shahpura) ##Bijey Singh ##Sir Jamaetjee Jejeebhoy, Bart., C.S.I., Miss Jejeebhoy, Mr. Jejeebhoy ##Mr. and Mrs. Powrala ##Major J. G. Turner and Mrs. Turner ##Mr. A. R. Wood and Mrs. Wood #The "officers of the Imperial Service Troops, with British officers and ladies"<ref name=":1" /> (4, Col. 2b) ##Captain Mir Hashim Ali Khan Hyderabad-Resaldar ##Major Sunayat Singh, Kashmir ##Commandant Abdul Ganny, Gwalior ##Commandant Gooind, Rao Matkar, Indore ##Commandant Mirza Kurim Beg, Bhopal ##Rai Bahadur Dhunpat Rai, Jeypore ##Commandant Nand Singh, Patiala ##Commandant Rai Bahadur Thakur Dip Sing, Bikanir ##Commandant Chatru Singh, Bhartpur ##Resaldar Abdul Majid Khan, Babawalpur ##Commandant Daud Khan, Ulwar ##Commandant Nazir Khan, Rampur ##Risalda-Major Didar Singh, Sindi ##Risaldar-Major Kishan Singh, Nabha ##Risaldar Hara Singh, Karpurthala ##Risaldar Dhan Singhi, Bhavnagar ##Colonel H. Melliss, C.S.I., and Mrs. Melliss ##Major F. H. R. Drummond and Mrs. Drummond ##Captain F. Angelo ##Lieutenant H. Coape-Smith ##Captain G. F. Chenevix-Trench #The "officers of Native Cavalry Corps with British officers and ladies"<ref name=":1" /> (4, Col. 2b) ##Risaldar-Major Baha-ud-din-Khan ##Sardar Bahadur, A.D.C. to Viceroy ##Risaldar-Major Sayyid Abdul Aziz ##Risaldar-Major Khan Bahadur ##Risaldar-Major Izzat Khan ##Risaldar-Major Hukam Singh ##Risaldar-Major Sher Singh ##Risaldar-Major Husain Khan ##Risaldar-Major Mangal Singh ##Risaldar-Major Kesar Singh ##Risaldar- Major Faiz Khan ##Risaldar-Major Muhammad Umar Khan ##Risaldar-Major Ali Mahomed Khan ##Risaldar-Major Mihrab Ali Khan ##Risaldar Kaddam Khan ##Risaldar Jahanzir Khan ##Risaldar Nadir Khan ##Risaldar Mir Haidar Shah Khan ##Risaldar Makbul Khan ##Risaldar Net Ram ##Ressaidar Gurdatt Singh ##Subadar Muhammed Beg Junadar ##Abdul Karin Khan ##Lieutenant-Colonel J. C. H. Gordon and Mrs. Gordon ##Major A. Phayre and Mrs. Phayre ##Captain C. F. Campbell ##Captain P. Melville, in attendance on his Highness Thakur Sahib of Morvi ##Captain M'Cartney Filgate, in attendance on their Highnesses the Thakur Sahib and Maharani of Gondal ##Mr. Nowroz ##M. Parveez ##Sir M. Mansherjee Bhownaggree, M.P. ##Mr. Percy Armytage and Mrs. Armytage ##Mr. Frank Cook, C.I.E., and Mrs. Frank Cook #The "commanding officers of Colonial contingents, with the ladies accompanying them"<ref name=":1" /> (4, Col. 2b) ##Colonel the Hon. M. and Mrs. Aylmer, Canada ##Colonel and Mrs. Lassetter, New South Wales ##Major Reay, Victoria ##Colonel Pitt, New Zealand ##Major and Miss King, Queensland ##Lieutenant and Mrs. Phillips, Cape of Good Hope ##Lieutenant-Colonel Rowell, South Australia ##Major Strickland, Western Australia ##Captain Shepstone, Natal ##Major and Miss Reeves, Ceylon ##Mr. Badeley, Hong Kong ##Colonel Walker, C.M.G., and Mrs. Walker, Straits Settlements ##Captain Lucie Smith, Jamaica ##Lieutenant-Colonel E. B. M'lnnis, C.M.G., and Mrs. M'lnnis, British Guiana ##Major Rooks, Trinidad ##Captain Bernard, Malta ##Captain Kershaw, Cyprus ##Captain and Mrs. Middlemist, Gold Coast ##Inspector Hook, Lagos ##Captain Blakeney, Sierra Leone ##Lieutenant Festing, Royal Niger Company ##Captain Flint, British North Borneo Company ##The Hon. M. Gifford, Rhodesian Horse ##The following British officers attached: Lieutenant-Colonel Boulton, Lieutenant-Colonel Prior, Lieutenant-Colonel Tucker, Lieutenant-Colonel Domville, Lieutenant-Colonel Gibson, and Lieutenant-Colonel Tyrwhitt #The "gentlemen representing the various races in the Island of Ceylon"<ref name=":1" /> (4, Col. 2c) ##Maha Mudaliyar don Solomon Dias Bandaranaihe ##The Hon. Alexander Dealius Sonewiratne ##M. E. Rowland Goonoratne ##M. Charles de Soysa Dessanayaka ##Panabokko Jikiri Banda ##Nugawela Kuia Banda ##Kobbokeduwe Loku Banda ##M. E. S. W. Senathi rajah [sic] and Mrs. Senathi ##M. J. H. de Saram and Miss de Saram ##M. P. Ramanathan ##M. Saunders and Miss Saunders #The "members of the Corps Diplomatique and other foreigners of distinction"<ref name=":1" /> (4, Col. 2c) ##The Russian Ambassador, Madame de Staal, Madlle. de Staal, Madame de Stoeckl, Princess de San Donato, Madame Yermoloff, Madlle. Yermoloff, the Councillor, three Secretaries, and four Attachés of Embassy ##The German Ambassador, Countess Paul Hatzfeldt-Wildenburg, her Serene Highness Princess Hans Hohenlohe-Oehringen, Baroness yon Eckardtstein, the Councillor, two Secretaries, three Attachés of Embassy, and the Director of the Chancery ##The Austro-Hungarian Ambassador, Countess Deym, Countess Isabella Deym, Countess Clary Aldringen, Baroness Ferstel, the Councillor, two Secretaries, and four Attachés of Embassy ##The French Ambassador, Baroness de Courcel[,] Madlle. de Courcel, Madame Geoffray, the Minister Plenipotentiary, five Secretaries, and three Attachés of Embassy ##The Italian Ambassador, Princess Ruspoli, three Secretaries, and three Attachés of Embassy ##The Spanish Ambassador, Countess de Casa Valencia; Mesdlles. de Alcala Galiano (2), Marquise de Guiria, Donna de Zea Bermudez, Countess de Morella, Donna de Ia Camara y Livermore, three Secretaries, and four Attachés of Embassy ##The Turkish Ambassador, Madame Antbopoulos, the Councillor, and two Secretaries of Embassy ##The United States Ambassador, Mrs. Hay, Miss Hay, Mrs. Henry White, Mrs. Carter, Mrs. Colwell, two Secretaries, one Attaché of Embassy, and the Private Secretary to the Ambassador ##The Argentine Minister, Madame Dominguez, Mesdlles. Dominguez (3), and the Secretary of Legation ##The Persian Minister, and one Secretary of Legation ##The Danish Minister, Madame de Bille, Madame Gosch, and the Secretary of Legation ##The Siamese Minister, Mrs. Verney, Miss Verney, Mrs. Loftus, the Councillor, the Secretary, the Attaché, and the Interpreter to the Legation ##The Liberian Minister ##The Roumanian Minister and the Councillor of the Legation ##The Netherlands Minister, Baroness de Goltstein d'Oldenaller, Baroness Schimmelpenninck van der Oye, and the Councillor of Legation ##The Belgian Minister, the Councillor, and two Secretaries of Legation ##The Mexican Minister, Madame Yturbe, Madame Romero, Madame Farias, Madame Garcia, two Secretaries and three Attachés of Legation ##The Japanese Minister, Madame Kato, two Secretaries, and three Atachés [sic] of Legation ##The Minister for Sweden and Norway, Countess Lewenhaupt, and the Attaché of Legation ##The Chinese Minister, Lady Macartney, the English Secretary, three Secretaries, and four Attachés of Legation ##The Portuguese Minister, Madlle. de Quilinan, three Secretaries, and one Attaché of Legation ##The Swiss Minister, Madame Bourcart, Madame de Salis, the Secretary, and the Attaché of Legation ##The Haytian Chargé d’Affaires ##The Chargé d’Affaires of Greece, Madame Metaxas, and the Attaché ##The Chargé d’Affaires of Chile and Madame Bascunan ##Two Secretaries and one Attaché of the Brazilian Legation ##Count E. van Rosen ##Mr. Hippolyte de Aranjo ##Vice-Admiral Montt ##Mr. Pinto, Mrs. Pinto ##Mr. and Mrs. Scaramanga ##Vicomte de Galard ##Dr. Arnold, and Madlle. von Rappoport ##Mrs. John Meiggs, Miss Meiggs ##Miss Margaret Butler ##Mrs. Henry Morgan ##Hon. Chauncey Depew ##Mr. and Mrs. James Taylor ##Mr. and Mrs. Charles Marshall ##Mr. and Mrs. Edmund Bayliss ##Mrs. Colgate ##Miss Furniss ##Miss Wells ##Miss Harris ##Hon. Levi P. Morton, Mrs. Morton, and the Misses Morton ##The Bishop of Illinois and Mrs. Leonard, Miss Leonard ##The Bishop of Albany and Mrs. Doane ##The Bishop of New York and Mrs. Potter ##the Bishop of Minnesota and Mrs. Whipple ##Mr. and Mrs. Walter Burns ##Mrs. Douglas Grant ##Miss Scott ##Mrs. Grace, Miss Margarita Grace ##Mrs. Wentworth ##Miss van Wart ##M. Valentin de Courcel ##Madame la Marquise de Talleyrand Perigord ##Comte Boson de Perigord ##Vicomte d'Espenilles ##Madame and Madlle. Thierry Delanoue ##Madlle. de la Cherè ##M. Cellerier ##M. and Madame Delawarre ##Madame Evelina Fenzi ##Count A. Zannini ##M. and Madame Jules Cottran ##Chevalier E. Mazzuechi ##Signor A. Tedeschi ##Signor A. Mariotti ##Captain Lucian von Ziegler ##Chevalier Lieutenant von Barry ##Baron Georg Rothschild ##Privy Councillor Count Berchtold ##Baron G. E. Levi, Baroness Levi ##Commander E. Philipson, Mrs. E. Philipson ##The Duke and Duchess of San Germano Calabritto ##The Marquis of San Vito ##Donna Lidia Serramezzana ##Donna Margherita Chigi ##Marchioness Vitelleschi ##Chevalier Elia ##Count de Franqueville ##Count Urbain Chevrau ##M. Marcel Fonquier ##M. Baudon de Mony, Madame Baudon de Mony ##Duchess de Rohan ##Marquis de Lastorgrie, Marchioness de Lastorgrie ##Count de Boisgelin, Countess L. de Boisgelin ##M. Stern, Madame Stern, Madlle. Stern ##Count Charles du Luart ##General de Saucy ##M. E. Seydoux ##Count Jean de Madre ##M. de Monbrison ##Baron de la Chevrelière ##Count de la Villestreux, Countess de la Villestreux ##Count Urbain de Maille, Countess Urbain de Maille ##General Faveret de Kerbrich ##Monsieur de la Haye Jousselin ##Baronne Faveret de Kerbrich ##Colonel Matton ##M. Ferinier Didet ##Madame Ferinier Didet ##Donna Isabella Colonna, Donna Victoria Colonna ##Pom-k-Soh ##Madame Reyntiens ##Marquis de Fuente Hermosa ##Herr Rudolf Swobody ##M. Lauritz Tuxen ##Duchesse de Baiten ##M. de Marcoarti ##Comte de Heeren, Madlle. de Heeren ##Monsieur M. de Mauny Talvande ##Senor Don Nicolas Campero ##Lieutenant Charny ##Lieutenant Sanders ##Madame and Madlle. de Mouni ##Comtesse de Montsoulmin #"Foreign Admirals and Commanding Officers and Staffs"<ref name=":1" /> (4, Col. 3a / Col. 3b) ##Austrian Admiral Baron von Spaun, Commander von Ziegler, Lieutenant Retter yon Barry, Lieutenant Mitchell, R.N. (attached) ##Danish Admiral H. H. Koch, Captain Waudel, Lieutenant Middelboc, Lieutenant Majendie, R.N. (attached) ##French Admiral C. F. E. De Courthille, Captain Germinet, Commander Poidlone, Lieutenant Perdriel, Sub-Lieutenant de Caqueray, Lieutenant Phillimore, R.N. (attached) ##Italian Admiral C. E. Morin, Commander Count Prasca, Lieutenant Lunghetti, Lieutenant Count Morano, Lieutenant Henderson. R.N. (attached) ##German Admiral his Royal Highness Prince Henry of Prussia, Captain Muller, Lieutenant von Spee, Sub-Lieutenant Wittman, Lieutenant Garforth, R.N. (attached) ##Japanese Admiral H.I.H. Prince Arizugawa, Captain Miura, Commander Tsuda, Lieutenant Stewart, R.N. (attached) ##Netherlands Admiral F. K. Englebrecht, Captain de Groot, Lieutenant Baron von Hardenbrock, Lieutenant Woolcombe, R.N. (attached) ##Norwegian Rear-Admiral von Krogh, Captain Muller, Lieutenant Petersen, Lieutenant Kerr Pearse, R.N. (attached) ##Portuguese Captain Barreto de Vascomellos, Captain de Cartillo, Lieutenant Trye, R.N. (attached) ##Russian Admiral Nicholas Skrydloff, Captain Domojiroff, Lieutenant Stetsenkoff, Lieutenant Twisleton Wykeham Fiennes, R.N. (attached) ##Spanish Admiral Don Segismundo Bermijo y Merelo, Captain Don Antonio Eulate y Fery, Lieutenant Don Juan Romero, Lieutenant Don Antonio Romero, Lieutenant Fair, R.N. (attached) ##Swedish Admiral A. F. H. Klintberg, Captain Ingelman, Commander Flack, Lieutenant Alton, R.N. (attached) ##United States Admiral J. N. Miller, Lieutenaut Richmond (attached) ##Captain de Mar E. Guerra ##Captain R. S. D. Cumins #The Lord Lieutenant of Ireland and Countess Cadogan #The Right Hon. the Speaker and Mrs. Gully, Miss Gully, and Miss Shelly Gully #Cardinal Vaughan #Right Hon. the Lord Mayor and Lady Mayoress, and Misses Faudel Phillips (2) #The Gold Stick in Waiting, Silver Stick in Waiting, Silver Stick Adjutant in Waiting #Officer Commanding 1st Life Guards and five officers #Officer Commanding 2nd Life Guards and four officers #Officer Commanding Royal Horse Guards and four officers #Officer Commanding 2nd Dragoons and three officers #Field Officer in Brigade Waiting, Adjutant in Brigade Waiting #Commanding Officer Grenadier Guards #Commanding Officer Coldstream Guards #Commanding Officer Scots Guards, a Regimental Adjutant #Commanding Officer 1st, 2nd, and 3rd Battalions Grenadier Guards and three officers of each Battalion #Commanding Officer 1st and 2nd Battalions Coldstream Guards and three officers of each Battalion #Commanding Officer 1st and 2nd Battalions of Scots Guards and three officers of each Battalion #Commanding Officer Woolwich District and six officers #Commanding Officer R.H.A. Home District and two officers #Commanding Officer R.E. and four officers #Commanding Officer 2nd Battalion Lincolnshire Regiment and three officers #Commanding Officer Royal Marines (Chatham) and four officers #Commanding Officer Royal Marines (Portsmouth) and two officers #Four officers of the Honourable Corps of the Gentlemen at Arms #Archbishops — Canterbury, York, Armagh, Ontario, Rupertsland #Dukes and Duchesses ##The Duke and Duchess of [[Social Victorians/People/Argyll|Argyll]] ##The Duke and Duchess of [[Social Victorians/People/Abercorn|Abercorn]] ##The Duchess of De Baileu ##The Duke and Duchess of [[Social Victorians/People/Buccleuch|Buccleuch]] ##The Duchess of [[Social Victorians/People/Cleveland|Cleveland]] ##The Duke and Duchess of [[Social Victorians/People/Devonshire|Devonshire]] ##The Duchess of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Hamilton]] ##The Duke and Duchess of [[Social Victorians/People/Leeds|Leeds]] ##The Duke and Duchess of [[Social Victorians/People/Marlborough|Marlborough]] ##The Duke and Duchess of [[Social Victorians/People/Manchester|Manchester]] ##The Duke and Duchess of [[Social Victorians/People/Montrose|Montrose]] ##The Duke and Duchess of [[Social Victorians/People/Newcastle|Newcastle]] ##The Duke of [[Social Victorians/People/Norfolk|Norfolk]] ##The Duke of [[Social Victorians/People/Northumberland|Northumberland]] ##The Duke and Duchess of [[Social Victorians/People/Portland|Portland]] ##The Duke of [[Social Victorians/People/Richmond and Gordon|Richmond and Gordon]] ##The Duke and Duchess of [[Social Victorians/People/Roxburghe|Roxburghe]] ##The Duke and Duchess of [[Social Victorians/People/Somerset|Somerset]] ##The Duke and Duchess of [[Social Victorians/People/Sutherland|Sutherland]] ##The Duke and Duchess of St. Albans ##The Duke and Duchess of Wellington ##The Duchess of [[Social Victorians/People/Westminster|Westminster]] #Marquises and Marchionesses ##The Marquis of Abergavenny ##The Marchioness of Ailesbury ##The Marquis and Marchioness of Ailsa ##The Marquis of Anglesey ##The Marquis and Marchioness of [[Social Victorians/People/Breadalbane|Breadalbane]] ##The Marchioness of [[Social Victorians/People/Marlborough#Marchioness of Blandford|Blandford]] ##The Marquis and Marchioness of Bristol ##The Marquis of [[Social Victorians/People/Camden|Camden]] ##The Marquis and Marchioness of Conyngham ##Dowager [Marchioness of] Conyngham ##The Marchioness of Cassar de Sai[n] ##The Marquis and Marchioness of Cholmondeley ##The Marquis of D'Auerstadt ##The Marquis and Marchioness [[Social Victorians/People/Stonor|D'Hautpoul]] ##The Marquis and Marchioness of Downshire ##Dowager [Marchioness of] Downshire ##The Marquis and Marchioness of [[Social Victorians/People/Hamilton Temple Blackwood|Dufferin and Ava]] ##The Marquis and Marchioness of [[Social Victorians/People/Exeter|Exeter]] ##The Marquis and Marchioness of Granby ##The Marchioness of [[Social Victorians/People/Florence Rawdon-Hastings Chetwynd|Hastings]] ##The Marquis and Marchioness of [[Social Victorians/People/Bective|Headfort]] ##The Marquis and Marchioness of Hertford ##The Marquis and Marchioness of Huntly ##The Marquis and Marchioness of [[Social Victorians/People/Abercorn#James Hamilton, Marquess of Hamilton|Hamilton]] ##The Marquis and Marchioness of [[Social Victorians/People/Lansdowne|Lansdowne]] ##The Marquis and Marchioness of Lothian ##Dowager (Marchioness of) [[Social Victorians/People/Londonderry|Londonderry]] ##The Marquis and Marchioness of [[Social Victorians/People/Londonderry|Londonderry]] ##The Marquis and Marchioness of [[Social Victorians/People/Ormonde|Ormonde]] ##The Marchioness of [[Social Victorians/People/Queensberry|Queensberry]] ##The Marquis and Marchioness of [[Social Victorians/People/Ripon|Ripon]] ##The Marquis and Marchioness of [[Social Victorians/People/Salisbury|Salisbury]] ##The Marquis and Marchioness of [[Social Victorians/People/Tweeddale|Tweeddale]] ##Dowager (Marchioness of) [[Social Victorians/People/Tweeddale|Tweeddale]] ##John Stewart-Murray, [[Social Victorians/People/Atholl|Marquess of Tullibardine]] ##Lawrence, [[Social Victorians/People/Zetland|Marquess of Zetland]] and Lilian, [[Social Victorians/People/Zetland|Marchioness of Zetland]] #Earls and Countesses ##Countess of Aberdeen and Dowager Countess of Aberdeen ##Earl and Countess of Albemarle and Dowager Countess of Albemarle ##Earl and Countess of Ancaster ##Earl and Countess of Amherst ##Earl of Ava ##Earl and Countess of Antrim ##Earl and Countess of Aylesford ##Earl and Countess of Annesley ##Earl and Countess of Airlie ##Earl and Countess of Arran ##Earl of Aberdeen ##Earl and Countess of Bandon ##Countess of Bantry ##Earl and Countess of Beauchamp ##Earl and Countess of Bathurst and Dowager Countess of Bathurst ##Countess of Bective ##Earl and Countess of Belmore ##Earl of Bradford ##Countess of Bremer ##Earl and Countess of Brownlow ##Earl and Countess of Buckinghamshire ##Earl of Burford ##Earl and Countess of Cairns ##Earl and Countess of Caledon ##Earl of Camperdown ##Earl of Cardigan ##Earl and Countess of Carnarvon and Dowager Countess of Carnarvon ##Earl of Carnwath ##Earl and Countess of Carrington ##Earl and Countess of Carysfort ##Earl and Countess of Castlestuart ##Earl and Countess of Cathcart ##Earl and Countess of Cavan ##Earl and Countess of Chesterfield ##Earl and Countess of Chichester ##Dowager Countess of Clancarty ##Countess of Clanwilliam ##Earl and Countess of Compton ##Countess of Cottenham ##Earl of Courtown ##Earl and Countess of Cowper ##Earl and Countess of Cranbrook ##Earl and Countess of Craven and Dowager Countess of Craven ##Earl and Countess of Crawford ##Earl of Crewe ##Earl and Countess of Cork and Orrery ##Earl and Countess of Coventry ##Countess of Cromartie and Dowager Countess of Cromartie ##Earl and Countess of Dalkeith ##Earl and Countess of Dartmouth ##Earl and Countess of De Grey ##Dowager Countess of De La Warr ##Earl and Countess of Denbigh ##Earl and Countess of Derby ##Earl and Countess of Donoughmore ##Earl and Countess of Drogheda ##Earl of Ducie ##Earl and Countess of Dudley and Dowager Countess of Dudley ##Earl and Countess of Dundonald ##Earl and Countess of Dunmore ##Earl and Countess of Dunraven ##Earl of Durham ##Earl and Countess of Eglinton and Winton ##Earl of Eldon ##Earl and Countess of Ellesinere ##Earl and Countess of Enniskillen ##Earl and Countess of Erne ##Earl and Countess of Errol ##Earl and Countess of Essex and Dowager Countess of Erroll ##Earl of Euston ##Earl and Countess of Feversham ##Earl and Countess of Fingall ##Earl of Fortescue ##Earl and Countess of Gainsborough ##Earl and Countess of Galloway ##Earl and Countess of Glasgow ##Countess of Gosford ##Earl and Countess of Granard ##Countess of Granville ##Earl and Countess of Grey ##Countess of Grosvenor ##Countess of Guilford ##Earl and Countess of Harewood and Dowager Countess of Harewood ##Earl and Countess of Harrington ##Earl and Countess of Hopetoun ##Earl and Countess of Huntingdon ##Earl and Countess of Harrowby ##Countess of Hohenau ##Countess of Howe ##Earl and Countess of Iddesleigh ##Earl and Countess of Jersey ##Earl and Countess of Kenmare ##Earl of Kerry ##Earl and Countess of Kilmorey ##Earl of Kimberley ##Earl and Countess of Kingston ##Earl of Kinnoull ##Josephine, Countess Kinsky ##Earl and Countess of Kintore ##Countess of Leitrim ##Earl and Countess of Lanesborough ##Countess of Lathom ##Earl and Countess of Lauderdale ##Countess of Leicester ##Earl and Countess of Leven and Melville ##Earl and Countess of Lichfield ##Earl and Countess of Limerick ##Earl and Countess of Lindsay ##Earl and Countess of Lisburne ##Earl and Countess of Listowel ##Earl and Countess of Londesborough ##Earl and Countess of Longford ##Earl and Countess of Lonsdale and Dowager Countess of Lonsdale ##Earl and Countess of Loudoun ##Earl and Countess of Lovelace ##Earl and Countess of Lucan ##Countess of Lytton ##Countess of Macclesfield ##Earl and Countess of Malmesbury and Dowager Countess of Malmesbury ##Earl and Countess of Mar ##Earl and Countess of Mar and Kellie and Dowager Countess of Mar and Kellie ##Earl and Countess of Mayo and Dowager Countess of Mayo ##Countess of Meath ##Countess of Metaxas ##Earl and Countess of Mexborough ##Earl and Countess of Minto ##Earl of De Montalt ##Earl and Countess of Morley ##Earl and Countess of Morton and Dowager Countess of Morton ##Earl of Nelson ##Earl and Countess of Norbury ##Earl of Northbrook ##Earl and Countess of Northesk and Dowager Countess of Northesk ##Earl and Countess of Onslow ##Earl of Orford ##Countess of Oxford ##Earl and Countess of Pembroke ##Countess of Percy ##Earl and Countess of Portarlington ##Earl and Countess of Portsmouth ##Earl and Countess of Powis ##Earl and Countess of Radnor ##Earl and Countess of Ravensworth ##Earl and Countess of Roden ##Earl and Countess of Romney ##Lawrence, [[Social Victorians/People/Zetland|Earl of Ronaldshay]] ##Earl of Rosebery ##Earl and Countess of Rosse ##Earl and Countess of Rosslyn and Dowager Countess of Rosslyn ##Earl of Sandwich ##Earl of Scarbrough ##Earl and Countess of Selborne ##Countess of Selkirk ##Countess of Shaftesbury ##Dowager Countess of Shrewsbury and Talbot ##Earl and Countess of Spencer ##Earl and Countess of Stamford ##Earl and Countess of Stanhope ##Earl and Countess of St. Germans ##Earl of Stradbroke ##Earl of Strafford ##Earl and Countess of Suffolk and Berkshire ##Earl and Countess of Temple (of Stowe) ##Earl and Countess of Verulam ##Earl and Countess of Waldegrave ##Earl and Countess of Warwick ##Earl and Countess of Westmeath ##Earl and Countess of Wharncliffe ##Elizabeth, Dowager Countess of Wilton and Isabella, Dowager Countess of Wilton ##Earl and Countess of Winchilsea and Nottingham ##Earl and Countess of Winterton ##Earl and Countess of Yarborough and Dowager Countess of Yarborough #Viscounts<ref name=":1" /> (4, Col. 3c / Col. 4a) and Viscountesses ##Viscount and Viscountess of Boyne ##Viscountess of Cantelupe ##Viscount and Viscountess of Castlerosse ##Viscount and Viscountess of Chelsea ##Viscount and Viscountess of Chetwynd ##Viscountess of Chewton ##Viscount and Viscountess of Clifden ##Viscount and Viscountess of Cobham ##Viscount and Viscountess of Coke ##Viscount of Corry ##Viscount and Viscountess of Cranborne ##Viscount of Crichton ##Viscount and Viscountess of Cross ##Viscount of Curzon ##Viscount and Viscountess of Dalrymple ##Viscount and Viscountess of Deerhurst ##Viscount and Viscountess of De Vesci ##Viscount and Viscountess of Dillon ##Viscount of Doneraile ##Viscount and Viscountess of Duncannon ##Viscount of Dungarvan ##Viscount and Viscountess of Ebrington ##Viscount and Viscountess of Emlyn ##Viscount of Encombe ##Viscount and Viscountess of Exmouth ##Viscount and Viscountess of Falkland ##Viscount and Viscountess of Falmouth ##Viscount of Fitz Harris ##Viscount and Viscountess of Folkestone ##Viscount and Viscountess of Frankfort de Montmorency ##Viscount and Viscountess of Gage ##Viscount and Viscountess of Galway ##Viscount and Viscountess of Garnock ##Viscount and Viscountess of Gough ##Viscount of Gort ##Viscount and Viscountess of Halifax ##Viscount and Viscountess of Hardinge ##Viscount of Harrington ##Viscount and Viscountess of Hood ##Viscount and Viscountess of Kilcoursie ##Viscount and Viscountess of Knutsford ##Viscount and Viscountess of Lifford ##Viscount of Llandaff ##Viscount and Viscountess of Maitland ##Viscount and Viscountess of Marsham ##Viscount and Viscountess of Massereene and Ferrard ##Viscount and Viscountess of Melville ##Viscount and Viscountess of Midleton ##Viscount and Viscountess of Milton ##Viscount and Viscountess of Monck ##Viscount and Viscountess of Morpeth ##Dowager Viscountess of Mountmorres ##Viscount and Viscountess of Newark ##Viscount and Viscountess of Newport ##Viscount and Viscountess of Oxenbridge ##Viscount of Parker ##Viscount of Peel ##Viscount and Viscountess of Portman ##Viscount and Viscountess of Powerscourt ##Viscount and Viscountess of Raincliffe ##Viscountess of Sherbrooke ##Viscount of Sidmouth ##Viscount of St. Cyres ##Viscount of Southwell ##Viscount of Suirdale ##Viscount and Viscountess of Templetown ##Viscountess of Torrington ##Viscount and Viscountess of Trafalgar ##Viscount and Viscountess of Valentia ##Viscount of Valletort ##Viscount of Villiers ##Viscountess of Wolseley #Bishops — Auckland, Barry, Bath and Wells, British Colombia, Chichester, Durham, Ely, Exeter, Gloucester and Bristol, Gibraltar, Hereford, London, Lichfield, Lincoln, Manchester, Newcastle, Norwich, Oxford, Peterborough, Rochester, Ripon, Stepney, Southwark, St. Albans, Salisbury, Sodor and Man, Southwell, Sydney, Sierra Leone, Worcester, Winchester, Wellington #Baronesses — Burdett-Coutts, Macdonald #Lords and Ladies<ref name=":1" /> (4, Col. 4b / Col. 5a) — ##Lord and Lady Abercromby ##Lord and Lady Aberdare ##Lord Aberdour ##Lady Abinger ##Lady Alexandra Acheson ##Lady Adam ##Lady Adderley ##Lord and Lady Addington ##Lady Adye ##Lady Agnew ##Lady Alderson ##Lord and Lady Alington ##Lady Alison ##Lady Mildred Allsopp ##Lord and Lady Amherst of Hackney ##Lady Heathcoat Amory ##Lord and Lady Ampthill ##Lady Agnes Anderson ##Lady Bertha Anson ##Lady Arbuthnot ##Lady Alice Archer Houblon ##Lord Ardee ##Lord and Lady Ardilaun ##Lady Armstrong ##Lady Arnold ##Lady Arnott ##Lord and Lady Ashbourne ##Lord and Lady Ashburton and Dowager Ashburton ##Lord and Lady Ashcombe ##Lady Alice Ashley ##Lady Edith Ashley ##Lady Ashmead-Bartlett ##Lord and Lady Ashton ##Lord and Lady Ashtown ##Lady Florence Astley ##Lady Gertrude Astley-Corbett ##Lady Austin ##Lord Bagot ##Lady Bailey ##Lady Blanche Baillie ##Lady Baird ##Lady Baker ##Lord Balcarres ##Lord and Lady Balfour of Burleigh, Lady Nina Balfour and Lady Betty Balfour ##Lord Balvaird ##Lord Bangor ##Dowager Lady Barclay ##Lord and Lady Barnard ##Lady Florence Barnardiston ##Lady Constance Barne ##Lady Barran ##Lady Barrington ##Lord and Lady Basing ##Lord and Lady Bateman ##Lady Evelyn Bathurst ##Lord and Lady Battersea ##Lady Steuart Bayley ##Lady Violet Beauchamp ##Lord Osborne Beauclerk and Lady Beauclerk (2) ##Lady A. Beaumont ##Lady Bedford ##Lord and Lady Belhaven and Stenton and Dowager Belhaven and Stenton ##Lord and Lady Bellew and Dowager Bellew ##Lord and Lady Belper ##Lady Charles Beresford ##Lady William Beresford (Lilian Duchess of Marlborough) ##Lady Bergne ##Lord and Lady Bertie and Lady Elizabeth Bertie ##Lady Biddulph, Lady Elizabeth Biddulph and Lady Wilfreda Biddulph ##Lady Bigge ##Lord and Lady Bingham ##Lord and Lady Binning ##Lord Blackwood, Lord Basil Blackwood. Lady Hermione Blackwood and Lord Terence Blackwood ##Lady Bloomfield ##Lady Blythswood ##Lord and Lady Bolton ##Lady Maud Bootle-Wilbraham, Lady Bertha Bootle-Wilbraham and Lady Edith Bootle-Wilbraham ##Lord Borthwick ##Lady Margaret Boscawen ##Lord and Lady Boston ##Lady Boughey ##Lady Albreda Bourke and Lady Florence Bourke ##Lady Bowen ##Lady Bower ##Lady Muriel Boyle and Lady Boyle (2) ##Lady Mary Brabazon ##Lady Brackenbury ##Lady Braddon ##Lady Bramwell ##Lady Bramston ##Lord Brassey, Lady Idina Brassey and Lady Violet Brassey ##Lord and Lady Braye ##Lady Mary Bridgeman ##Lady Eleanor Brodie ##Lady Hilda Brodrick ##Lady De Capel Brooke and Dowager Brooke ##Lady Cunliffe Brooks ##Lord and Lady Brougham and Vaux ##Lord and Lady Ulick Browne, Lady Browne and Lady Crichton Browne ##Lady Brownlow ##Lord and Lady F. Brudenell-Bruce ##Lady Brunner ##Dowager Buchanan-Riddeil ##Lady Audrey Buller ##Lady Burdett ##Lord and Lady Burghclere ##Lord Burghley ##Lady Agnes Burne ##Lady Burrell ##Lord and Lady Burton ##Lady Butler and Lady Butler (2) ##Lord and Lady Arthur Butter ##Lady Buxton and Lady Victoria Buxton ##Lady Susan Byng ##Lord and Calthorpe ##Lady C. Cameron and Lady Margaret Cameron ##Lord and Lady Archibald Campbell and Lady A. Campbell ##Lord and Lady George Campbell ##Lady Campbell-Bannerman ##Lord and Lady Camoys ##Lord and Lady Carbery and Dowager Carbery ##Lady Carbutt ##Lady Cardon ##Lord and Lady Cardross ##Lord and Lady Carew ##Lady Carmichael ##Lord and Lady Carnegie ##Lord and Lady Castlemaine ##Lord and Lady Castletown ##Lady Eva Cathcart and Lady R. Cathcart ##Lady Frederick Cavendish, Lady Myra Cavendish, Lady Evelyn Cavendish and Lady Harriet Cavendish ##Lord Charles Cavendish-Bentinck, Lord and Lady Henry Cavendish-Bentinck, Lord William Cavendish-Bentinck, Lady Ottoline Cavendish-Bentinck ##Lord and Eustace Cecil, Lord Hugh Cecil, Lord and John Cecil, Lord and Edward Cecil, Lord and Lady Robert Cecil, Lord W. Cecil, Lady Gwendolen Cecil, Lady Florence Cecil, Lady William Cecil, Lady Louisa Cecil ##Lady Francis Cecil-Dallas ##Lady Chamberlain ##Lady Chelmsford ##Lord and Lady Chesham ##Lady Chetwode ##Lord Cheylesmore ##Lord and Lady Fitzwarine Chichester ##Lady Chitty ##Lady Cholmeley ##Lady Henry Cholmondeley ##Lady Clements (2) ##Lady Churchill, Lady Randolph Churchill, Dowager Churchill, Lady Spencer Churchill (2) ##Lord Edward Spencer-Churchill, Lady Alfred Spencer-Churchill ##Lord and Lady Churston ##Lord and Lady Clifford of Chudleigh ##Lady Marshal Clarke, Lady E. Clarke ##Lady Isabel Clayton ##Lord and Lady Clinton ##Lord and Lady Clonbrock ##Lord Cloncurry ##Lady Muriel Close ##Lady Evelyn Cobbold ##Lady Cochrane, Lady Gertrude Cochrane, Lady Adela Cochrane ##Lady Coddington ##Lady Mabel Coke ##Lord and Lady Colchester ##Lady Cole (2) ##Lady Colebrooke ##Lord and Lady Coleridge ##Lady Collins ##Lady Colomb ##Lady Colvile, Lady Colville ##Lord and Lady Colville of Culross ##Lady Jane Seymour Combe, Lady Constance Combe ##Lady Commerell ##Lord and Lady Alwyne Compton ##Lady Dowager Congleton ##Lord and Lady Connemara ##Lady Conyers ##Lady Blanche Conyngham ##Lady Cooper ##Lady Evelyn Cotterell ##Lord and Lady Cottesloe ##Lady Couch ##Lord and Lady Courtenay ##Lady Coventry (2) ##Lady Cowell ##Lady Helen Craven ##Lord and Lady Crawshaw ##Lady Evelyn Crichton, Lady Emma Crichton ##Lord Crofton ##Lady Cromer ##Lady Mary Crosse ##Lady Crossley ##Lady Mary Cuffe ##Lady Culme-Seymour ##Lady Cunliffe ##Lady Georgiana Curzon ##Lady Elizabeth Cust ##Lady Ida Dalzell ##Lady Mary Dashwood ##Lord and Lady Davey ##Lady Victoria Dawnay, Lady Evelyn Dawnay, Lady Adelaide Dawnay ##Lady Decies ##Lord and Lady De Freyne ##Lord and Lady De L’Isle and Dudley ##Lord De Manley ##Lady Mildred Denison, Lady Elinor Denison ##Lord Deramore ##Lord and Lady De Ramsey ##Lady Dering ##Lady De Ross ##Lord and Lady De Saumarez ##Lady Des Voeux ##Lady De Trafford, Lady Agnes De Trafford ##Lady De Winton ##Lord and Lady Digby ##Lady Dorchester ##Lady Dorington ##Lady Margaret Douglas, Lady Edith Douglas ##Lady H. Douglas-Hamilton ##Lady Dowell ##Lady Drummond, Lady Edith Drummond ##Lady Du Cane ##Lady Duckworth ##Lady Eva Dugdale ##Lord Dunally ##Lady Florence Duncombe, Lady Ulrica Duncombe, Lady Caroline Duncombe ##Lady Alice Dundas ##Lord and Lady Dunleath ##Lord Dunglass ##Lady Dunn ##Lord Dunsandle and Clanconal ##Lady Durand ##Lord Dynevor ##Lord Ebury ##Lady Edmonstone ##Lady Edwards, Lady J. B. Edwards, Lady Blanche Edwards ##Lady Ernestine Edgcumbe ##Lady Egerton (2) ##Lord Egerton of Tatton ##Lady Grey-Egerton ##Lord and Lady Elcho ##Lord and Lady Elibank ##Lady Ellenborough ##Lady Ellis ##Lord and Lady Elphinstone ##Lady Winifred Cary-Elwes ##Lady Engleheart ##Lord Erskine, Lady Erskine (2), Lady Horatia Erskine, Lady Erskine ##Lord and Lady Esher ##Lady Evans ##Lady Evelyn Ewart, Lady Mary Ewart ##Lady Evelyn Eyre ##Lady Fairbairn ##Lady Fairfax ##Lady Anne Fane, Lady Augusta Fane ##Lady Farquhar ##Lord and Lady Farrer ##Lady Fayrer ##Lady Louisa Feilding ##Lady Helen Munro Ferguson ##Lady Fergusson ##Lady Ffolkes ##Lady Finlay ##Lady Fisher ##Lady Dorothea Fitz-Clarence, Lady Maria Fitz-Clarence, Lady Dorothy Fitzclarence ##Lord and Lady Henry Fitz-Gerald, Lady B. Fitz Gerald, Lady M. FitzGerald, Lord Seymour Fitz-Gerald ##Lady Beatrix Fitzmaurice ##Lord and Lady F. FitzRoy, Lady C. Fitz-Roy ##Lady Mary Fitzwilliam ##Lady FitzWygram ##Lady Fletcher ##Lady Flower, Lady Flower ##Lord Foley, Lady Mary Foley ##Lady Gertrude Foljambe ##Lady Angela Forbes, Lady Forbes (2), Dowager Helen Forbes ##Lord and Lady Forester ##Lady Forrest ##Lady Susan Fortescue ##Lady Forwood ##Lady Foster ##Lady Fowler ##Lady Edith Franklin ##Lady Fremantle, Lady Fremantle ##Lady Frere ##Lady Fulton ##Lady Gardiner, Lady Lynedoch Gardiner ##Lord Garioch ##Lady Galton ##Lady Katharine Gathorne-Hardy ##Lady Garvagh ##Lord and Lady Gerard ##Lady Gilbey ##Lady Gillford ##Lady Susan Gilmour ##Lady Gipps ##Lord and Lady Glamis ##Lord and Lady Glenesk ##Lady Glyn, Lady Mary Carr Glyn ##Lady D'Arcy Godolphin-Osborne ##Lady Gordon ##Lady Margaret Ormsby Gore, Lady Constance Gore ##Lady Gore Langton (2) ##Lord Walter Gordon-Lennox, Lord Algernon Gordon-Lennox ##Lady Evelyn Goschen ##Lord R. S. Gower ##Lady Graham, Lady Margaret Graham, Lady Helen Graham ##Lady Charlotte Graham-Toler ##Lady Grant, Lady Florence Grant ##Lady Grant-Duff ##Lady Green ##Lord Greenock ##Lady Grenfell ##Lady Frances Gresley ##Lady Victoria Grey, Lady Grey ##Lady Jane Grey-Trefusis ##Lady Griffin ##Lady Helen Grimston ##Lord and Lady Arthur Grosvenor, Lady Grosvenor (2) ##Lady Gull ##Lady Haldon ##Lady Haliburton ##Lady Basil Hall ##Lady Halle ##Lord and Lady Halsbury ##Lord and Lady E. Hamilton, Lord F. Hamilton, Lady F. Douglas Hamilton, Lady Alexandra Hamilton, Lady Baillie Hamilton (2), Lady C. Hamilton, Lady Victoria Hamilton, Lady George Hamilton ##Lady Hanson ##Lady Harcourt ##Lady Cicely Hardy, Lady Hardy ##Lady Beatrice Hare ##Lord Harlech ##Lady Constance Harris, Lady Harris ##Lady Harrison, Lady Harriet Harrison ##Lady Hart ##Lady Emily Hart-Dyke ##Lady Dixon-Hartland ##Lady Hartopp ##Lord and Lady Hastings ##Lord and Lady Hatherton ##Lady Alice Havelock-Allan ##Lady Hawke ##Lord and Lady Hawkesbury ##Lady John Hay, Lady Hay ##Lady Blanche Haygarth ##Lady Hayter ##Lady Hely-Hutchinson (2) ##Lady Hemming ##Lord and Lady Heneage ##Lord and Lady Henley ##Lord Henniker ##Lady Beatrix Herbert, Lady Herbert (2) ##Lord and Lady Herries ##Lord and Lady Herschell ##Lord Francis Hervey, Lady Augustus Hervey ##Lady Hervey-Bathurst ##Lady Fermor Hesketh ##Lady Hibbert ##Lady Lucy Hicks-Beach ##Lord and Lady Arthur Hill, Lady Clement Hill, Lady Stock Hill ##Lord and Lady Hillingdon ##Lord and Lady Hindlip ##Lord and Lady Hobhouse ##Lady Norah Hodgson ##Lady Holdich ##Lady Mary Holland ##Lady Beatrix Douglas Home ##Lady Maria Hood ##Lady Hood of Avalon ##Lady Hooker ##Lady Mary Hope ##Lady Hoskins ##Lord and Lady Hotham ##Lord and Lady Hothfield ##Lady Houldsworth ##Lady Eleanor Howard, Lady Agnes Howard, Lady Howard (2), Lady Mabel Howard, Lady Rachel Howard ##Lord and Lady Howard of Glossop ##Lady Howarth ##Lady Mary Hozier ##Lady Florentia Hughes ##Lady Seager Hunt ##Lady Hunter ##Lord Hyde ##Lady Hylton ##Lord and Lady Inchiquin ##Lord Inverurie ##Lord and Lady Iveagh ##Lady Jackson ##Lord James of Hereford ##Lady Margaret Jenkins, Lady Jenkins ##Lady Jenner ##Lady Jephson ##Dowager Jessel, Lady Jessell ##Lady Jeune ##Lady Hill Johnes ##Lady Joicey ##Lady Alice Jolliffe ##Lady Burn Jones ##Lady Caroline Lister Kaye, Lady Beatrice Lister Kaye, Lady Lister Kaye ##Lady Isabella Keane ##Lady Keith-Falconer (2) ##Lord and Lady Kelvin ##Lady Kemball ##Lady Beatrice Kemp ##Lady Kennard ##Lady Kennaway ##Lady Aline Kennedy ##Lady Kennett-Barrington ##Lord Kenyon ##Lady Mabel Kenyon-Slaney ##Lord Kensington ##Lady Mary Stuart Keppel ##Lady Innes-Ker (2) ##Lady Kerr (2) ##Lord Kilmarnock ##Lady King ##Lady Florence King King ##Lady Emily Kingscote ##Lady Edith King-Tenison ##Lord and Lady Kinnaird ##Lady Kitson ##Lady Laking ##Lady Frances Lambart, Lady Ellen Lambart ##Lady Victoria Lambton ##Lady Adela Larking ##Lady Isabel Larnach ##Lady Mary Lascelles ##Lord and Lady Lawrence ##Lady Lawson ##Lord and Lady Leconfield ##Lady Elliott Lees, Lady Lees ##Lady Leese ##Lady Legard ##Lord and Lady Leigh ##Lady Henry Gordon-Lennox, Lady Walter Gordon-Lennox, Lady Algernon Gordon-Lennox, Lady Caroline Gordon-Lennox ##Lady Katharine Le Poer Trench ##Lady Constance Leslie ##Lady Susan Leslie-Melville ##Lady Lewis ##Lady Lilian Liddell ##Lady Lindley ##Lady Harriet Lindsay, Lady Jane Lindsay, Lady Jane Lindsay ##Lord and Lady Lingen ##Lord and Lady Lister ##Lady Gwendolen Little ##Lady Margaret Littleton ##Lord and Lady Llangattock ##Lady Llewelyn ##Lord and Lady Loch ##Lady Lockwood ##Lady Louise Loder ##Lady Catherine Loftus ##Lady Doreen Long ##Lady Longley ##Lady Albertha Lopes ##Lady Loraine ##Lord and Lady Lovat ##Lady Drury Lowe, Lady Lucy Drury Lowe ##Lady Lowry-Corry (2) ##Lady Mary Loyd ##Lady Lubbock ##Lord and Lady Lurgan and Dowager Lurgan ##Lady Lyall ##Lady Lyell ##Lady Mary Lygon ##Lady Lyons ##Lady Lysons ##Lady Lyttelton ##Lady Emily Lytton ##Lady MacCormac ##Lord and Lady Macdonald ##Lady Macgregor, Lady MacGregor, Lady Helen MacGregor ##Lady Mackenzie, Lady Mackenzie ##Lady Mackworth ##Lady Maclean ##Lord and Lady Macnaghten ##Lady Macpherson-Grant ##Lady Caroline Madden, Lady Madden ##Lady Louisa Magenis ##Lady Magheramorne, Dowager Magheramorne ##Lady Nora Maitland ##Lady Margaret Crichton-Maitland ##Lady Margaret Majendie ##Lord Cecil Manners, Lord Edward Manners, Lord Manners, Lady Victoria Manners, Lady Manners ##Lady Blundell Maple ##Lady Mappin ##Lady Marjoribanks ##Lady Markham ##Lady Marriott ##Lady Martin, Lady Martin ##Lady Evelyn Mason ##Lady Maude (2) ##Lady H. Maxwell, Lady Maxwell, Lady Maxwell, Lady Maxwell ##Lady Heron-Maxwell ##Lady M'Clintock ##Lady Evelyn M'Donnell ##Lady Meade (2) ##Lord and Lady Medway ##Lady Methuen ##Lady Meysey-Thompson ##Lord and Lady Middleton, Lady Middleton ##Lady Mary Milbanke ##Lady Miller ##Lady Milner ##Lady Clementina Mitford ##Lady Lady M'lver ##Lady Hilda M'Neile ##Lady Monckton ##Lord Moncreiff, Lady Scott Moncrieff ##Lady Moncreiffe ##Lord and Lady Monkswell ##Lady Monson ##Lord Charles Montagu, Lady Cecil Scott Montagu, Lady S. Montagu, Lady Agneta Montagu ##Lord Montagu of Beaulieu ##Lord and Lady Monteagle ##Lady Edith Montgomerie, Lady Sophia Montgomerie ##Lady Charlotte Montgomery ##Lady More-Molyneux ##Lord and Lady Moreton ##Lady Morgan ##Lord and Lady Morris ##Lady Blanche Morris ##Lady Mary Morrison ##Lady Moseley ##Lord and Lady Mostyn ##Lord and Lady Mowbray and Stourton, Dowager Mowbray and Stourton, Lady Mowbray ##Lord and Lady Muncaster ##Lady Anne Murray ##Lady Murray (2) ##Lady Georgiana Mure, Lady Georgiana Mure [sic] ##Lord and Lady Napier and Ettrick ##Lord and Lady Napier of Magdala and Dowager Napier of Magdala ##Lady Naylor-Leyland ##Lady Nelson ##Lord and Lady Henry Nevill ##Lord and Lady Newton ##Lord and Lady Newtown-Butler ##Lady Nicolson ##Lady Augusta Noel, Lady Agnes Noel ##Lady Norman ##Lord and Lady Norreys ##Lord and Lady North, Lady Muriel North ##Lady Northcote, Lady Northcote (2) ##Lord Norton ##Lady Elizabeth Nugent ##Lady O'Brien, Lady O'Brien [sic] ##Lady O'Hagan ##Lady Olpherts ##Lord and Lady O'Neill ##Lady Gwendoline O'Shee ##Princep [sic] Alice Packe ##Lord and Lady Berkeley Paget ##Lady Alfred Paget ##Lady Paget of Cranmore ##Lady Katherine Pakenham ##Lady Palgrave ##Lady Sophia Palmer, Lady Palmer ##Lady Evelyn Parker ##Lady Parratt ##Lady Maude Parry ##Lady Muriel Parsons ##Lord and Lady Pearson, Lady Pearson ##Lady Peel, Lady Georgiana Peel ##Lady Constance Childe-Pemberton ##Lord and Lady Penrhyn ##Lady Mary Pepys ##Lady Perceval ##Lady Percy (2) ##Lady Petre ##Dowager Lady Peyton ##Lady Phillimore ##Lady William Phipps ##Lord and Lady Pirbright ##Lord and Lady Playfair ##Lady Chichele Plowden ##Lady Anna Chandos-Pole ##Lady Pollock ##Lord and Lady Poltimore ##Lady Pontifex ##Lady Alice Portal ##Lady Powell, Lady Powell [sic] ##Lady Baden-Powell ##Lady Dickson-Poynder ##Lady Poynter ##Lord and Lady George Pratt ##Lady Priestley ##Lady Probyn ##Lady Eva Wyndham-Quin, Lady Wyndham-Quin (2) ##Lord and Lady Raglan, Dowager Raglan ##Lady Ramsay ##Lord and Lady Rathdonnell ##Lady Rathmore ##Lord and Lady Rayleigh, Dowager Rayleigh ##Lord and Lady Reay ##Lady Reid ##Lord and Lady Rendel ##Lord Rendlesham ##Lady Jane Repton ##Lord Revelstoke ##Lord and Lady Ribblesdale ##Lady Laura Ridding ##Lord and Lady Robartes ##Lady O. Roberts ##Lady Roberts of Kandahar ##Lady Robinson ##Lord and Lady Rodney ##Lord Romilly ##Lord and Lady Rookwood ##Lord and Lady Rossmore ##Lord Rowton ##Lady Roxburgh ##Lord and Lady Rothschild ##Lady Victoria Russell, Lady Arthur Russell, Lady G. Russell, Lady W. H. Russell, Lady Alexander Russell ##Lord and Lady Russell of Killowen ##Lord and Lady Ruthven ##Lady Jane Ryan ##Lady Mary Sackville ##Lady Salmon ##Lord and Lady Saltoun ##Lady Samuelson, Lady S. Samuel ##Lady Mary Saurin ##Lord and Lady Savile, Lady Marie Savile ##Lady Savory ##Lord George Scott, Lord Henry Scott, Lord Herbert Scott, Lady Sophie Scott, Lady Charles Scott, Lady Louisa Scott, Lady Scott (2) ##Lord and Lady Seaton ##Lord and Lady Settrington ##Lady Seymour, Lady Albert Seymour, Lady William Seymour, Lady Seymour (2) ##Lord and Lady Shand ##Lady Shaw ##Lady Constance Shaw-Lefevre ##Lady Octavia Shaw-Stewart, Lady Alice Shaw-Stewart ##Lady Mary Shelley ##Lord and Lady Sherborne ##Lady Shippard ##Lady Shute ##Lady Kay-Shuttleworth ##Lady Simeon ##Lady Simmons ##Lady Simpson of Windsor ##Lord and Lady Sinclair ##Lord and Lady Skelmersdale ##Lady Esther Smith, Lady Barbara Smith, Lady Smith, Lady Blanche Smith, Lady Sybil Smith, Lady Euan Smith, Lady D. Smith ##Lady Smyth ##Lady Catherine Somerset, Lady Geraldine Somerset, Lady Henry Somerset ##Lord and Lady Southampton, Dowager Southampton ##Lady Edward Spencer-Churchill ##Lady Margaret Spicer ##Lady Sprigg ##Lady Stafford ##Lord Stalbridge ##Lady Stanhope (2) ##Lord Stanmore ##Lord Stanley, Lady Alice Stanley, Lady Isobel Stanley ##Lady Stansfield ##Lord Stavordale ##Lady Stephenson ##Lady Stevenson ##Lady Helen Stewart, Lady Mary Stewart, Lady Mark Stewart, Lady Stewart, Lady Houston Stewart, Lady Stewart [sic], Lady Isabel Stewart ##Lady Stewart of Grantully ##Lady Edith St. Aubyn ##Lord and Lady St. Levan ##Lady St. Leonards ##Lord and Lady St. Oswald ##Lady Stone ##Lady Charlotte Stopford ##Lord and Lady Stratheden and Campbell ##Lady Mary Stuart-Richardson ##Lord Suffield ##Lady Sutherland ##Lady Evelyn Sutton, Lady Susan Sutton ##Lord and Lady Swansea ##Lady Swinnerton Dyer ##Lady Kathleen Swinnerton-Pilkington ##Lord and Lady E. Talbot, Lady Emma Talbot ##Lady Jane Taylor ##Lady Taylour (2) ##Lady Tatton Sykes ##Lord Herbert Vane-Tempest, Lord Henry Vane-Tempest ##Lord and Lady Templemore ##Lady Tennant ##Lord and Lady Tennyson ##Lady Tenterden ##Lord Tewkesbury ##Lord and Lady Teynham ##Lord and Lady Thring ##Lady E. Thornton ##Lady Thursby ##Lady Ulrica Thynne ##Lord and Lady Tollemache ##Lady Agnes Townshend ##Lady Mary Trefusis ##Lady Tredegar ##Lady Trevelyan, Lady Trevelyan [sic] ##Lord and Lady Trevor ##Lady Troubridge ##Lady Turner ##Lady Henrietta Turnor ##Lady Tuson ##Lord and Lady Tweedmouth ##Lady Tyler ##Lady Emily Van De Weyer ##Lady Jane Van Koughnet ##Lord and Lady Ventry ##Lady Villiers (2), Lady Edith Villiers ##Lady Howard Vincent, Lady Helen Vincent, Lady Vincent ##Lady Vivian, Lady Jane Vivian ##Lady Mary Waldegrave ##Lady F. F. Walker, Lady James Walker ##Lady Walrond ##Lady Clementine Walsh ##Lord Wandsworth ##Lady Wantage ##Lord Warksworth ##Lady Leucha Warner ##Lady Warrender ##Lord and Lady Watson ##Lady Cecilia Webb ##Lady Rose Weigall ##Lord Welby ##Lady Willes ##Lady Willis ##Lady Arthur Wellesley ##Lord and Lady Wenlock ##Lord and Lady Westbury and Dowager Westbury ##Lady Isabella Whitbread ##Lady White ##Lady Whitehead ##Lady Whiteway ##Lady Elizabeth Williamson ##Lady Williams-Wynn ##Lady Willoughby (2) ##Lord Willoughby de Broke ##Lord Willoughby de Eresby ##Lady Willshire ##Lady Wilson, Lady Sarah Gordon Wilson ##Lord and Lady Wimborne ##Lady Windeyer ##Lord and Lady Windsor ##Lady Winnington ##Lady Constance Wodehouse ##Lord and Lady Wolverton ##Lady Julia Wombwell ##Lady Wood, Lady Mary Wood ##Lady Woods ##Lord Wrottesley ##Lady Hugh Wyndham ##Lady Barbara Yeatman ##Lady Lilian Yorke ##Lord Zouche #Right Honourables ##H. H. Asquith ##E. Ashley ##A. H. Dyke Acland ##J. Atkinson ##J. B. Balfour ##Sir G. Bowen ##G. W. Balfour ##Sir Hicks-Beach ##A. J. Balfour ##James Bryce ##Sir H. Campbell-Bannerman ##A. H. Smith-Barry ##E. Carson ##H. Chaplin ##Sir J. Chitty ##Jesse Collings ##Sir R. Couch ##G. N. Curzon ##J. Chamberlain ##L. Courtney ##Sir M. Grant-Duff ##A. Akers-Douglas ##Sir W. Hart Dyke ##Sir H. Elliot ##F. Foljambe ##Sir H. Fowler ##Sir A. B. Forwood ##Sir J. Fergusson ##Herbert Gladstone ##Sir J. Gorst ##G. J. Goschen ##W. E. Gladstone ##Sir G. Grey ##C. H. Hemphill ##Charles Seale-Hayne ##R. W. Hanbury ##Lord George Hamilton ##Staveley Hill ##Sir J. T. Hibbert ##Sir W. Harcourt ##lon Hamilton ##Sir Arthur Hayter ##Sir F. Jeune ##W. L. Jackson ##Sir John Kennaway ##G. Shaw-Lefevre ##W. Lidderdale ##Sir Massey Lopes ##James Lowther ##Sir J. Lubbock ##Sir H. Lopes ##Walter Long ##Sir N. Lindley ##J. W. Mellor ##Sir G. O. Morgan ##John Morley ##Arnold Morley ##Sir J. Mowbray ##A. J. Mundella ##J. H. Macdonald ##F. Max Müller ##Sir W. Marriott ##Graham Murray (the Lord Advocate) ##Sir E. Monson ##Sir P. O'Brien ##Sir A. Otway ##Sir F. Peel ##Sir R. Paget of Cranmore ##W. J. Pirrie ##J. P. Robertson ##Sir. J. Rigby ##C. T. Ritchie ##Sir S. H. Strong ##Sir B. Saunderson ##Sir J. Stansfeld ##Sir A. Smith ##C. R. Spencer ##Sir C. Kay-Shuttleworth ##Sir R. Temple ##Sir R. Thompson ##Sir E. Thornton ##Lord Henry Thynne ##Sir G. O. Trevelyan ##C. P. Villiers ##Sir Algernon West ##Sir C. L. Wyke ##C. B. Stuart-Wortley ##S. J. Way #Honourables<ref name=":1" /> (4, Col. 5a / Col. 5b) and Honourable Ladies<ref name=":1" /> (4, Col. 5b / Col. 5c) ##Mrs. Acland ##Mrs. Alexander ##H. Allsopp, Mrs. Allsopp, George Allsopp ##Mrs. Anstruther ##Mrs. Armytage ##[Hon. Lady] Vere Annesley ##Mrs. Bagot, Mrs. Bagot [sic 2x] ##Mrs. Baillie of Dochfour ##Mrs. Balfour ##[Hon.] Coplestone and [Hon.] Mrs. Bampfylde ##John Baring, Susan Baring, Lilian Baring ##Mrs. Barker ##Mrs. Barlow ##Eric Barrington, Mrs. Barrington ##Mrs. Hamar Bass ##Misses Bateman-Hanbury (2) ##Allen B. Bathurst ##Mrs. Benyon ##[Hon. Lady] Beresford ##[Hon.] R. Chetwynd ##Arthur Chichester ##Lady Biddulph ##C. E. Bingham, Mrs. Bingham, Albert Bingham, Mrs. Bingham [sic x2] ##Lady Birkbeck ##Ivo Bligh, Mrs. Bligh ##Diana Sclater-Booth ##O. Borthwick ##J. Boscawen ##Henry Bourke, Mrs. H. Bourke, Charles Bourke, Terence Bourke, Mrs. T. Bourke, Algernon Bourke, Mrs. A. Bourke, Mrs. E. R. Bourke ##Charles Brand, Arthur Brand, Mrs. Brand, Mrs. T. Brand ##T. Brassey, Mrs. A. Brassey ##Mrs. Stapleton Bretherton ##Reginald Brett, Mrs. Brett ##Mrs. F. Bridgeman, Misses Bridgeman (2) ##Mrs. Britten ##W. St. John Brodrick, Albinia Brodrick ##Emmeline Brownlow ##Mrs. T. C. Bruce, Misses Bruce (2) ##Misses M'Clintock Bunbury (2) ##Mary Byng ##T. J. Byrnes ##Arthur Cadogan, Mrs. A. Cadogan, Mrs. C. Cadogan, Ethel Cadogan ##Mrs. Gough-Calthorpe, Rachel (Gough) Calthorpe, Misses Gough Calthorpe (2) ##Mrs. Candy ##G. H. Campbell, K. Campbell, Hugh Campbell, Mrs. H. Campbell, Mrs. Ronald Campbell, Misses Campbell (2), Mrs. J. B. Campbell, Mildred Campbell ##Mrs. Carington ##Mrs. Carpenter ##Emily Cathcart ##W. Cavendish, Mrs. W. Cavendish, Mrs. Cavendish ##Eleonora Chetwynd, Mrs. R. Chetwynd ##Mrs. A. Chichester, Hilda Chichester ##Mrs. Clowes ##T. H. Cochrane ##Audrey Coleridge ##George Colville ##Mrs. Corbett ##Mrs. H. Corry ##Caroline Courtenay ##Henry Coventry ##Osbert Craven ##Misses Cross ##Mrs. P. Crutchley ##Henry Cubitt, Mrs. Cubitt ##Hamilton Cuffe, Mrs. Otway Cuffe ##Lady Cunningham ##Montagu Curzon, Darea Curzon, Mrs. Curzon ##Hew Dalrymple ##John Dawnay, Eustace Dawnay, W. Dawnay, Mrs. Dawnay (2) ##Misses de Montmorency (2) ##Mrs. H. Dennison ##R. C. Devereux, Mrs. R. C. Devereux ##Mrs. Digby ##Conrad Dillon, Mrs. C. Dillon, Edith Dillon ##Misses Douglas-Pennant (2) ##A. Hay Drummond, Mrs. Hay Drummond, Frances Drummond, Mrs. M. Drummond ##Hubert V. Duncombe, Cecil Duncombe, Mrs. C. Duncombe ##C. T. Dundas, Mrs. C. T. Dundas, W. Dundas, Mrs. W. Dundas, Mrs. John Dundas ##Lady Du Cane ##Herbert Eaton, Mrs. H. Eaton ##F. Egerton, Mrs. A. F. Egerton, Lady Grey Egerton, Tatton Egerton, Mrs. T. Egerton ##Arthur Elliot, Mrs. Arthur Elliot, Lady Elliot, Mrs. Eliot ##Lilian Elphinstone ##Mrs. Ellis ##Muriel Erskine ##H. Escombe, Mrs. Escombe ##Mrs. Evans ##Mrs. C. Keith-Falconer ##Sir S. Ponsonby Fane ##Mrs. W. Farquhar ##Ailwyn Fellowes, Mrs. A. Fellowes ##Mrs. Ferguson of Pitfour ##Everard Fielding ##N. Fitzgerald, Mrs. N. Fitzgerald, Mrs. Fitzgerald, , Mrs. F. G. FitzGerald, Lady FitzGerald ##R. Fitzwilliam, W. H. Fitzwilliam ##Mary Forester ##Sir John Forrest ##Mrs. W. H. Forster ##Mrs. Lionel Fortescue ##Sir C. Fremantle, Mary Fremantle ##Sir Malcolm Fraser, Misses Fraser (2) ##Mrs. Charles Keith-Fraser ##Violet Gibson ##Evelyn Giffard ##Mrs. Henry Gladstone ##Lady Godley ##George Ormsby Gore ##F. Leveson-Gower ##Mrs. Gough ##Mrs. Alaric Grant ##Ronald Greville, Mrs. R. Greville, Louis Greville, Mrs. L. Greville, Sidney Greville, Mrs. A. Greville, Mrs. A. H. F. Greville ##Robert Grosvenor, Algernon Grosvenor, Mrs. A. Grosvenor, Maud Grosvenor, Elizabeth Grosvenor ##Lady Hamilton Gordon, [Hon. Lady] Nevil Gordon ##Misses Guest (2) ##Geoffrey Browne Guthrie ##Mrs. Gye ##Mrs. A. Haig ##Mrs. Halford ##, Misses Hamilton (2) ##Mrs. North Dalrymple-Hamilton ##Mrs. Hobart Hampden ##Mrs. Assheton Harbord, Mrs. C. Harbord, Judith Harbord, Bridget Harbord, Mrs. Harbord ##C. Hardinge, Mrs. C. Hardinge, A. Hardinge ##A. E. Gathorne-Hardy, Nina Gathorne-Hardy ##Misses Hawke (2) ##C. G. Hay ##Misses Heneage (2) ##Helen Henniker, Mrs. Henniker ##Robert Herbert, Sir Robert Herbert, Mrs. R. Herbert, Mrs. Herbert ##A. Holland Hibbert, Mrs. A. Holland Hibbert ##Lady Higginson ##Mrs. Hill ##Lionel Holland, Sydney Holland ##Grosvenor Hood, Dorothy Hood ##Lady Acland-Hood ##Fanny Hood of Avalon ##Mrs. Curzon Howe ##[Hon.] Evelyn Hubbard, Mrs. E. Hubbard, Alice Hubbard ##Mary Hughes ##Mrs. Meynell Ingram ##G. Jolliffe, Sydney H. Jolliffe, Mrs. Jolliffe ##Lady Johnston ##G. Keppel, Mrs. Keppel, Derek Keppel, Mrs. William Keppel ##Mrs. Alfred Ker ##Constance Kerr ##Mrs. Kingscote ##C. C. Kingston ##Lady Knollys ##Bertha Lambart ##F. W. Lambton, Mrs. Lambton ##Mary Lascelles ##Charles Laurence, Herbert Laurence ##Wilfrid Laurier ##Mrs. Lawley ##Mrs. C. Lawrence, Misses Lawrence (2), Mrs. H. Lawrence ##Mrs. Legge ##T. W. Legh, Mrs. Legh, Sybil Legh ##F. D. Leigh, Mrs. F. D. Leigh, E. Chandos Leigh, Mrs. E. C. Leigh, Cordelia Leigh ##C. Hanbury Lennox, Mrs. Hanbury Lennox ##G. W. Leslie ##R. l’Estrange ##Atholl Liddell, Mrs. A. Liddell ##Mrs. H. Gore-Lindsay ##Reginald Lister ##Henry Littleton, Misses Littleton (2) ##Misses Loch (2) ##William Lowther, Mrs. W. Lowther, L. Lowther, Mrs. L. Lowther ##Mrs. E. H. Loyd ##Mrs. Lumley ##Alfred Lyttelton, Mrs. A. Lyttelton, Misses Lyttelton (2), Mrs. Lyttelton ##Flora Macdonald, Lady Macdonald ##Mrs. Mackinnon ##Mrs. Maclagan ##Mrs. Magniac ##Mrs. Maguire ##W. Massey-Mainwaring, Mrs. Massey-Mainwaring ##Mrs. Fuller-Maitland ##Aline Majendie ##Misses Henniker Major (2) ##Mrs. Mallet ##Archibald Marjoribanks ##Misses Constable Maxwell (2) ##Mrs. M'Calmont ##Schomberg M'Donnell ##Charles Mills, Violet Mills, Mrs. Mills ##Mrs. Percy Mitford ##Maud de Moleyns ##Mrs. C. Molyneux ##Annette Monck, Mrs. Monck ##Violet Monckton ##Mrs. Monson ##John Scott Montagu ##[Hon.] Evelyn Moore ##R. Moreton, Mrs. R. Moreton ##Mrs. Mostyn, Misses Mostyn (2) ##Mrs. G. H. Murray, Alice Murray ##Lady Musgrave ##[Hon. Lady] Napier, Emilia Napier, Mrs. Scott Napier ##Mrs. Neeld ##Sir Hugh Nelson ##[Hon.] R. Nevill ##Mrs. Newdigate ##Sir H. S. Northcote ##Misses O'Brien (2) ##Mary O'Hagan ##Mrs. Okeover ##Mrs. Oliphant ##R. Terence O'Neill, Henrietta O'Neill ##Misses Palk (2) ##Cecil Parker, R. Parker, F. Parker, Mrs. F. Parker, Mrs. Parker ##Mabel Parnell ##[Hon.] C. B. Parsons, Mrs. Parsons ##Mrs. W. Paton ##[Hon.] Sydney Peel, Misses Peel (2) ##Mrs. Anderson Pelham ##E. S. Douglas-Pennant, Mrs. E. S. Douglas-Pennant ##Mrs. Heber Percy ##Albert Petre, Mrs. A. Petre ##Harriet Phipps ##Mrs. Pirie ##Thomas Playford ##Horace C. Plunkett ##[Hon.] Ashley Ponsonby, Mrs. Ponsonby, Misses Ponsonby (2) ##H. Orde Powlett, Mrs. Orde-Powlett, Myra Orde-Powlett ##E. W. B. Portman, Mrs. Portman, Mary Portman ##Mrs. Pretyman ##C. Ramsay, Mrs. C. Ramsay ##G. H. Reid ##Misses Rendel (2) ##Misses Rice (2) ##Lady White Ridley ##Mrs. Ritchie ##F. Roberts, Mrs. Phillips Roberts ##Misses Roberts (of Kandahar) (2) ##J. M. Rolls, Eleanor Rolls ##W. Rothschild, Evelina Rothschild ##W. Rowley, Mrs W. Rowley, Lady Thelluson Rowley ##A. Russell, Misses Russell (2) ##Gustavus Hamilton-Russell, Misses Hamilton Russell (2) ##the Master of Ruthven, Mrs. Ruthven ##Mrs. J. D. Ryder ##Sir Saul Samuel ##A. Saumarez, Mrs. A. Saumarez ##Mrs. E. J. Saunderson ##J. Maxwell Scott, Mrs. Maxwell Scott ##R. J. Seddon ##Mary Sidney ##Lady Simeon ##Misses Skeffington (2) ##Sir Donald Smith, Mrs. A. H. Smith, [Hon.] W. F. D. Smith ##Granville Somerset, Mrs. G. Somerset, Arthur Somerset, Mrs. A. Somerset, R. Somerset, Violet Somerset ##Mrs. C. R. Spencer ##Sir J. Gordon Sprigg ##Lyulph Stanley, F. C. Stanley, George Stanley, Mrs. E. J. Stanley, Mrs. Stanley, Mrs. V. A. Stanley, Maude Stanley ##Lady Cowell-Stepney ##Randolph Stewart, Mrs. R. Stewart, FitzRoy Stewart, Mrs. Stewart ##Mabel St. Aubyn ##Misses St. Clair (2) ##Mrs. Stirling ##Horatia Stopford ##[Hon. Lady] Alison Stourton ##Mrs. Strutt, Misses Strutt (2) ##Hilda Sugden ##Alfred Talbot, Mrs. Talbot, Mrs. R. A. J. Talbot ##Sir D. Tennant ##S. R. Thayer ##Misses Thellusson (2) ##Edward Thesiger, Mrs. E. Thesiger, Frederick Thesiger, Mrs. F. Thesiger, Mary Thesiger ##Lady Thorold ##Katharine Thring ##Misses Tollemache (2) ##R. Marsham-Townshend, Mrs. Marsham-Townshend ##Alice Hanbury-Tracy ##Charles Grey Trefusis, Misses Trefusis (2) ##Mrs. Trelawny ##Mrs Tremayne ##Mrs. W. le Poer Trench ##Charles Trevor ##George Hill-Trevor, Marcus Hill-Trevor, Mrs. Hill-Trevor, Misses Hill-Trevor (2) ##Mrs. C. W. Trotter ##Lady Tryon ##Rosamond Tufton ##Sir G. Turner ##Rev. L. Tyrwhitt ##Misses Tyssen Amherst (2) ##Misses Vereker (2) ##R. Greville-Verney, Mrs. R. G. Verney, Misses Verney (2) ##F. Villiers, Mrs. F. Villiers ##Misses Vivian (2) ##Arthur Walsh ##Mrs. P. E. Warburton ##Robert Ward, Mrs. Dudley-Ward ##Mrs. West ##Mrs. Whateley ##Sir W. Whiteway ##F. Bootle-Wilbraham ##Ella Williamson ##Tatton Willoughby ##Lady Wilson ##[Hon.] Armine Wodehouse, Mrs. Wodehouse ##Frances Wolseley ##F. Wood, Misses Wood (2) ##Mrs. G. Wrottesley, Evelyn Wrottesley ##Percy Wyndham, Mrs. P. Wyndham, Misses Wyndham (2) ##Maud Wynn ##Lois Yarde-Buller ##Alex. G. Yorke, Mrs. J. Yorke, Mrs. E. C. Yorke #Sirs<ref name=":1" /> (4, Col. 5c–6a) ##Augustus Adderley ##Edwin Arnold ##John Austin ##George Arthur ##John Heathcoat-Amory ##A. Armstrong ##Andrew Agnew ##Frederick Abel ##Henry Acland ##A. Arnold ##Alexander Arbuthnot ##John Barran ##G. Bower ##J. W. Bonser ##J. Crichton-Browne ##Joseph Bailey ##E. Ashmead-Bartlett ##Henry Barkly ##R. Beauchamp ##Raymond Burrell ##Charles Barrington ##David Baird ##Arthur Birch ##Edward Birkbeck ##W. Cunliffe Brooks ##A. de Capel Brooke ##Courtenay Boyle ##F. Burton ##F. Buxton ##Steuart Bayley ##John Bramston ##John Baker ##H. Bullard ##J. T. Brunner ##H. Bellingham ##Henry Bergne ##Thomas Boughey ##F. J. Bramwell ##E. Burne-Jones ##James Blyth ##Seymour Blane ##Henry Chamberlain ##Roderick Cameron ##Hugh Cholmeley ##John Conroy ##Edward Clarke ##C. Cameron ##E. Carbutt ##W. Coddington ##Marshal Clarke ##Reginald Cathcart ##Savile Crossley ##Edward Colebrooke ##Reginald Cust ##Charles Crosthwaite ##John Colomb ##Daniel Cooper ##F. Astley-Corbett ##Donald Currie ##Henry Cunningham ##Robert Cunliffe ##Henry Cotterell ##T. D. Gibson Carmichael ##F. Curden, ##George Dallas ##James Drummond ##Mortimer Durand ##G. Des Vieux ##Henry Dering ##J. N. Dick ##Dyce Duckworth ##T. Swinnerton Dyer ##E. Hastings Doyle ##John Dorington ##William Dunn ##Humphrey de Trafford ##Charles Dalrymple ##G. Dashwood ##Gardner ##Engleheart ##Francis Evans ##A. Edmonstone ##Whittaker Ellis ##W. H. Flower ##Horace Farquhar ##Joseph Fayrer ##H. Fletcher ##William Ffolkes ##William Fraser ##Bartle Frere ##Gerald Seymour Fitz-Gerald ##Robert Finlay ##B. Walter Foster ##Gerald FitzGerald ##R. FitzGerald ##Maurice FitzGerald ##Forrest Fulton ##William Flower ##Andrew Fairbairn ##John Gilbert ##E. T. Gourley ##Edward Grey ##W. Gull ##Walter Gilbey ##Lepel Griffin ##G. Macpherson-Grant ##Reginald Graham ##Philip Grey Egerton ##Douglas Galton ##R. Glyn ##Arthur Godley ##Charles Grant ##R. Gresley ##Alexander Acland-Hood ##T. G. Fermor Hesketh ##Arthur Haliburton ##Brydges Henniker ##F. Dixon-Hartland ##R. Hanson ##Alfred Hickman ##W. Houldsworth ##Henry Howorth ##F. Seager Hunt ##Charles Hall ##E. W. Hamilton ##Reginald Hardy ##Clement Hill ##Basil Hall ##Joseph Hooker ##Charles Hunter ##Charles Hartopp ##Victor Houlton ##Augustus Hemming ##Henry Irving ##Frederic Johnstone ##W. Jenner ##J. Jenkins ##James Joicey ##Charles Jessell ##Harry Johnston ##Edward Jenkinson ##James Hill Johnes ##John Jackson ##H. Seymour King ##James Kitson ##J. Lister-Kaye ##V. Kennett-Barrington ##George Kekewich ##John Leslie ##Thomas Dick Lander ##T. Villiers Lister ##James Linton ##Charles Lees ##Charles Legard ##Thomas Lea ##Wilfrid Lawson ##Elliott Lees ##A. C. Lyall ##J. T. D. Llewelyn ##Joseph Leese ##Leonard Lyell ##F. Laking ##Godfrey Lushington ##F. Lockwood ##Henry Longley ##George Lewis ##F. Milner ##Herbert Maxwell ##Francis Montefiore ##Graham Montgomery ##Robert Moncreiffe ##Musgrave ##Colin Scott Moncrieff ##Francis Mowatt ##Evan MacGregor ##J. G. Miller ##F. D. Maclean ##J. Blundell Maple ##Allan Mackenzie ##Lewis M'lver ##F. Mappin ##Theodore Martin ##Samuel Montagu ##William MacCormac ##Hubert Miller ##Lewis Morris ##Clements Markham ##A. C. Mackenzie ##John Monckton ##J. Stirling-Maxwell ##J. Heron Maxwell ##Kenneth Matheson ##J. S. Montefiore ##Acquin Martin ##W. Maxwell ##Oswald Moseley ##Arthur Nicolson ##Terence O'Brien ##Reginald Ogilvy ##Herbert Oakeley ##Hush Owen ##G. G. Petre ##Walter Parratt ##Frederick Pollock ##Herbert Perrott ##Douglas Powell ##Weetman Pearson ##Joseph Pease ##Francis S. Powell ##Reginald Palgrave ##W. Priestley ##E. G. Poynter ##G. S. Baden-Powell ##Charles Pontifex ##J. Dickson-Poynder ##James Paget ##C. M. Palmer ##C. Lennox Peel ##James B. Peile ##Westby Perceval ##Charles Pigott ##John Puleston ##W. Plowden ##Richard Quain ##George Russell ##C. Lister Ryan ##W. H. Russell ##J. Ramsay ##Owen Roberts ##R. T. Reid ##Charles Robinson ##J. Thellusson Rowley ##James Reid ##C. Euan-Smith ##J. Barrington Simeon ##J. B. Stone ##M. Shaw-Stewart ##Edward Sieveking ##T. H. Sanderson ##Augustus K. Stephenson ##Thomas Sutherland ##Mark Stewart ##Andrew Scoble ##Joseph Savory ##Douglas Straight ##Charles Shelley ##S. Shippard ##E. Sassoon ##A. Condie Stephen ##E. Sullivan ##Arthur Sullivan ##S. Scott ##H. Simpson ##E. Stafford ##Ernest Satow ##Tatton Sykes ##John Tyler ##Charles Tennant ##John Tenniel ##J. Thorold ##John Thursby ##Thomas Troubridge ##Charles Turner ##H. Meysey-Thompson ##W. Vincent ##Edgar Vincent ##Arthur Vicars ##W. Williams-Wynn ##James Walker ##R. Webster ##George Wombwell ##C. Rivers Wilson ##W. H. Wills ##Donald Mackenzie Wallace ##George Warrender ##F. Winnington ##James Whitehead ##Arthur Willshire ##Henry Wood ##Hugh Wyndham ##W. White ##Sidney Waterlow ##Hedworth Williamson ##Jacob Wilson ##W. Windeyer ##Albert Woods (Garter) ##Allen Young #Chairman of County Council (Dr. Collins) #Counts and Countesses ##Count Cassini ##Count and Countess De Ganay ##Count Gurowski ##Count Hohenau ##Count Theodor Bolesta Koziebrodski ##Count Leon Mniszeek ##Count and Countess Potocki ##Count and Countess Raben #Barons and Baronesses ##Baroness Emile Beaumont d'Erlanger ##Baroness De Brienen ##Baron De Onethau and Baroness D’Onethan [sic] ##Baron and Baroness Alphonse de Rothschild ##Baron Ferdinand Rothschild ##Baron and Baroness Schröder ##Baron and Baroness von Deichmann ##Baron von Heeckeren van Wassenaer ##Baroness von Hügel, Baroness Gertrud von Hügel [sic] ##Baron and Baroness Campbell von Laurentz ##Baroness Wilhelm von Rothschild #Rev. the Moderator of the General Assembly of the Church of Scotland #Deans — Christ Church, St. Paul's, Westminster, Windsor #The Provost of Eton #Master of Trinity (Mr. Butler) #The Sub-Dean of the Chapels Royal #Canons — Blundell, Dalton, Duckworth, Fleming, Hervey, Teignmouth Shore, Wilberforce #Dr. Adler (Chief Rabbi) #Dr. M'Cormick #Chaplain of the Fleet #Chaplain General #Reverend Doctors — Edmund Warre, C. J. Welldon #Reverends — Prebendary Hawkshaw, Albert Baillie, W. H. Bliss, M. Ebrington Bisset, Lord W. Cecil, Lord Charles Fitzroy, J. H. Ellison, H. Haweis, W. R. Jolly, G. J. Martin, Newton Mant, Marquis of Normanby, A. Robins. W. Gunion Rutherford, Clement Smith, Montagu Villiers #Doctors — Lennox Browne, J. V. Bridge, Barlow, Robert Farquharson, J. F. Fox, Surgeon-Major Kilkelly, John Lowe, C. H. H. Parry, G. V. Poore, Dorrien Smith, S. Wilks #Messieurs<ref name=":1" /> (4, Col. 6b–7a), Mesdames (4, Col. 7a–b) and Misses<ref name=":1" /> (4, Col. 7c – 5, Col. 1a) ##Mme Abdy ##Mr C. T. Dyke-Acland, Mme A. H. Dyke Acland, Mme Dyke Acland ##Mme Adair ##Misses Adam (2) ##Mr and Mme Adeane ##Misses Adye [?] (2) ##Mme Agar ##Mr Hamilton Aidé ##Mr John Aird, Misses Aird (2) ##Miss Akers-Douglas ##Mr Edward Alderson ##Mr George Alexander, Mme Alexander, Miss Alexander ##Miss Alison ##Mr and Mme Allhusen ##Mme Alma-Tadema ##Mr W. Ambrose ##Miss Heathcoat-Amory ##Mr R. Anderson, Miss Florence Anderson ##Mr E. H. Anson ##Mr H. T. Anstruther, Miss Rosomond Anstruther ##Mme Antrobus ##Mr Arbuthnot, Miss Arbuthnott [sic] ##Miss Archer-Houblon ##Mme Argles ##Mme Arkwright, Miss Arkwright ##Misses Armytage (2) ##Miss Arnott ##Mr and Mme Ascroft, Miss Ascroft ##Mr Arthur Ash ##Mr A. Asher ##Mme Ashton ##Mme Asquith ##Mr Astor, Mr W. Astor ##Mr B. F. Astley ##Mme Evelyn Atherley ##Mr and Mme Alfred Austin, Misses Austin (2) ##Mr and Mrs C. H. Babington ##Mr and Mrs Bagge ##Mrs Charles Bagot, Mrs J. F. Bagot, Miss Alice Bagot ##Mr James Bailey, Mrs J. Bailey, Mrs Bailey, Misses Bailey (2) ##Mrs Duncan Baillie, Misses Duncan Baillie (2) ##Mr Baillie of Dochfour ##Mr and Mrs W. A. Baillie-Hamilton ##Mr E. Bainbridge ##Mr and Mrs H. R. Baird, Mr and Mrs J. G. A. Baird, Misses Baird (2) ##Mr and Mrs Baldwin ##Mr and Mrs E. Balfour, Mr and Mrs Charles Balfour, Miss Balfour ##Mr and Mrs Banbury, Miss Banbury ##Mr and Mrs S. B. Bancroft [actor "Bancroft and his wife accepted with becoming grace the congratulations with which they were well-nigh overwhelmed"<ref name=":3" /> (5, Col. 6b)] ##Bandanaratke [?] ##Mrs Bankes ##Mr Banks ##Mr and Mrs Walter Baring, Miss Baring ##Miss Barker ##Mr J. Emmott Barlow, Mrs Barlow, Mrs Barlow [sic 2x] ##Misses Barnardiston (2) ##Miss Barne ##Mr and Mrs F. G. Barnes, Mr and Mrs Barnes, Misses Barnes (2) ##Miss Barran (2) ##Mr and Mrs J. Wolfe Barry, Mr and Mrs F. Tress Barry, Mrs A. Barry ##Misses Bartlett (2) ##Mr and Mrs D. P. Barton, Mr and Mrs Barton ##Mr Hamar Bass ##Mrs Bates, Miss Bates ##Mr and Mrs H. Bathurst, Misses Bathurst (2) ##Mr and Mrs Baxendale, Miss Baxendale ##Miss Mariot [?] Bayley ##Mr and Mrs W. W. Beach, Miss Beach ##Misses Hicks-Beach (2) ##Mr R. M. Beachcroft ##Mr and Mrs Wentworth Beaumont, Mr Wentworth B. Beaumont, Mrs Beaumont, Miss Hilda Beaumont ##Mr and Mrs Rupert Beckett, Mr E. W. Beckett ##Mr and Mrs Beer ##Mr and Mrs F. F. Begg ##Mr Charles Bell, Mr and Mrs Bell, Misses Bell (2) ##Miss Bellingham ##Mr and Mrs R. Benson, Mr and Mrs Benson ##Miss Berens ##Mr and Mrs Beresford, Miss Beresford ##Miss Berkeley, Misses Berkeley (2) ##Mr and Mrs Bertier, Miss Bertier ##Mr and Mrs Cosmo Bevan, Mr and Mrs F. Bevan, Miss Bevan ##Mr M. M. Bhownaggree ##Mr and Mrs F. Bibby ##Mr Leonard Biddulph, Mr Biddulph, Mr Victor Biddulph, Mr M. Biddulph, Mrs H. M. Biddulph, Misses Biddulph (2), Miss Biddulph, Miss Freda Biddulph ##Mr and Mrs Bigham ##Mr Bigwood ##Mrs C. Bill, Miss Bill ##Miss Birch ##Mrs Birch-Reynardson, Misses Birch-Reynardson (2) ##Mr A. Birrell, Mrs Birrell ##Mr and Mrs Bischoffsheim ##Mrs Ebrington Bissett ##Misses Blackwood (2) ##Mr and Mrs R. G. Blennerhassett ##Mrs W. H. Bliss ##Mrs Blundell, Miss Blundell ##Misses Blyth (2) ##Mr and Mrs Bolitho, Miss Bolitho ##Mr H. C. O. Bonsor, Mrs Bonsor, Miss Bonsor ##Mrs W. Borsel ##Mrs Griffith-Boscawen ##Mr and Mrs Boulnois ##Miss Bourke ##Mr W. R. Bousfield ##Mrs Bowden-Smith, Misses Bowden-Smith (2) ##Miss Bowen (2) ##Mr T. G. Bowles, Mrs Bowles ##Mr Edmund R. Boyle ##Miss Mabel Brackenbury ##Mrs Bradley, Miss Bradley ##Miss Beryl Bradford ##Miss Braddon ##Miss Bramwell ##Mr H. L. C. Brassey, Mrs H. A. Brassey, Misses Brassey (2), Misses Brassey (2) [sic 2x] ##Mr Stapleton Bretherton, Misses Stapleton Bretherton (2), Mr F. Stapleton Bretherton ##Mrs Bridge ##Mr G. Bridgman, Mr and Mrs C. G. O. Bridgeman ##Mr Brigg ##Mrs Brocklehurst ##Misses Brodie (2) ##Mr and Mrs Brookfield, Miss Brookfield ##Miss Bromley-Davenport ##Miss Brooke ##Miss Rhoda Broughton ##Mr and Mrs A. H. Brown, Miss Brown ##Mrs Browne, Misses Browne (2), Misses Browne (2) [sic 2x] ##Mrs Brownrigg, Miss Brownrigg ##Mr A. O. Bruce, Mrs A. C. Bruce [sic], Misses Bruce (2) ##Miss Brunner ##Mrs Bryce ##Mr Brymer ##Mr and Mrs Buchanan ##Mrs C. E. Buckle ##Mr Bucknill ##Miss Budgett ##Miss Mary Bulteel ##Miss Burdett ##Mr and Mrs Burges, Misses Burges (2) ##Mrs C. K. Burn ##Mr and Mrs F. C Burnand ##Miss Evelyne Burne ##Mr and Mrs W. Burns, Miss Burns ##Misses Burrell (2) ##Mr J. G. Butcher ##Mrs Butler, Mrs Butler, Miss Butler ##Mr Sydney Buxton, Mrs S. Buxton, Misses Buxton (2) ##Mr P. H. Calderon ##Mrs Calley ##Mrs Archibald Calvert, Miss Calvert ##Mr Cameron, Miss Cameron, Misses Cameron (2) ##Mr and Mrs J. D. Campbell, Mr J. A. Campbell, Miss J. A. Campbell, Mrs F. Campbell, Mrs W. Campbell, Mrs Hastings Campbell, Mrs W. Campbell [sic 2x], Mrs F. L. Campbell, Mrs D. B. O. Campbell, Miss Lilah Campbell, Miss Campbell, Miss Ronald Campbell, Misses Campbell (2) ##Miss Grace de Capell-Brooke ##Miss Carden ##Miss Carleton ##Mr and Mrs W. W. Carlile, Miss Carlisle ##Mrs Rivett Carnac ##Mrs Carnegy ##Mrs Boyd Carpenter, Misses Boyd Carpenter (2) ##Mrs Carson ##Mr and Mrs D'Oyly Carte ##Mrs Carter ##Mrs Castance ##Mr R. K. Causton, Mrs Causton, Miss Causton ##Mrs Cavaye ##Mr and Mrs C. Tyrall Cavendish, Mr Victor Cavendish, Mr Henry Cavendish, Mr Cavendish, Mrs Cavendish ##Mr and Mrs F. Cavendish-Bentinck, Mr Cavendish-Bentinck, Mrs W. G. Cavendish-Bentinck ##Mr F. Cawley ##Mr and Mrs Cayzer, Miss Cayzer ##Mr and Mrs W. M. Cazalet ##Mr F. Cazenove ##Mr Evelyn Cecil, Miss Cecil ##Mrs Chaine ##Mrs Chaloner ##Mr Austen Chamberlain, Mrs Chamberlain, Misses Chamberlain (2) ##Misses Chaning (2) ##Mr and Mrs Channing ##Mr and Mrs Cecil Chaplin, Misses Chaplin (2), Miss Edith Chaplin, Miss Chaplin ##Mrs Chapman ##Misses Chetwode (2) ##Mrs W. Chetwynd, Miss Chetwynd (2) ##Mr Childe-Pemberton ##Miss Chitty ##Miss Leila Crichton ##Miss Cholmeley (2) ##Miss Cholmondeley ##Miss Chrichton-Maitland ##Mrs H. Churchill ##Miss Spencer Churchill ##Mr J. D. Clark, Mr and Mrs Atkinson Clark, Mr Clark, Mrs B. F. Clark, Mrs G. D. Clark, Stanley Clark, Miss Clark ##Mr Purdon Clarke, Mr Ernest Clarke, Miss Clarke, Miss Stanley Clarke ##Mrs Clerk ##Mr and Mrs Henry Pelham Clinton ##Mrs Clive, Misses Clive (2) ##Mrs Close ##Mr Clough ##Mr Clowes, Misses Clowes (2) ##Mr Cobbold ##Mr T. B. Cochrane, Miss Cochrane ##Mr and Mrs W. A. Cockerell, Miss Cockerell, Miss Cockerell [sic 2x] ##Mr and Mrs D. Coghill ##Mr B. Cohen ##Mr Wentworth Cole ##Miss Colomb ##Mr and Mrs Colston ##Miss Colville ##Mr Richard Combe ##Miss Commerell, Miss Commerell [sic 2x] ##Mr and Mrs Compton ##Mr and Mrs Consett, Miss Vera Consett ##Mr and Mrs F. L. Cook, Mr Ward Cook, Miss Cook ##Mr and Mrs Kinloch Cooke, Mr Cooke, Mr and Mrs C. Kinloch Cooke ##Mr and Mrs Daniel Cooper, Mrs E. H. Cooper, Misses Cooper (2), Miss Cooper ##Mr and Mrs Cameron Corbett, Miss Corbett ##Mr and Mrs V. Seymour Corkran, Miss Corkran ##Mr and Mrs F. S. W. Cornwallis ##Mr and Mrs Cory ##Mrs Armar Corry, Mrs Clifford Corry, Miss Corry ##Mr J. R. G. Cotterell, Miss Cotterell (2) ##Mrs Stapleton Coton ##Mr and Mrs George Courroux ##Mrs Courtney ##Mr Burdett-Coutts ##Mrs Coventry ##Miss Cowell ##Miss Cowell-Stepney ##Mr and Mrs R. Cox, Mrs Cox, Miss Cox ##Mrs Crabbe, Misses Crabbe (2) ##Mrs Craik ##Mr and Mrs Crawshay ##Mrs Creignton, Miss Lucia Creighton ##Mr C. A. Cripps, Mr and Mrs Wilfrid Cripps ##Mr and Mrs Critchett ##Mr and Mrs Croombie ##Mrs A. B. Crosbie ##Mr and Mrs Shepherd Cross, Mr A. Cross, Miss Crosse ##Mr and Mrs Cruddas, Misses Cruddas (2) ##Mr and Mrs Percy Crutchley, Misses Crutchley (2) ##Miss Cuffe ##Miss Culme-Seymour ##Mrs Cuninghame ##Miss Cunliffe ##Mrs Dick-Cunynghame ##Mrs Curzon ##Misses Cust (2) ##Miss Custance ##Mrs Dalbiac ##Miss Gladys Dalgety [?] ##Mr C. B. Dalison ##Miss Dalrymple ##Mrs Dalton ##Mrs Denis Daly ##Mr and Mrs Darling ##Miss Dashwood ##Mr W. Bromley-Davenport ##Miss Davey ##Mr and Mrs Louis Davidson, Mrs Randall Davidson ##Mr W. Rees Davies, Mr Ben Davies, Mr and Mrs Vaughan Davies ##Mrs Davis ##Miss Dawnay (2) ##Mrs de Arcos ##Misses De Brienen (2) ##[Miss] La Baronne de Friesen ##Mrs R. C. de Grey Vyner ##[Miss] La Baronne Sirtema de Grovestins [?] ##Mr and Mrs J. de la Cour ##Mr and Mrs Edwin de Lisle ##Mr W. E. Denison ##Mrs Denny ##Miss De Perpigna ##Mrs de Salis ##Mr de Soria ##Mr De Trafford, Miss De Trafford ##Mr Deverell, Miss Deverell ##Mr and Mrs W. de Winton, Miss De Winton ##Mr and Mrs Gerard Dicconson ##Mr and Mrs Dicken ##Mr and Mrs C. S. Dickson, Mrs Dickson ##Mr J. K. Digby, Kenelm E. Digby, Mrs Digby, Misses Digby (2), Miss Digby ##Mr and Mrs J. Diggle ##Mr Lee Dillon, Misses Dillon (2) ##Mr and Mrs Coningsby Disraeli, Mr and Mrs R. Disraeli, Miss Disraeli ##Mrs Domvile, Miss Domvile ##Mr Greville Douglas, Mrs A. L. Douglas, Misses Douglas (2) ##Mrs Akers-Douglas ##Miss Dowell ##Mr and Mrs Doxford, Miss Doxford ##Mrs Geoffrey Drage ##Mr A. Drummond, Mr and Mrs G. Drummond, Mrs A. Hay Drummond, Mrs Lawrence Drummond, Mrs Drummond, Miss Edith Drummond, Misses Drummond (2), Miss Mary Drummond, Miss Adelizs [?] Drummond, Misses Drummond (2) [sic 2x] ##Misses Du Cane (2) ##Miss Du Chair ##Mr W. H. Dudley-Ward, Miss Sybil Dudley-Ward ##Mr F. Dugdale ##Misses Duncombe (2) ##Mrs Dundas, Miss May Dundas ##Miss Dunn ##Mrs Dunne, Miss Marion Dunne ##Mr Du Plat Taylor, Mrs G. Du Plat Taylor ##Mrs Durnford ##Mr and Mrs Thiselton Dyer ##Mrs East, Misses East (2) ##Mr F. Eaton ##Mr R. Edgcumb ##Mrs Edis, Misses Edis (2) ##Mr Bevan Edwards, Miss Bevan Edwards (2), Mr C. C. Edwards, Mrs Edwards ##Mrs Egerton, Miss Egerton (2), Miss Egerton ##Miss Grey Egerton ##Mr and Mrs M. Eliot, Misses Eliot (2) ##Miss Ellaby ##Mrs Ellicott, Miss Ellicott ##Mr and Mrs F. Elliot, Mr T. H. Elliott, Miss Gertrude Elliot ##Mr T. E. Ellis, Miss Ellis (2), Miss Evelyn Ellis ##Mrs Ellison, Miss Ellison ##Misses Elphinstone (2) ##Mr Cary-Elwes ##Mr Erskine, Miss Rachel Erskine ##Mr Maurice Euphrussi ##Mr W. H. Evans, Misses Evans (2) ##Mr H. P. Ewart, Mrs C. B. Ewart ##Mr Eyre ##Mr Cecil Fane, Mr G. H. Fane, Mr Fane ##Mr Dyafer Fakhry ##Misses Keith Falconer (2) ##Mrs Fane ##Mrs Fanshawe, Miss Fanshawe ##Mr and Mrs Fardell, Misses Fardell (2) ##Mr and Mrs Farmer, Mrs Lancelot Farmer, Miss Farmer ##Mrs Farnham ##Mr Alfred Farquhar, Mr W. Farquhar, Mr and Mrs E. Farquhar, Mrs G. M. Farquhar ##Mr J. N. Farquharson, Miss Amelia Farquharson, Miss Henrietta Farquharson ##Mr and Mrs Farquharson of Invercauld, Misses Farquharson of Invercauld (2) ##Misses Feilding (2) ##Mrs Fellowes ##Mrs Fenn ##Mrs Fenwick, Misses Fenwick (2) ##Mr and Mrs Johnson-Ferguson ##Mr Munro-Ferguson ##Misses Ferguson of Pitfour (2) ##Miss Fergusson ##Miss Dorothy Ffolkes ##Mrs Field ##Mr and Mrs Fielden, Misses Fielden (2) ##Mrs G. H. Finch, Mrs Wynne Finch, Misses Finch (2) ##Mr and Mrs Firbank ##Mr Herbert Fisher, Mr and Mrs Hayes Fisher, Misses Fisher (2) ##Mr and Mrs Fison, Miss Fison ##Miss FitzClarence (2) ##Mrs FitzGeorge, Miss Olga FitzGeorge ##Mr Fitzgerald, Mr F. G. Fitzgerald, Miss Fitz Gerald ##Mr and Mrs Almeric Fitzroy, Miss Ethel Fitz-Roy ##Mrs R. Fitzwilliam, Misses Fitzwilliam (2) ##Mr Flannery ##Mr E. Flower, Miss Flower, Miss Flower [sic 2x] ##Mrs Floyd ##Mrs H. Fludyer ##Mr H. St. George Foley ##Mrs Barrington Foote ##Mr J. S. Forbes, Mr Forbes ##Mr John Ford ##Mr H. W. Forster ##Mr and Mrs Arnold-Forster ##Mr and Mrs Bevill Fortescue ##Misses Forwood (2) ##Mr W. S. Foster, Mrs W. H. Foster, Mrs H. S. Foster, Miss Foster ##Misses Fowler (2) ##Mr Franklin ##Mrs Houston French ##Misses Frere (2) ##Mr L. Fry ##Mrs Fullerton, Misses Fullerton (2) ##Mr Gadson ##Mr Wilhelm Ganz ##Miss Gardiner, Miss Gardiner [sic 2x] ##Mrs Gardner ##Mr and Mrs Garfit [?] ##Miss Gathorne-Hardy ##Mr Hamilton Gatliff ##Mr and Mrs Scott Gatty ##Mr and Mrs Sydney Gedge ##Mr Geoffrey Drage [sic; does this belong here?] ##Mr F. W. Gibbs, Misses Gibbs (2) ##Mr and Mrs Walter Gibson ##Miss Gilbey ##Mr and Mrs Tyrell Giles ##Mr W. Gillett ##Mr and Mrs Gilliat, Misses Gilliat (2) ##Mr Henry Gladstone, Mrs Gladstone, Miss Helen Gladstone ##Miss Glyn ##Misses Godley (2) ##Mrs Godson ##Mr and Mrs Goelet, Miss Goelet ##Mr Charles Gold, Miss Gold ##Mr G. P. Goldney ##Mr and Mrs S. Hoffnung Goldnung Goldsmid ##Mrs A. Goldsmid, Miss Goldsmid ##Mr Otto Goldsmidt ##Mrs Goldsworthy ##Mrs Goodden, Miss Gurrney Goodden ##Mrs Goodenough ##Mr and Mrs John Gordon, Mr and Mrs J. E. Gordon, Mrs Gordon, Mrs G. G. Gordon, Mrs S. Gordon, Mrs Gordon [sic 2x], Miss Hamilton Gordon, Misses Gordon (2) ##Mr and Mrs Frank Gore, V. Gore, Mr and Mrs S. W. Gore, Mrs Gore, Miss Gore ##Mr and Mrs Goschen, Misses Goschen (2) ##Mr and Mrs A. Gosling, Miss Gosling ##Mr and Mrs F. R. Gosset ##Misses Gough-Calthorpe (2) ##Mr E. A. Goulding ##Mr G. Leveson-Gower ##Mr F. Graham, Mr Graham, Mr H. R. Graham, Mr and Mrs C. C. Graham ##Mrs Grant, Miss Grant ##Miss Victona Grant-Duff ##Mr and Mrs Henry Graves, Miss Graves ##Mr Ernest Gray ##Mrs Green ##Mr H. D. Greene, Mr W. R. Greene ##Mrs Gregory, Miss Gregory ##Mr and Mrs W. H. Grenfell, Mrs H. Grenfell, Miss Maud Grenfell ##Mr J. A. Gretton ##Mr Howard of Greystoke ##Mr Grifflth-Boscawen ##Mr and Mrs W. H. Kendal Grimston ##Mr George Grossmith ["George Grossmith was not a little lionised by titled ladies"<ref name=":3">“The Queen’s Garden Party. Buckingham Palace Grounds. A Brilliant Scene. The Queen’s Cup of Tea.” ''Daily News'' (London) 29 June 1897, Tuesday: 5 [of 10], Col. 6a [of 7] – 6, Col. 2a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970629/021/0005. Print pp. 5–6.</ref> (5, Col. 6b)] ##Mr Montagu Guest ##Mrs Gunter, Misses Gunter (2) ##Mr Gurdon ##Mrs Gurney ##Mrs Guy-Pym ##Mr and Mrs Gye ##Mr and Mrs Carl Haag, Miss Carl Haag ##The Munshi Abdul Hafiz Karim ##Mr and Mrs Haggard ##Miss Haig ##Mr R. B. Haldane ##Mr Halford, Misses Halford (2) ##Mr and Mrs Lewis Hall, Mrs Hall, Miss Hall, Miss (Lewis) Hall ##Mr and Mrs Thomas Halsey, Misses Halsey (2) ##Mr Francis Hamilton, Mrs R. W. Hamilton, Mrs Ian Hamilton, Misses Hamilton (2), Miss Hamilton ##Mrs Hammet ##Mr and Mrs Hanbury, Miss Dora Hanbury ##Miss V. Hanson ##Mr L. V. Harcourt ##Mr and Mrs Hardcastle, Misses Hardcastle (2) ##Mr and Mrs Hardy, Misses Hardy (2) ##Mr Cozens-Hardy ##Mr T. Hare, Mr Augustus Hare, Mr and Mrs John Hare, Mrs Marcus Hare, Mrs Marcus Hare [sic 2x], Miss Hare, Misses Hare (2), Misses Hare (2) [sic 2x] ##Mrs Harford ##Mrs Hargreaves-Rogers ##Mr C. Harrison, Miss Harrison ##Miss Hart ##Misses Hart-Dyke (2) ##Mr and Mrs Hartmann ##Mr George Harwood, Misses Harwood (2) ##Mr Hatch ##Mrs Hatton ##Mrs Haweis ##Mr and Mrs Claude Hay, Misses Hay (2), Misses Hay (2) [sic 2x] ##, , , , , , , , , , , , , , , , , , , , , , , , , , . , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , A. C. Humphreys-Owen, G. M. Hutton, Heseltine, , , E. Hope, Philip Henriques, C. D. Hohler, Hornyold, , , , , R. K. Hodgson, Adrian Hope, Beresford-Hope, J. Hozier, E. S. Howard, Horner, Maurice Holzman, A. C. Howard, , , , James Hope, R. Hallett Holt, , , , , J. Henniker Heaton, Hermon-Hodge, E. Brodie Hoare, G. Hoare, H. Hobhouse, VV. H. Hornby, R. P. Houston, G. B. Hudson, John Hutton, A. E. Hutton, , , , G. T. Hertslet, , H. Higgins, Hughes, , , Hungerford, Joseph Howard, Hope, R. R. Holmes, H. Howard, , Cecil Higgins, J. C. Horsley, , Wootton Isaacson, E. R. Jenkins, Arthur James, Jacobs, Jebb, A. F. Jeffreys, J. H. Johnstone, Brynmor-Jones, H. C. Jervoise, W. James, Atherley-Jones, Philip Burne Jones, Henry Joslin, George Kemp, Nigel Kingscote, C. Kempe, W. Kenny, A. Kennard, J. Kenyon, Kearley, King King, Lees Knowles, Knowles, Kimber, Kuhe, T. Kingscote, Landon, Reginald Lucas, Letchworth, Lyon, Henry Gore Lindsay, E. H. Loyd, Leonard Lindsey, Drury Lowe, Fairfax Lucy, E. Law, Cecil Lister-Kaye, H. B. Lindsay, H. T. Lopes, J. Grant Lawson, H. Lubbock, W. A. Lindsay, A. K. Loyd, Lecky, W. F. Laurence, Edwin Laurence, J. W. Lowther, Luttrell, Loder, S. Leighton, W. C. F. Luttrell, E. Lloyd, Heathcote Long, L’Estrange, T. C. March, C. J. Murray, Mount, Morrell, R. J. More, Moon, E. P. Monckton, Monk, F. Bingham Mildmay, Beresford Melville, M’Laren, M'Ewan, Martin, H. H. Marks, lan Z. Malcolm, H. L. B. MCalmont, J. W. Maclure, Campbell Munro, J. Maclean, J. C. Macdona, W. G. E. Macartney, Muir Mackenzie, Hugh Morrison, G. H. Murray, P. C. Milbank, Bingham Mildmay, Alpin Macgregor, M. Myther, V. Montagu, Frederick Macmillan, C. M’Neill, Arundel St. John Mildmay, C. Maud, Fuller Maitland, A. Milman, W. A M’Arthur, Marjoribanks, W. H. Myers, F. W. Maude, Muntz, Charles Morley, Murdoch, A. B. F. Mitford, B. Mallet, Mure, Madden, W. J. Mure, R. Maguire, Mackinnon, Montgomerie, Maxwell-Lyte, Mason, Ronald Moncrieffe, Milvain, T. G. Menzies, G. Manners, Nicol, F. A. Newdigate, G. Noel, T. W. Nussey, Charles Orde, R. A. Oswald, M. Oldroyd, J. C. O'Dowd, Oswald, Oppenheim, Arthur Oliphant, C. L. Orr-Ewing, J. L. Pattison, J. Balfour Paul, Paton, A. Peckover, Archibald Peel, Perks, J. Pender, J. Penn, Price, Powell, Paoli, Constantine Phipps[,] Charles Phipps, Leslie Probyn, B. Faudel-Phillips, Wilton Phipps, L. Faudel-Phillips, Joseph Pease, Pollock, Arthur Pease, Roland Protheroe, Walter Peace, J. M. Paulton, Platt-Higgins, Pennefather, Provand, Guy Pym , A. E. Pease, Godfrey Pearse, Algernon Peel, A. V. Pryor, Montagu Price, Phelips, John Ponsonby, Hussey Packe, Wyndham Portal, Henry Petre, Lort Phillips, H. W. Primrose, E. Parkes, Herbert Praed, Heber Percy, Quilter, J. Rankin, Renshaw, J. A. Rentoul, H. C. Richards, Read, T. Richardson, A. T. Phillips Roberts, Hugo von Ruffer, Alexander Ramsay, Alderman and Sheriff Ritchie, Richardson, Rebow, G. L. Ryder, G. A. Redford, G. W. E. Russell, H. J. H. Russell, Pandeli Ralli, John Rutherford, J. Rennell Rodd, Leopold Rothschild, T. W. RusseII, Forbes Robertson, Alfred Rothschild, Brooke Robinson, Edmund Robertson, Repton, James Round, Royds, Henry Raikes, Bowen Rowlands, J. D. Ryder, Sheriff Hargreaves Rogers, Skeffington Smyth, Augustus Spalding, H. H. Shaw, E. Strachey, J. Murray Scott, J. Stern, P. L. Sclater, R. Sassoon, W. Sidebottom, Abel Smith, Louis Sinclair, C. H. Seely, Lucas Shadwell, W. E. T. Sharpe, C. E. Shaw, E. B. Sparke, T. H. Sidebottom, Steward, Stibbert, H. Somerset, H. S. Samuel, J. P. Smith, Horace Seymour, A. H. Smith, H. M. Stanley, J. A. Swettenham, A. Spicer, Stevenson, J. H. Stock, J. Sturgis, H. C. Smith, C. J. Stewart, Leslie Stephen, T. Smith, Senhouse, Eames Storey, Christopher Sykes, H. Seton-Karr, Philip Somers-Cocks, T. Skewes-Cox, Shelley-Bontein, Salting, Leo Schuster, Smith, Arthur Sassoon, G. D. Smith, Shaw, Michael Shaw-Stewart, E. J. Stanley, Albert Sandeman, Scaramanga, Sant, F. Sutton, Dudley Smith, C. E. Tritton. W. E. M. Tomlinson, H. F. Tollemache, A. M. Torrance, Tarleton, Edward Tighe, Alma-Tadema, W. H. Wilson-Todd, P. Thornton, F. Taylor, Beerbohm Tree, Dan Tupper, Montagu Tharp, Abel Thomas, Algernon Turnor, Tudway, C. W. Trotter, H. J. Tennant, J. C. Thynne, H. D. Trelawny, C. E. Thynne, F. J. Thynne, Montagu Thorold, Tremayne, H. Graham Toler, John Taylor, A. J. R. Trendell, Tosti, Christopher Tower, T. Usher, A. Ure, T. Usborne, Chas van Raalte, Graham Vivian, R. C. de Grey Vyner, Hope Vere, F. E. Villiers, Von André, Venning, L. Van Loon, Van De Weyer, Val Prinsep, Walter, Thomas Wayman, Hwfa Williams, Cornwallis West, R. G. Webster, Sackville West, Wanklyn, A. S. Wiison, G. Fleetwood Wilson, A. F. Warr, F. W. Wilson, Piers Egerton Warburton, S. Wombwell, Weigall, Powell Williams, John Welby, Wingfleld, Whitbread, J. W. Wilson, Walton, D’Arcy Wyvill, Wodehouse, Wylie, A. Wilson, John Wilson, C. H. Wilson, Herbert Whiteley, Wynne, Lee Warner, W. West, G. Whiteley, Spencer Walpole, H. C. Woods, M.D., Deputy Inspector-General, Charles Wyndham, J. Humphrey Ward, F. Walker, Whateley, W. Woodall, Wyndham, Godfrey Webb, J. Welby, Charles Waldstein, H. Yorke and Yerburgh #Mesdames<ref name=":1" /> (4, Col. 7a–b) — , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Staveley Hill, James Hope, Heneage, Heseltine, E. Hope, Arthur Heath, Philip Henriques, C. D. Hohler, Hornyold, Holmes, Beresford Hope, Adrian Hope, Hanbury, Horner, A. C. Howard, Heath, Hermon Hodge, Brodie Hoare, S. Hoare, H. Hobhouse, W. H. Hornby, G B. Hudson, Platt-Higgins, Hildyard, G. M. Hutton, G. Hutton, W. G. G. Hutchinson, Higgins, Hungerford, Humphreys-Owen, Hills, Hippisley, Herbert, Henderson, Hervey, J. Howard, H. Howard, Gathorne-Hardy, Howard, , Inglefield, Wootton Isaacson, Joicey, Jenkinson, Inigo Jones, Jackson, A. James, Cotton-Jodrell, Jacoby, Jebb, A. F. Jeffreys, Jessel, Brynmor Jones, Pryce Jones, J. E. Jameson, H. C. Jervoise, W. James, Atherley Jones, G. Johnstone, J. H. Johnstone, A. Kennard, Kearley, Kimber, Hegan Kennard, Kitching, Kennion [?], Kennison, Knowles, W. Kenny, Kennedy, Keeley, Kuhe, Kingston, Kilkelly, Colin Keppel, Hanning Lee, A. K. Loyd, Lyon, Long, Lane, Lucas, Lockwood, S. Leighton, Lecky, E. Lawrence, Lawrie, Luck, Lloyd. A. P. Lake, J. W. Lowther, Lowe, Lidderdale, Liddell, Lascelles, Luttrell, H. Lubbock, Leslie, Lucas-Shadwell, Laurier, Naylor Leyland, Langenbach, E. Law, Fairfax Lucy, Lockhart, Lewis, Maxwell-Lyte, F. A. Lucas, G. Manners, Beresford Melville, Morrell, Victor Milward, Marshall, Maclure, J. Maclean, M'Laren, M'Ewan, R. B. Martin, Marks, Markham, J. M'Calmont, F. W. Maude, Napier Miles, M’Neill, Max Muller, Meeking, Manvers Moorson, Arundel St. John Mildmay, Frederick Macmillan, Mount, Muntz, Murdoch, Wyndham Murray, W. J. Mure, Graham Murray, Montefiore, W. C. F. Molyneux, Newton Mant, Millett, Malet, Ashurst Morris, May, Maurice, Milvain, Marjoribanks, J. C. Macdona, Moorhouse, Muir Mackenzie, G. Moncrieff, J. Murray, Montgomery, Milbank, Bingham Mildmay, Mellor, C. Maude, T. G. Menzies, J. M'Donald, W. A. M'Arthur, M'Neile, M'N'eill, Campbell Munro, Mostyn, A. Milman, Majoribanks, Noel, H. F. Nicholson, F. Neville, Nicol, Nevul [?], Nugent, Newhouse, Oppenheim, M. Oldroyd, Charles Orde, H. H. Oldham, R. A. Oswald, Oswald. A. Oliphant, Oakley, J. L Pattison, Price, Perowne, Perks, Constantine Phipps, Peacocke [?], R. Prothero, Powell, Leslie Probyn, Pitman, Upton Prior, Lort Phillips, Primrose, Powlett, Pakenham, Peyton, Parkes, Wyndham Portal, Pipon, Pender, Phillpotts, Pollock, Montagu Price, Phellps, John Ponsonby, Fox Pitt, A. Peel, Aldrich Pelham, J. Pease, Poe, G. Pearse, A. Paget, A. Pease. N. G. Philips, Pirie [?], Dampier Palmer, F. Post, Pakenham, Paget, H. Parr, Wilton[?]-Phipps, Quilter, Rebow, J. C. Russell, Rolfe, Rutherford, I[?]. Richardson. James Ronand, Robins, Rennell Rodd, W. W. Russon[?], Alexander Ramsay, Robinson, J. Rennell Rodd, Redford, Harcourt Rose, Royds, H. Raikes, Carl Rosa, Ronalds, Arrnold Royle [? Royce?], Rice, Leopold Rothschild, Raikes, J. Rankin, Renshaw, F. Russell, Ricardo, Riddel, Robertson, G. Royle, Teignmouth [?] Shore, Sandeman, Stopford, Graham Smith, Salting, Brinsley Sheridan, Salmon [?], Salmond, Edgar Shephard, Sant. A. Sandeman, H. Seymour, H. S. Samuel, St. Clair, AbeI Smith, J. P. Smith, H. M. Stanley, A. Spicer, Stevenson, Swaine, Sullivan, J. H. Stock. E. B. Sparke, J. Sturgis, Louis Sinclair, H. Seton-Karr, Slade, J. Stern, Skefflngton Smyth, P. L. Slater, A. C. Stewart, R. Sassoon, C. Smith, E. Strachey, Napier Sturt, Steward, Eames Storey, Starkie, Senhouse, Bridgman Simpson, Seddon, T. Smith, Leslie Stephen, Settle, Scaramanga [?], Arthur Sassoon, L. Seymour, Shaw. R. F. Synge, T. Skewes-Cox, Stevenson, H. C. Smith, Sterling, T. H. Sidebottom, C. H. Seely, Shelley-Bontem [?], Sandford, Hawley Smart, Sergison [?], Frederick Slade, Scobell, Graves Sawle, Scott, Settle, Smith-Barry, Stewart, J. A. Swettenham, Surtees, Synge, Dudley Smith, Thomson, M. Thorold, H. Graham Toler [?], J. W. Taylor, Christopher Tower, Tosti, Temple, Beerbohm Tree, Dan Tupper, R. T. Thynne, Montagu Tharp, Trotter, Anstruther Thomson, Tupper, Taylor, C. E. Tritton, C. F. Anstruther Thomson, Edward Tighe, F. Taylor, Tillard, Tillbrook, Brook Taylor, Tudway, C. E. Thynne, J. C. Thynne, H. Thomas, Thwaites, Tarleton, A. Ure, Usher, R. Vivian, Val Prinsep, Edmund Vaughan, E. Villiers, C. van Raalte, Von André, Verschoyle, F. E. Villiers, Vance, Hope Vere, Villiers, Venning, Sackviile West, Whatman, Williams Wynn, Watson, Wharton, John Wilson, Williams, Stuart Wortley, Wood, C. H. Wilson, S. J. Way, Walton, H. Whiteley, G. Whiteley, Ellis Williams, Wilson, Weywan, E. F. Wodehouse, John Welby, Wray, Wickham, Whatley, Spencer Walpole, Hwfa Williams. J. Woodford, Charles Wyndham, Wingfield, Charles Wood, Lee Warner, Warre, Humphrey Ward, Wallis, Wilberforce, Wynne, J. Welby, Eardley Wilmot, A. S. Wilson, C. [?] E. Ward, Walter, Warner, R. G. Webster, Wells, Cornwallis-West, F. Charteris Wemyss, Yerburgh #Misses<ref name=":1" /> (4, Col. 7c – 5, Col. 1a) — , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Hood (2), Heneage, M. Carew Hunt, Hickman, Adrian Hope (2), Agnes Beresford Hope, , Heseltine, Hope, Heron Maxwell (2), Howard (2), Herbert Margaret Acland Hood, Mary Hope, Hill, Hemming (2), Jean Hotham, Gladys Higginson, (Brydges) Henniker, (Stock) Hill (2), (Brodie) Hoare (2), Hoare (2), Houldsworth (2), Hervey, Hutchinson, Hughes (2), Hooker, Hervey-Bathurst (2), Louisa Heathcote, Howard, Howard (2), Hornby (2), Satyendra Bala Tagore, Grace Jackson, Jolliffe (2), Helena James, Jenner, Joicey, Cotton-Jodrell, Jenkins (2), Johnstone, Jameson (2), Jessel, Jervoise, S. L. Johnstone, Keith-Falconer (2), Ker (2), Ethel Kenny, Kennedy, Kennard, King King, Kennard (2), Kitson (2), Kerr (2), Kimber, Kennaway (2), Nona Kerr, Keppel, Kemball, C. Lees, Lyson, Gore Lindsay, Linton (2), Lindley (2), Lubbock, Aline Lambton, Lambart (2), Liddell, Alice Loch, Lindsay, Lucas-Shadwell, Hanning Lee (2), Emily Loch, Larking, Leese, Llewelyn, Leighton (2), Lawrence (2), Lopes (2), J. Lawson, Laurie (2), Lyte (2), Lloyd (2), Lyall, Luttrell, Lockwood, Lister, Lidderdale (2), Violet Leigh, Liddell (2), Drury Lowe, Lewis (2), Loftus, Lindsay, Lyell (2), Aimee Lowther, l’Estrange, Graham Murray, Mellor (2), Milward, Monk (2), Maxwell (2), Massey-Mainwaring, Mackworth (2), Markham (2), Macdonald, More-Molyneux, Cicely Monson, Maclure, Lena Milman, Morris (2), Macnaghten, Mowatt (2), Margaret Muir MacKenzie, Murray, Mundella, Mowbray, Ethel Morris, Beatrice Mildmay, May Milbank, Evelyn Moreton, Magniac (2), Mackenzie, M'Clintock, Madden, MacGregor(2) Mount (2), Muntz, Murdoch, Mitford, Montagu (2), Mure, Menzies, Macpherson-Grant, Malet, Moseley (2), Meeking, Macgregor, Mary Moore, Montgomery, St. John Mildmay, Madden (2), Milman (2), Constance Maude, Martyn, Campbell Munro, Nevill (2),Noel (2), Nevill, Nicol, Neville, Nelson (2). Olpherts [?] (2), Oakley, Ogilvy, Humphreys Owen, V. A. Okeover, O’Brien (2), Linda Oppenheim, Phoebe Otway, Alina O'Shee, Anderson Pelham, Pole, Pereira, Peyton, Pattison, Orde Powlett, Powlett, Pelly (2), Perowne, Charlotte Probyn, Julia Ponsonby, Peekover, Peel (2), Penn, Peace, Baden Powell, Powell (2), Pease, Priestley, Palgrave, Post, Parker, Pease, Palmer, Packe (2), Alice Paget, Paget, Paget of Cranmore (2), [?] Phillips, Phipps, Cecilia Peel, Chandos Pole, Pollock Phellps, Parry, Ponsonby, Wilton Phipps, Quain, Quilter (2), Russell of Killowen(2), Ritchie (2), Robins, Sibyl Robertson, Round (2), Royds (2), Russell, Rebow, Jane Ryan, Ramsay, Ricardo (2), Rigby, Russell (2), Lucy Raikes, Rankin, Frances Rod, Beatrix Rice, Russell (2), K. Reiss, Ricardo, Smith, Stafford, Stevenson, Stopford (2), M. Seymour, Kay Shuttleworth [?] (2), Seymour (2), Shaw, Shaw-Stewart, Evelyn Starling[?], Maxwell Scott, Abel Smith, Sartorius (2), Maud S[?]hey, Stewart, Magaret Stanley, Dorrien Smith, Smith (2), [?]-Smith, Saurin, Salmond (2), Sandeman (2), Sant, Dudley Smith[?], Swaine, Stephenson (2), Stewart (2), Dora Stone, Sparkes, [Stanley?], Nita Houston Stewart, Lily Severn, Evelyn Stanley, [Sheppart?], Saumarez Smith, Truda Saunderson, Swinburne, [Sullivan?], Mabel Seymour, Shute, R. Sterling, Stern (2), Sar[?] (2), Sassoon (2), P. L. Sclater, Sparke, Smith (Clement), [Sanderson?], Hilda Stewart, Seddon (2), Shelley, Sprigg (2), [?] Stephen, Ruby Spencer Churchill, Rachel Smith, [?], Tremayne (2), Ellen Terry, Ethel Thomas, Muriel [?], Taylor, Mary Talbot, Tomlinson, G. le M. Tupper, [?], Ella Taylor, Thorold, Taylor (2), E. Tuson, Trelawny [?], Adela[?] Trefusis, Rachel Thynne, Tritton (2), Thomson (2), [?], Thesiger, Thynne, I. C. (2), Thynne (2), Thornton (2), [Temple?], Turner, Talbot, Thynne, Usher, Van de Weyer (2), [Vivian?] (2), Dorothy Villiers, Freda Villiers, Verschoyle, Van [der Byl?], Villiers, Venning, Hilda von Deichmann, Wood[ford?], Fleetwood Wilson, Eardley-Wilmot, Maud Walpole, [?hend?] Wilson, Wilson, Wilberforce, Warren (2), [W?vil?] (2), Wills (2), Warrender (2), Walrond (2), Wynd[ham?] (2), Webster (2), Watson, Wombwell, Whitehead (2), [W?Ieyer?] (2), Evelyn Wellesley, Cornwallis West, Whatman {2), [?] (2), Rachel Weigall, F. Walker, Smart Walker, Wood (2), de la Wood[?], Ward, Wilbraham, Wilberforce (2), Walker, Williams, [Workham?] (2), Yeatman #Admirals of the Fleet [initial large caps, rest sm caps] — Earl of Clanwilliam, Lord John [Hay?], the Hon. Sir H. Keppel #Admirals — H. G. Andoe, C. E. Buckle, Sir F. Bedford, Britten, the Hon. W. Carpenter, H. F. Cleveland, Sir H. Chads, Close, [?], Carr, E. J. Church, Sir W. Dowell, R. G. Douglas, A. L. [?], C. E. Domvile, A. T. Dale, D’Eyncourt, Field, Sir A. [Farquhar?], Fitzgerald, Fellowes, Fanshawe, Sir H. Fairfax, Sir [?] Fisher, C. J. Fane, Fullerton, the Hon. Sir E. Fremantle, [?] FitzGeorge, Woods Pasha, Sir W. Hunt-Grubbe, Sir Anthony [?] Hoskin, Lord Hood of Avalon, Sir Leopold Heath, Sir [?] [F.?] Hotham, Sir Algernon Heneage, R. H. Hamond, the Right Hon. Sir [J.?] Hay, St. G. C. D’Arcy Irvine, Jones, Kennedy, Sir A. [?s], A. P. Lake, R. M. Lloyd, Sir L. Loraine, A. H. Markham, [Sir?] R. More-Molyneux, Sir F. L. M'Clintock, Sir R. Macdonald, [the?] Hon. V. Montagu, Nicholson, Noel, Marquis of Northampton, Sir E. Ommaney [?], Sir Augustus Phillimore, A. T. Powlett, [?], [?. ?.] Rowley, Sir F. Richards, Lord Charles Scott, [? St.? John?], W. H. C. St. Clair, Bowden Smith, Sulivan, E. H. Sey[mour?], H. Stephenson, Sir Nowell Salmon, Sir W. Houston [Stewart?], Sir M. [Cuhne?]-Seymour, E. W. Turnour, E. W. Van[?] Wharton, Sir G. Willes, the Hon. W. J. Ward #Captain, R.N. — W. A. D. Acland, C. J. Barlow, F. R. Board[?], H. Bainbridge, Hon. T. Brand, Bickford, Lord Charles [B?ford?], B. F. Clark, Colville, Carter, Hon. S. Cecil Colville, [?ford?], A. G. Douglas, Sir C. Domville, Hon. A. Hay Dru[?], [?] [W.?] [?] Gordon, Hammet, Hon. Curzon Howe, Hender[?], [?] Ingles, Jellicoe, Jephson, Johnstone, Jeffreys, H. C. [?], Hon. A. Littleton, Hon. Hedworth Lambton, Moore, May, [? Net?], Poe, Pipon [?], Aldrich Pelham, Alfred Paget, [Bi.idcl?], Rolleston, John Sinclair, Bridgeman Simpson, [?], Van Koughnet [?], Burges Watson, Eardley-Wilmot, [?ham, Winsloe, Hon. J. Yorke #[Lieutenants???] — Anson, G. R. Bethell, Blair, Bayley, Cave[?], [?] Cave,Hon. Cecil Cadogan, de Salis, Fraser, Floyd, Hon. [?] [F?], Alaric Grant, Morgan, Moore, Marescaux, [?] Stuart, Tupper, Wells, Williams, G. J. S. Warrender #[Lieutenants?] R.N. — Alton, Murray Aynsley, Boyle, Bather, [?], [R. F.?] Boyle, Chaytor, Sir Charles Cust, G. W. Davy, [?] Wyndham-Fiennes, Fair, Godfrey Faussett, Garforth, [L?]ord Clifford, Hopkinson, Henderson, Keyes, Keppel, [?] Lloyd, Majendie, Mitchell, Morant, Kerr-Pearse, [?] Richmond, Rae, Stewart, Hon. Victor Stanley, [?] [Calta?]-Seymoar, Trye, Thring, Hon. Cyril Ward, W[?], R. E. Wemyss, Woolcombe #[Captain?] Trinity House, Sir J. Sydney Webbe #[Field?] Marshall — Sir F. P. Haines, Sir Lintorn Simmons, Sir [?] Stewart, Lord Roberts of Kandahar, Viscount Wolseley #[Generals?] —Sir J. Ardagh, Sir A. Alison, Sir H. J. Alderson, [?n] Annesley, J. Alleyne, Sir J. M. Adye, Sir C. G. [Arbuth?]not, Sir H. Havelock-Allan, R. Bateson, Sir W. F. [B?er, Sir H. Brackenbury, H. M. Bengough, the Right Hon. [?] Buller, Sir Owen Tador-Burne, H. J. Buchanan, Sir C. H. [Brown?low], Sir S. Browne, Sir M. Biddulph, Viscount Bridport, [?. O.?] Barnard, E. F. Chapman, Lord Clarina, C. F. Clery, the Hon. S. Gough-Calthorpe, E. H. Clive, Godfrey Clerk, Lord [Ch?]sford, the Hon. Sir Andrew Clarke, Sir E. Du Cane, Crutchley [?], Lord de Ros, Sir John Donelly, J. H. Dunne, Sir Martin Dillon, Sir Collingwood Dickson, Sir H. de Bathe, Davis, Sir F. de Winton, Sir T. Dennehy, Sir H. Ewart, Sir J. B. Edwards, C. B. Ewart, Cecil East, Arthur French, Sir T. Fitz-Wygram, the Hon. Sir P. Feilding, Sir T. E. Gallwey, Sir T. Goldsmid, Sir R. Gipps, Sir R. Grant, Sir F. W. Grenfell, Coleridge Grove, Goldsworthy, J. J. H. Gordon, Sir E. A. Holdich, Sir E. W. Higginson, Sir R. J. Hay, Sir R. Harrison, Julian Hall, Earl Howe, the Hon. W. Home, J. Jameson, Sir Arnold Kemball, Kelly-Kenay, Lord Mark Kerr, F. T. Lloyd, Sir D. Lysons, Sir Drury Lowe, G. Luck, J. W. Laurie, F. Marshall, the Hon. R. Monck, Crichton Maitland, Sir J. M'Neill, Montgomery, the Hon. S. Mostyn, G. Moncrieff, E. Markham, Sir W. A. Mackinnon, Bryan Milman [?], H. M’Calmont [?], M'Donnell, W. C. F. Molyneux, Lord [Methuen?], J. F. Maurice, Sir F. Middleton, O. H. Nicolls, Sir E. [?] Newdegate, Sir H. N[orman?], Sir W. Olpherts, F. Peyton [?], G. [?] Upton Prior, T. H. Pakenham, G. W. T. Rich, Lord [?der] Russell, Robinson, Rowlands, J. C. Russell, F. [Russell?], A. C. Stewart, Sir Henry Smyth, Sterling, Sir C. [?] Shute, N. Stevenson, Swaine, Lord William Seymour, [?] [Sahmond?], Sir Frederick Stephenson, Sir John Stokes, Sir R. [?], Sir H. B. Tuson, the Hon. R. A. J. Talbot, G. le M. [Tupper?], Taylor, Hon. C. Thesiger, R. T. Thynne, Upperton, [?]H. Utterson, Sir J. Watson, Sir C. W. Wilson, Sir F. F. Walker, Sir Evelyn Wood, Sir C. Warren, Albert Williams, the Hon. G. Wrottesley, Sir G. H. Willis, Sir H. Wilmot #Colonels — Armytage, Arkwright, Pat Boyle, Burges, the Hon. [?] Byng, H. B. H. Blundell, M. S. Brownrigg, Sir E. Bradford, Sir A. [Blyge? Bigge?], the Hon. F. Bridgeman, Brassey, Lord William Beresford, St. John Barne, N. Barnardiston, Lord Blythswood, [?] Cunynghame, F. H. Custance, Clayton, Sir Henry Colville, [?] Carnac [?], Cavaye, Seymour Corkran, the Hon. Charles [?], W. Campbell, Chaloner, Archibald Calvert, the Hon. [?] Campbell, the Hon. Wenman C. Coke, the Hon. W. [?ton], the Hon. Sir W. Colville, Chaine, A. B. Crosbie, [T.?] [R?] Crosse, Lord Edward Pelham Clinton, the Hon. Henry [C?hton], E. H. Cooper, the Hon. H. Corry, John Clerk, Lord Dorchestcr, C. R. Dease, the Hon, Lewis Dawnay, [the?] Hon. H. Denison, Denny, Dalbiac, A. Davidson, the Hon. Cathbert Edwards, the Right Hon. Sir F. Edwards, [?son], R. Edis, the Hon. Charles Edgecumbe, Aubone Fife, [?], Wynne Finch, Ferguson of Pitfour, Forster, Lancelot [?r] H. Frudyer, Barrington Foote, Goldsmid, Gore, Grenfell, [?n], C. G. Gordon, R. Gunter, Alan Gardner, Hon. G. Gough, [?] [?iton], the Hon. A. Hood, the Earl of Home, Lord Claud [Hamilton?], Harford, Herbert, the Earl of Haddington, Haygarth, G. Hatton [?], Hillyard, Arthur Haig, Sir E. Stock Hill, R. Hennell, Archer Houblon [?], the Hon. Cospatrick Home, the Hon. C. Gathorne-Hardy, Johnstone, Cotton-Jodrell, Hegan, [H?nard], Sir N. Kingscote, H. A. Lascelles, the Hon. Heneage [L?], Hanning Lee, F. A. Lucas, the Hon. H. Lyttelton, Lockwood, L. V. Loyd, C. W. Long, Ronald Lane, Lucas, J. Leslie, the Hon. Caryl [?]Molyneux, John Murray, Sir A. W. Mackworth, J. M'Calmont [?], Milward, the Hon. F. C. Morgan, J. J. Mellor, Meeking, Manvers [?], Moorsom, H. Malet, the Earl of Mount Edgecumbe, the [Earl?] of March, Wyndham Murray, Sir V. Majendie, the Hon. G. [Napper?], H. H. Oldham, L. J. Oliphant, A. Paget, Dampier Palmer, [Earl?] Percy, George Paget, C. D. Patterson, Arthur Peel, [Birch?] [Richardson?], the Hon. F. W. Stopford, Sir W. G. Stirling, E J. [Sanderson?], T. M. Sandys, H. Smith, J. F. Sandeman, Renyon-[Surrey?], C. E Stewart, E. H. Sartorius, the Hon. Walter [Stewart?], L. Seymour, Settle, Stevenson, Starkie, C. H. Seafe, the Hon. Sir W. P. Talbot, J. Du Plat[?] Taylor, H. Thomas, A. W. [T?], the Hon. W. Ie Poer Trench, H. P. Vance, Sir C. E. Howard Vincent, M.P.; R. Vivian, A. P. Vivian, E. Villiers, the Duke of Westminster, the Earl of Wemyss, Lord Wantage, Ward, [Waring?], [Earle?] Welby, Lord Arthur Wellesley, Robert Williams, the Hon. H. L. Wood, Sir W. H. Walroud, F. Smart Walker, A. [Williams?] Wynn, Wardrop #Majors — Anne, Atherley, Ashton, F. H. Bowles, the Hon. [?] R. Bourke, Carnegy, H. Candy, Close, the Hon. F. Colborue, the Hon. Wenman Coke, Lawrence Drummond, Alfred [Edgecombe?], G. Egerton, E. H. Elliot, the Hon. A. Henniker, J. [H?a?h], the Hon. Assheton Harbord, the Hon. North Dalrymple [Hamilton?], Jameson, Pryce Jones, Larnach, the Hon. Osbert [Lumley?], C. Little, Marindin, the Hon. J. Scott Napier, Wyndham Quin, F. C Rasch, the Hon. A. Sidney, the Hon. J. T. St. Aubyn, Sir Edgar Sebright, Stirling, T. E. M. Swinnington-Parkington, [?.] M. Temple, Tillbrook, Anstruther Thomson, [E.?] [L.?] Woodhouse, and the Marquis of Winchester #Captains — O. Ames, J. Acland, Alan Boisragon, Bates, H. M. [Biddulph?], the Hon. Baring, Butler, the Hon. J. Byng, the Hon. [N.?] Yarde-Butler, E. W. Blunt, J. F. Bagot, the Hon. W. Bagot, Seymour Combe, W. Chetwynd, Dundas, Denis Daly, Cecil Drummond, M. Drummond, Ellison, Houston French, Gye, R. G. [Gilmour?], P. Green, W. G. Grice-Hutchinson, Ahmed Hussain, G. [L.?] Holford, Jessel, the Hon. W. Lambton, the Hon. G. H. [L?], Sir H. Naylor-Leyland, G. Lister, Matthews, A. D. Miller, [?],M. M'Neill, C. Norton, Phillpotts, N. G. Philips, Prety[man?], Duncan Pirie, Pitman, Fox Pitt, Petre, Harcourt Rose, [W.?] [J.?] Stopford, Sir Eyre Shaw, H. G. D. Shute, Spicer, the Hon. [?.] St. Aubyn, Sutton, Tillard, Webbe, Wray, and Gordon [Watson?] #Lieutenants — Baun, A. Cowell, the Hon. E. C. Lennox, F. Ponsonby, J. Ponsonby, Vandeleur, the Hon. C. Willoughby, and the Hon. C. S. H. D. Willoughby ===Entertainment=== "The Bands of the 1st Life Guards, Grenadier Guards, and Royal Artillery played a selection of music during the afternoon."<ref name=":1" /> (4, Col. 2c) ==Anthology== ====Quote Intro==== <quote></quote> () == Notes and Questions == # ==References== * <references /> h7angmpuyw2wbludl7ws4r9ph66g88v 0 308349 2690250 2663014 2024-12-04T12:25:03Z Bocardodarapti 289675 2690250 wikitext text/x-wiki {{Linear algebra (Osnabrück 2024-2025)/Part I/Lecture design|12| {{Subtitle|Invertible matrices}} {{:Invertible matrix/Field/Similar/Introduction/Section|}} {{Subtitle|Properties of linear mappings}} {{ inputfactproof |Linear mapping/Matrix to basis/Several properties/Fact|Lemma|| }} {{Subtitle|Elementary matrices}} We want develop effective methods to determine the rank of a matrix, and to decide whether a matrix is invertible. For this, elementary matrices are helpful. {{:Elementary matrices/Introduction/Section|}} {{ inputfactproofhere |Matrix/Staircase form by elementary manipulations/Fact|Theorem||Proof text=This rests on the corresponding manipulations as in the elimination procedure, see [[Linear algebra (Osnabrück 2024-2025)/Part I/Lecture 5|Lecture 5]]. }} {{ inputfactproof |Invertible matrix/Staircase form/Identity matrix/Fact|Corollary|| }} In particular, for an invertible matrix {{mat|term=M|pm=,}} there are elementary matrices {{mathl|term=E_1 {{commadots|}} E_k|pm=}} such that {{ Math/display|term= E_k {{circdots|}} E_1 \circ M |pm= }} is the identity matrix. {{Subtitle|Finding the inverse matrix}} {{ inputmethod |Invertible matrix/Find inverse matrix/Table/Method|| }} {{ inputexample |Invertible matrix/Find inverse matrix/Table/Method/131/412/011/Example|| }} {{Subtitle|Rank of matrices}} {{:Matrix/Column rank/Row rank/Introduction/Section}} }} q65ja4or7qpmvfxpstr7viops5vgle7 0 309727 2690248 2675571 2024-12-04T12:07:43Z Bocardodarapti 289675 2690248 wikitext text/x-wiki {{ Mathematical section{{{opt|}}} |Content= {{ inputdefinition |Invertible matrix/Field/Definition|| }} {{ inputdefinition |Inverse matrix/Field/Definition|| }} The product of invertible matrices is again invertible. Due to {{ Factlink |Factname= Base change/Three bases/Composition/Fact |Nr= |pm=, }} the matrix describing a base change is invertible, and the matrix of the reversed base change is its inverse matrix. {{ inputdefinition |Matrix/K/General linear group/Definition|| }} {{ inputdefinition |Matrix/Similarity/Definition|| }} For a linear mapping {{ Mapping |name= \varphi |V|V || |pm=, }} the describing matrices with respect to two bases are similar to each other, due to {{ Factlink |Factname= Endomorphismus/Finite dimensional/Change of basis/Fact |pm=. }} |Textform=Section |Category= |}} 60jficjdozygh913nc2f5iktk2mjuom 0 310523 2690252 2672050 2024-12-04T13:18:35Z Bocardodarapti 289675 2690252 wikitext text/x-wiki {{Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet design|12| {{Subtitle|Exercise for the break}} {{ inputexercise |Elementary matrices/Invertible/Explicit/Exercise|| }} {{Subtitle|Exercises}} {{ inputexercise |Invertible matrix/No zero row/Exercise|| }} {{ inputexercise |Matrix/Left-and right inverse/Same/Exercise|| }} {{ inputexercise |Invertible matrices/K/Product/Exercise|| }} {{ inputexercise |Upper triangular matrix/Invertible/Inverse matrix/Exercise|| }} {{ inputexercise |Upper triangular matrix/Invertible/Diagonal elements/Exercise|| }} {{ inputexercise |Linear mapping/Matrix for basis/Surjective and column generating system/Exercise|| }} {{ inputexercise |Matrix/Elementary row operations/Elementary matrix from left/Fact/Proof/Exercise|| }} {{ inputexercise |Elementary matrices/Multiplication from right/Action/Exercise|| }} {{ inputexercise |Inverse matrix/2/Linear system/1/Exercise|| }} {{ inputexercise |Matrix/2/Left inverse/Right inverse/Exercise||extra1=&nbsp; directly }} {{ inputexercise |Invertible matrix/Find inverse matrix/27/-49/Exercise|| }} {{ inputexercise |Inverse matrix/2 4 0 -1 0 3 0 1 1/Exercise|| }} {{ inputexercise |Invertible matrix/Find inverse matrix/123/6-1-2/037/Exercise|| }} {{ inputexercise |Invertible matrix/Find inverse matrix/C/2+3i 1-i/5-4i 6-2i/Exercise|| }} {{ inputexercise |Matrix/C/2+5i 1-2i 3-4i 6-2i/Invertible/Solution for 54+72i 0/Exercise|| }} {{ inputexercise |Matrix/Inverting algorithm/Breaks down/1/Exercise|| }} {{ inputexercise |Matrix/Elementary matrices/Identity matrix/4/Exercise|| }} {{ inputexercise |Matrix/Rank/Linear dependency/3 2 6/4 1 5/6 -1 3/Exercise|| }} {{ inputexercise |Elementary row operations/Column rank constant/Exercise|| }} {{ inputexercise |Matrix/Surjective/Right inverse/Exercise|| }} {{ inputexercise |Column rank/Product of matrices/Estimate/Exercise|| }} {{ inputexercise |Column rank/Product with invertible matrix/Exercise|| }} {{:Block matrix/2x2/Definition|}} {{ inputexercise |Block matrix/Rank/Exercise|| }} {{Subtitle|Hand-in-exercises}} {{ inputexercise |Invertible matrix/Find inverse matrix/232/504/1-23/Exercise|m| }} {{ inputexercise |Invertible matrix/Find inverse matrix/C/5+8i 3-7i/2-9i 4-5i/Exercise|m| }} {{ inputexercise |Matrix/Elementary matrices/Identity matrix/2/Exercise|m| }} {{ inputexercise |Matrix/44/Depending on k/Inverse to itself/Exercise|m| }} {{ inputexercise |Determinant/Inverting algorithm on general matrix/Exercise|m| }} {{ inputexercise |Linear mapping/Matrix and basis/Rank/Fact/Proof/Exercise|m| }} }} i2p9o9uw276v6cwlqg2if6jcc7ljh0t 0 310605 2690317 2688243 2024-12-04T23:09:11Z Atcovi 276019 declined, see here: https://en.wikiversity.org/wiki/Talk:Institute_for_Mental_and_Behavioral_Health_Research_(IMBHR)_at_Nationwide_Children%27s_Hospital#Please_do_not_speedy_delete! 2690317 wikitext text/x-wiki == Institute for Mental and Behavioral Health Research (IMBHR) == === Overview === The Institute for Mental and Behavioral Health Research is located in the state-of-the-art Big Lots Behavioral Health Pavilion. The mission of faculty within the institute is to conduct innovative translational, clinical and epidemiological research focused on etiology, prevention and treatment of mental, emotional and behavioral disorders. In addition to collaborative efforts across the Nationwide Children’s Hospital campus, institute members work in collaboration with the Ohio State University Department of Psychiatry and Behavioral Health and the OSU Institute for Behavioral Medicine Research. With 25,000 square feet of current research space and additional research space in development, the institute has flexibility for growth and offers ample room for staff and trainees. Embedded within the institute is the Center for Suicide Prevention and Research. Recently awarded a P50 grant from the National Institute of Mental Health, the CSPR hosts four principal investigators and a large staff to sustain the work of multiple NIMH-funded projects. '''[https://www.nationwidechildrens.org/research/awri-landing The Abigail Wexner Research Institute]''' is ranked among the top 10 for NIH funding among free-standing children's hospitals. === Center for Suicide Prevention and Research === '''''To save children's lives and reduce suicide in Ohio and beyond though prevention efforts and cutting-edge research.''''' The Center for Suicide Prevention and Research (CSPR) is a cornerstone of IMBHR. CSPR was created to address the growing problem of suicide among youth in central Ohio. Find out more about CSPR here: '''[[Institute for Mental and Behavioral Health Research (IMBHR) at Nationwide Children's Hospital/Center for Suicide Prevention and Research (CSPR)|Center for Suicide Prevention and Research (CSPR)]]''' === Current Research Showcase === [[File:RISE 2024 Cohort.jpg|thumb|This picture shows the 2024 Research at IMBHR Summer Experience (RISE) cohort.''Top (left to right): Emily Glatt, Maya Garg, Eric Youngstrom, Noreen Xu, Shannon Price.'' ''Bottom (left to right): Aarav Kukreja, Jeremy Baggs, Zachery Mondlak, Halle Deericks, Hannah Brockstein.'']] Congratulations to the RISE 2024 cohort! This past summer Nationwide Children's Hospital selected 9 undergraduate, post-baccalaureate, and graduate students from around the country to participate in the first annual installment of the Research at IMBHR Summer Experience (RISE) program. These students worked to further a diverse array of research efforts to NCH, building meaningful connections and developing their own careers along the way. Stay tuned for more information about RISE 2025! Check out this page for more information about RISE, including an in-depth recap of RISE 2024: '''[[Institute for Mental and Behavioral Health Research (IMBHR) at Nationwide Children's Hospital/Research at IMBHR Summer Experience (RISE)#Research at IMBHR Summer Experience (RISE)|Research at IMBHR Summer Experience (RISE)]]''' IMBHR Clinical Research Coordinators Marissa McClellan and Charles Sabgir are currently working on building a repository of notable publications authored by IMBHR members and collaborators on Zotero. For more information on this project, see the page below: '''[[Institute for Mental and Behavioral Health Research (IMBHR) at Nationwide Children's Hospital/Reference Management SOPs|Reference Management SOPs]]''' === Leadership === [[File:IMBHR Banyan Tree.png|thumb|This figure visually depicts the members of the Institute of Mental and Behavioral Health at Nationwide Children's Hospital using a Banyan tree to highlight the importance of collaboration at IMBHR.]] The Institute for Mental and Behavioral Health Research is led by Eric Youngstrom, PhD, a nationally renowned psychologist specializing in the relationship of mood and psychopathology, and the clinical assessment of children and families. Dr. Youngstrom’s research focuses on improving clinical assessment instruments for differential diagnoses and on predicting a child’s treatment progress, especially for bipolar disorder. In addition to being the institute’s inaugural director, he will be the first recipient of the DiMarco Family Endowed Chair in Mental and Behavioral Health Research at Nationwide Children’s Hospital. Dr. Youngstrom was twice elected President of the Society of Clinical Child and Adolescent Psychology, and was President of the Society of Quantitative and Qualitative Methods. He consulted on the 5th revision of the Diagnostic and Statistical Manual (DSM-5) and the International Classification of Diseases (ICD-11) and chaired the Work Group on Child Diagnosis for the International Society for Bipolar Disorders. He is the first recipient of the Early Career Award from the Society of Child and Adolescent Clinical Psychology; an elected full member of the American College of Neuropsychopharmacology; and a fellow of the American Psychological Association (Divisions 5, 12, and 53), as well as the Association for Psychological Science and the Association for Behavioral and Cognitive Therapies. === Contact === '''Jacqueline Pazaropoulos''' - Administrative Support Lead The Institute for Mental and Behavioral Health Research Big Lots Behavioral Health Pavilion 444 Butterfly Gardens Dr. Columbus, OH 42315 === IMBHR Talks and Conferences === <small>[insert OSF links here]</small> === Helpful Links and Graphics === Employee Recognition- eCards https://nationwidechildrens.sharepoint.com/sites/A10095/SitePages/ECard-Recognition-Program.aspx 7pezxnzd7tgehsme2a0qb1kzcnsym68 0 310617 2690319 2688265 2024-12-04T23:09:26Z Atcovi 276019 declined, see here: https://en.wikiversity.org/wiki/Talk:Institute_for_Mental_and_Behavioral_Health_Research_(IMBHR)_at_Nationwide_Children%27s_Hospital#Please_do_not_speedy_delete! 2690319 wikitext text/x-wiki == Research at IMBHR Summer Experience (RISE) == === Overview and Curriculum === This past summer, Nationwide Children's Hospital launched the RISE internship program. RISE provides exceptional summer opportunities for undergraduate, post-baccalaureate, and graduate students, particularly from backgrounds underrepresented in psychology and medicine. The RISE research mentorship faculty is comprised of over 9 NIH-funded principal investigators. The RISE curriculum includes: * An immersive summer experience in a research team working with neuropsychological, mental, sleep, and behavioral health data. * Learning about data capture (including in electronic medical records, as well as in structured survey tools and using AI). * Systematic reviews and meta-analyses, which are some of the most cited, high impact research one could do. * Assignment of a postdoctoral or faculty mentor to support readiness for further studies. * Weekly research seminars and professional development workshops on active research projects, psychology graduate school and medical school applications, test preparation, interviewing and other relevant topics. * Didactics about data wrangling, checking statistical assumptions, analysis, visualization, and presentation of findings. * Involvement in mentored projects that build skills and produce a suitable for presentation at the close of the internship. === RISE 2024 Recap === Between June 3rd and August 9th of 2024, RISE interns took an active role in groundbreaking research being conducted through IMBHR and NCH, involving the following: * Conducted self-directed research in psychology, culminating in the completion and presentation of 8 academic-style posters. ** Preliminary Gender-Based Differential Item Functioning of the BASC-3-PRS-C *** Authors: Aarav Kukreja, Emily Glatt, Hannah Brockstein, Jeremy Baggs, Kevin Stephenson, Eric Youngstrom *** [https://osf.io/fdpuv OSF Link] ** Assessing Content Overlap in Pediatric PTSD Scales: A Comparative Analysis *** Authors: Halle Deericks, Jeremy Baggs, Eric Youngstrom *** [https://osf.io/n2muv OSF Link] ** A Content Overlap Analysis of 7 Mania Rating Scales for Children and Adolescents *** Authors: Yinuo Xu, Phoebe Rodda, Jeremy Baggs, Eric Youngstrom *** [https://osf.io/53epk OSF Link] ** Brief Mental Health Screeners for Youth in Primary Care: An Exploratory Content Analysis *** Authors: Maya, Garg, Hannah Brockstein, Eric Youngstrom *** [https://osf.io/4ykax OSF Link] ** Seven Common Pediatric Depression Measures: An Item Content Overlap Analysis *** Authors: Shannon Price, Kevin Stephenson, Jeremy Baggs, Eric Youngstrom *** [https://osf.io/f5r69 OSF Link] ** Item Content Overlap of Catatonia Scales *** Authors: Zachery Mondlak, Eric Youngstrom, Musa Yilanli, Colleen Waickman *** [https://osf.io.vg8ay OSF Link] ** ADHD Assessment: Commonly Used Measures and When to Use Them *** Authors: Hannah Brockstein, Emily Glatt, Jeremy Baggs, Eric Youngstrom *** OSF Link: Not Public ** Latent Profiles of Manic and Depressive Symptoms and Their Associations with Eating Disturbances among Young Adults *** Authors: Yinuo Xu, Eric Youngstrom, Kevin Stephenson *** [https://osf.io/ruhvb OSF Link] * Worked to create a Nationwide Codebook for all data that can be exported from EPIC and REDCap. ** Obtained all measures utilized by the CDC, developed data dictionaries for each measure, and merged EPIC data to data dictionaries using a Shiny App created by Jeremy Baggs. ** Compiled LEAD codes with all possible DSM and ICD codes, which were later combined with diagnostic data to make answering research questions involving diagnostic data more feasible. * Helped to disseminate evidence-based psychology online though Wikipedia, Wikiversity, Creative Commons, Prospero, and the Open Science Framework. ** All interns learned how to use various Open Science tools to increase the accessibility and impact of their work, working alongside the 501(c)3 non-profit organization Helping Give Away Psychological Science. * Implemented Quarto documents, Git, Github, AI, Shiny Apps, and R to facilitate knowledge exchange between IMBHR projects and to beta test potential future adoption within IMBHR. ** Helped to create data visualizations for various research projects and converted 1000+ lines of SPSS to R code to enhance accessibility. * Worked with content experts to conduct mata-analyses on important topics in youth mental health, improving outcomes for youth and informing clinicians and researchers at NCH. ** Project 1: Working to establish the importance of the real relationship in psychotherapy by illustrating its impact on session or treatment outcome. ** Project 2: Re-examining the results from Sandbank et al. (2004), a mete-analysis that found intervention effects do not increase with increasing amounts of intervention for young autistic children. ** Project 3: Treatment of Pediatric Bipolar. {{multiple image|perrow = 4 | align = left | image1 = Description_of_Six_Brief_Youth_Mental_Health_Screeners.png | caption1 = Brief youth mental health screeners analyzed by Garg and colleagues. | image2 = Symptom_Content_Overlap_for_Six_Brief_Mental_Health_Screeners.png | caption2 = Symptom overlap for brief youth mental health screeners. | image3 = Symptom_Content_Percentages_for_Six_Brief_Youth_Mental_Health_Screeners.png | caption3 = Types of symptoms assessed by brief youth mental health screeners. | image4 = Pediatric_Depression_Measure_Overview.png | caption4 = Pediatric depression measures analyzed by Price and colleagues. | image5 = Pediatric_Depression_Measure_Item_Content_Overlap_Chart.png | caption5 = Symptom overlap for pediatric depression measures. | image6 = Pediatric_Depression_Measure_Item_Content_Overlap_Map.png | caption6 = Item content overlap for pediatric depression measures. | image7 = Graph_of_Jaccard_Index_for_PBD_Screening_Scales.jpg | caption7 = Correlation between pediatric mania measures analyzed by Xu and colleagues. | image8 = Symptom_Content_Overlap_Heatmap_for_7_Pediatric_Mania_Screening_Measures.jpg | caption8 = Symptom overlap for pediatric mania measures. | image9 = Symptom_Content_Overlap_Map_for_7_Pediatric_Mania_Screening_Measures.png | caption9 = Item content overlap for pediatric mania measures. | image10 = Jaccard_Index_Table_-_Pediatric_PTSD_Scales.png | caption10 = Correlation between pediatric PTSD measures analyzed by Deericks and colleagues. | image11 = Heatmap_of_Pediatric_PTSD_Symptoms_by_Scales.png | caption11 = Symptom overlap for pediatric PTSD measures. | image12 = Item_Content_Overlap_Map_of_46_Symptoms_Across_9_Pediatric_PTSD_Scales.png | caption12 = Item content overlap for pediatric PTSD measures. | image13 = Description_of_10_Commonly_Used_ADHD_Measures.png | caption13 = ADHD measures analyzed by Brockstein and colleagues. | image14 = Symptom_Content_Overlap_for_10_Commonly_Used_ADHD_Measures.png | caption14 = Symptom overlap for ADHD measures. | image15 = Content_Overlap_of_Catatonia_Scales.png | caption15 = Correlation between catatonia measures analyzed by Mondlak and colleagues. | image16 = Heatmap of Catatonia Symptoms by Scales.png | caption16 = Symptom overlap for catatonia measures. | footer = All figures from research projects led by 2024 RISE interns have been included above. For more information about the 2024 item content overlap projects, please see the [[OToPS/Item Overlap Methodology|Item Overlap Methodology]] page. }} m8yy7t4l5tnln6sxs69sp576yf00onv 0 312253 2690318 2688264 2024-12-04T23:09:19Z Atcovi 276019 declined, see here: https://en.wikiversity.org/wiki/Talk:Institute_for_Mental_and_Behavioral_Health_Research_(IMBHR)_at_Nationwide_Children%27s_Hospital#Please_do_not_speedy_delete! 2690318 wikitext text/x-wiki == Center for Suicide Prevention and Research (CSPR) == '''''"To save children's lives and reduce suicide in Ohio and beyond though prevention efforts and cutting-edge research."''''' The Center for Suicide Prevention and Research (CSPR) is a cornerstone of IMBHR. CSPR was created to address the growing problem of suicide among youth in central Ohio. Jeff Bridge, Ph.D. is the director of CSPR. === About CSPR === CSPR's strategic goals include the following: * Conduct research aimed at: ** Understanding the epidemiology of child and youth suicide and suicidal behavior. ** Examining risk and protective factors that contribute to youth suicide and attempted suicide. ** Developing and testing evidence-based intervention strategies that reduce suicide and suicide attempts for youth in healthcare settings. ** Implementing effective suicide prevention interventions and strategies in real world settings such as schools, community centers, and faith-based organizations. * Foster the development and implementation of school-based programs, such as the Signs of Suicide (SOS) prevention program, to prevent youth suicide and attempted suicide in Ohio and promote methods for evaluating outcomes. Nationally, suicide has emerged as the second leading cause of death for children ages 10-19 years old. * Nearly 1 in 6 teens has seriously contemplated suicide in the past year. * Suicide affects people of all backgrounds. * Early identification of risk factors can aid behavioral health specialists in implementing prevention strategies for youth at risk of suicide. * Suicide is complex and tragic, yet often preventable if communities are provided with the right tools. In response to NCH's behavioral health initiative, the Center for Suicide Prevention and Research (CSPR) was created in 2015 to address the growing problem of suicide among youth. CSPR is a joint partnership with Big Lots Behavioral Health Pavilion and the Abigail Wexner Research Institute, allowing for the development and implementation of evidence-based prevention strategies. === Prevention === The CSPR prevention team supports school, healthcare, and youth-serving community organizations in Ohio implement effective and sustainable suicide prevention programs. The CSPR prevention team increases community awareness and reduces mental health stigma through presentations, trainings, and actionable resources informed by the latest research on youth suicide prevention. The CSPR prevention team offers a wide variety of services, including: * Offering multiple evidence-based youth suicide prevention programs available to schools, healthcare, and community partners. * Training and education opportunities for community members and professionals. * Consultation and support for schools, healthcare, and community partners on best practices in suicide pre- and post-vention on the local and national level. [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/prevention/services View Programs and Services] [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/prevention Meet our Team] [https://www.nationwidechildrens.org/conditions/suicidal-behaviors Information about Suicidal Behaviors] [https://www.nationwidechildrens.org/specialties/behavioral-health/for-families/suicide-prevention-resources Suicide Prevention Resources] === Research === Researchers in the Center for Suicide Prevention and Research (CSPR) conduct epidemiological and intervention studies on child and youth suicide and suicidal behavior to inform policy, improve the delivery of services for suicidal youth, and ultimately prevent suicide and suicidal behavior. Learn more about the research projects and publications of our investigators and research teams, including our epidemiological studies and publications on the increase in suicide deaths after Netflix's release of ''13 Reasons Why'', noncompliance surrounding the guidelines for reporting suicide deaths in the media after the deaths of Kate Spade and Anthony Bourdaine and the disparities in black youth attempting and dying by suicide, and more. * [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/featured-research-topics/epidemiology Epidemiology (Risk Among Children With Certain Conditions)] * [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/featured-research-topics/health-service-use Health Service Use] * [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/featured-research-topics/special-populations Special Populations] * [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/featured-research-topics/racial-ethnic-disparities Racial/Ethnic Disparities] * [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/featured-research-topics/age-trends Age Trends] * [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/featured-research-topics/regional-differences Regional Differences] * [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/featured-research-topics/suicide-risk-screening-and-interventions Suicide Risk Screening and Interventions] * [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/featured-research-topics/media-and-suicide The Media and Suicide] ====== Research Labs ====== [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/research-labs/bridge-lab Bridge Lab] The Bridge Lab focuses on the epidemiology of suicidal behavior in young people and neurocognitive vulnerability to suicidal behavior, as well as improving the quality of care for suicidal youth and adolescents who have attempted suicide. [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/research-labs/fontanella-lab Fontanella Lab] Under the direction of Cynthia Fontanella, PhD, the Fontanella Lab is interested in examining and improving quality of care for children and youth. Their primary goal is to understand the relationship between suicide and health service use. [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/research-labs/ruch-lab Ruch Lab] The Ruch Lab is focused on improving behavioral health and suicide related outcomes across youth serving systems (juvenile justice, child welfare, healthcare, education) to better inform suicide prevention strategies. === CSPR Researchers Awarded P50 Center Grant Funding to Support New ASPIRES Center === The Center for Accelerating Suicide Prevention in Real-World Settings (ASPIRES) aims to accelerate the development and implementation of effective interventions to reduce suicide in children and adolescents. Supported by P50 Center grant funding from the National Institute of Mental Health (NIMH) of the National Institutes of Health (NIH), the Center for Accelerating Suicide Prevention in Real-World Settings (ASPIRES) aims to accelerate the development and implementation of effective interventions to reduce suicide in children and adolescents. Jeff Bridge, PhD, director of the Center for Suicide Prevention and Research (CSPR) in the Abigail Wexner Research Institute at Nationwide Children’s Hospital, and Cynthia Fontanella, PhD, a principal investigator in CSPR, lead ASPIRES and its investigators as co-directors. The goal of the ASPIRES pilot program, Practice-Based Research on Youth Suicide Prevention, is to fund small-scale, innovative or exploratory research focused on youth suicide prevention. === Publications === View all CSPR publications [https://www.nationwidechildrens.org/research/areas-of-research/institute-for-mental-and-behavioral-health-research/suicide-prevention-and-research/meet-our-team/publications here]. 1xwbjg71hsbjkwf3ym61rkw9nviqvwp 0 313222 2690257 2675562 2024-12-04T13:27:22Z Bocardodarapti 289675 2690257 wikitext text/x-wiki {{Number in course{{{opt|}}}|Exercise|12|7}} 1h4k050phcezcgzhq54b366op9cyi8c 0 313225 2690259 2675566 2024-12-04T13:27:49Z Bocardodarapti 289675 2690259 wikitext text/x-wiki {{Number in course{{{opt|}}}|Exercise|12|20}} ccyjbjmijirl1gc5p2pb452uk4yt18j 0 313226 2690260 2675567 2024-12-04T13:28:09Z Bocardodarapti 289675 2690260 wikitext text/x-wiki {{Number in course{{{opt|}}}|Exercise|12|30}} nvicj56lytzoul5k1v4fvm8dl928zys 0 313228 2690258 2675569 2024-12-04T13:27:32Z Bocardodarapti 289675 2690258 wikitext text/x-wiki {{Number in course{{{opt|}}}|Exercise|12|8}} 1txhiq8k8qov81tjxmzu6nn7l3pjfbl 0 313450 2690261 2676024 2024-12-04T13:30:59Z Bocardodarapti 289675 2690261 wikitext text/x-wiki {{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= This rests on the manipulations of the elimination procedure, and on the fact that elementary row manipulations are achieved, due to {{ Factlink |Factname=Matrix/Elementary row operations/Elementary matrix from left/Fact |Nr= |pm=, }} by multiplications with elementary matrices from the left. In doing this, it can not happen that a zero-column or a zero-row arises, because the elementary matrices are invertible, and, in each step, invertibility is preserved. If we have an invertible upper triangular matrix, then the diagonal entries are not {{mat|term= 0 |pm=}} by {{ Exerciselink |Exercisename= Upper triangular matrix/Invertible/Diagonal elements/Exercise |Nr= |pm=, }} and, by multiplication with a scalar, we can normalize them to {{mat|term= 1 |pm=.}} With this, we can further achieve, in every column, that all entries above the diagonal entry are {{mat|term= 0 |pm=.}} |Closure= }} |Textform=Proof |Category=See }} gr812w0rs3bzxrtmekutaulqmnf2w5r 2 313557 2690323 2679945 2024-12-05T01:12:08Z Dc.samizdat 2856930 /* 6-cell rings */ fix description of which great hexagons occur in which 6-cell rings in which fibrations 2690323 wikitext text/x-wiki {{Article info |journal=Wikijournal Preprints |last=Christie |first=David Brooks |abstract=The 24-cell is one of only a few uniform polytopes in which the edge length equals the radius. It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built solely from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the internal geometry of the 24-cell in detail as an expression of its rotational symmetries. |w1=24-cell }} == The unique 24-cell polytope == The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. == The 24-cell in the proper sequence of 4-polytopes == The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope. {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius. == Coordinates == The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. === Great squares === The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} === Great hexagons === The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the great [[#Great squares|squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} === Great triangles === The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts. == Hypercubic chords == [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. == Geodesics == The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} == Constructions == Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). == Relationships among interior polytopes == The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} == Boundary cells == Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} == Radially equilateral honeycomb == The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract. A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a closed spiral not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]], which may denote <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by four disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles). Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. Each 6-cell ring belongs to ''three'' fibrations. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four 6-cell rings, but the sets of four cell rings are not disjoint sets: the 24-cell has only four distinct 6-cell rings.{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object in the same orientation, indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Conclusions == .... We found that of all its geometric properties, the most useful single fact about the 24-cell is the observation we began with: it is radially equilateral. == Acknowledgements == ... == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built solely from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the internal geometry of the 24-cell in detail as an expression of its rotational symmetries. |w1=24-cell }} == The unique 24-cell polytope == The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. == The 24-cell in the proper sequence of 4-polytopes == The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope. {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius. == Coordinates == The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. === Great squares === The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} === Great hexagons === The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the great [[#Great squares|squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} === Great triangles === The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts. == Hypercubic chords == [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. == Geodesics == The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} == Constructions == Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). == Relationships among interior polytopes == The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} == Boundary cells == Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} == Radially equilateral honeycomb == The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract. A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a closed spiral not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]], which we denote by <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four 6-cell rings, but the sets of four cell rings are not disjoint sets. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object in the same orientation, indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Conclusions == .... We found that of all its geometric properties, the most useful single fact about the 24-cell is the observation we began with: it is radially equilateral. == Acknowledgements == ... == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built solely from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the internal geometry of the 24-cell in detail as an expression of its rotational symmetries. |w1=24-cell }} == The unique 24-cell polytope == The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. == The 24-cell in the proper sequence of 4-polytopes == The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope. {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius. == Coordinates == The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. === Great squares === The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} === Great hexagons === The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the great [[#Great squares|squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} === Great triangles === The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts. == Hypercubic chords == [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. == Geodesics == The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} == Constructions == Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). == Relationships among interior polytopes == The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} == Boundary cells == Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} == Radially equilateral honeycomb == The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract. A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a closed spiral not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]], which we denote by <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object in the same orientation, indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Conclusions == .... 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It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built solely from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the internal geometry of the 24-cell in detail as an expression of its rotational symmetries. |w1=24-cell }} == The unique 24-cell polytope == The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. == The 24-cell in the proper sequence of 4-polytopes == The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope. {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius. == Coordinates == The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. === Great squares === The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} === Great hexagons === The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the great [[#Great squares|squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} === Great triangles === The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts. == Hypercubic chords == [[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. == Geodesics == The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} == Constructions == Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). == Relationships among interior polytopes == The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} == Boundary cells == Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} == Radially equilateral honeycomb == The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract. A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram₂ geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram₂ geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram₂ geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a closed spiral not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical hexagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]₂, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}} Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram₂ path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram₂ has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells. At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12₄16₃, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram₂ geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object in the same orientation, indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration{{Efn|name=four hexagonal fibrations}} are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}} == Conclusions == .... We found that of all its geometric properties, the most useful single fact about the 24-cell is the observation we began with: it is radially equilateral. == Acknowledgements == ... == Notes == {{Regular convex 4-polytopes Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so" Mssrs. [https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Musk and Ramaswamy, WSJ, 11/20/2024]</blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1<nowiki> Trump test the constitutions limits 11/19/2024]</nowiki></ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}} == Reform the other Government Bureaus and Departments == === Department of Education === [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref>{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == NASA == In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} o3igzrsnvnozqyo3130xwmx1eq4rg3j 2690327 2690326 2024-12-05T02:27:18Z Jaredscribe 2906761 /* OMB */ Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director Russell Vought, which he did on 22nd Nov.[1] 2690327 wikitext text/x-wiki The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) will analyze the [[w:U.S._Federal_budget|U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the public interest, and in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], with non-partisan research, analysis, and criticism of forward-looking proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) might fulfill its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> The DOGE will work with through the [[w:Office_of_Management_and_Budget|Office of Management and Budget]] as its "policy vector". Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so" Mssrs. [https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Musk and Ramaswamy, WSJ, 11/20/2024]</blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1<nowiki> Trump test the constitutions limits 11/19/2024]</nowiki></ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}} == Reform the other Government Bureaus and Departments == === Department of Education === [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == NASA == In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} pt9284ck8s4daa9yla8tcmkeu5phk2g 2690328 2690327 2024-12-05T02:33:36Z Jaredscribe 2906761 /* Deregulate the Economy */ 2690328 wikitext text/x-wiki The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) will analyze the [[w:U.S._Federal_budget|U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the public interest, and in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], with non-partisan research, analysis, and criticism of forward-looking proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) might fulfill its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> The DOGE will work with through the [[w:Office_of_Management_and_Budget|Office of Management and Budget]] as its "policy vector". Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=November 20th, 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}} Mssrs. [https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Musk and Ramaswamy, WSJ, 11/20/2024] </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1<nowiki> Trump test the constitutions limits 11/19/2024]</nowiki></ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}} == Reform the other Government Bureaus and Departments == === Department of Education === [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == NASA == In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} 59hcsx7ztotpp1ghbpmbkx9cf0bu8hv 2690329 2690328 2024-12-05T02:36:46Z Jaredscribe 2906761 /* Deregulate the Economy */ 2690329 wikitext text/x-wiki The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) will analyze the [[w:U.S._Federal_budget|U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the public interest, and in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], with non-partisan research, analysis, and criticism of forward-looking proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) might fulfill its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> The DOGE will work with through the [[w:Office_of_Management_and_Budget|Office of Management and Budget]] as its "policy vector". Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1<nowiki> Trump test the constitutions limits 11/19/2024]</nowiki></ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}} == Reform the other Government Bureaus and Departments == === Department of Education === [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == NASA == In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} 39og2bfg2ullxrw8t2bow57n2cbacbr 2690330 2690329 2024-12-05T02:44:46Z Jaredscribe 2906761 2690330 wikitext text/x-wiki The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will analyze the [[w:U.S._Federal_budget|U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends work with through the [[w:Office_of_Management_and_Budget|Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1<nowiki> Trump test the constitutions limits 11/19/2024]</nowiki></ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}} == Reform the other Government Bureaus and Departments == === Department of Education === [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == NASA == In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} ayjedxk2l5apx5w54jeidudjo5vpe5v 2690331 2690330 2024-12-05T02:47:14Z Jaredscribe 2906761 2690331 wikitext text/x-wiki The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1<nowiki> Trump test the constitutions limits 11/19/2024]</nowiki></ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}} == Reform the other Government Bureaus and Departments == === Department of Education === [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == NASA == In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} 59dtd67bou8lao5u0ry96u8z5g5cmcz 0 316762 2690325 2689240 2024-12-05T01:59:10Z DavidMCEddy 218607 add video 2690325 wikitext text/x-wiki :''This is a discussion of an interview 2024-11-21 with Simon Fraser University professor Ahmed Al-Rawi<ref name=AlRawiSFU><!--SFU homepage of Ahmed Al-Rawi-->{{cite Q|Q131349551}}</ref> about his research into how to understand and counter the rise in political polarization and violence worldwide. A 29:00 mm:ss podcast excerpted from the companion video will be posted here after it is released to the fortnightly "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref>'' [[File:Media literacy to counter political polarization per Ahmed Al-Rawi.webm|thumb|Media literacy to counter political polarization: Interview with [[w:Simon Fraser University|Simon Fraser University]] professor Ahmed Al-Rawi about media and conflict.]] <!--[[File: ... .ogg|thumb|29:00 mm:ss extract from interview recorded 2024-10-25 regarding the legal concerns of Wikimedia Europe.]]--> [[w:Simon Fraser University|Simon Fraser University]] professor Ahmed Al-Rawi<ref name=AlRawiSFU/> discusses the media literacy laboratory he co-founded at the Lebanese American University in Beiruit<ref><!-- Author archives: Ahmed Al-Rawi, LLRX-->{{cite Q|Q131349668}}</ref> and his research into how to understand and counter the rise in political polarization and violence worldwide. He is interviewed by Spencer Graves. Al-Rawi is the author or co-author of a dozen books in the last dozen years plus co-editor of three others and author of dozens of articles.<ref name=AlRawiCV><!--Curriculum Vitae: Ahmed Al-Rawi-->{{cite Q|Q131349693}}</ref> Most of his publications describe the increase in political polarization and violence worldwide in recent decades and what might be done to counter it. His research has focused primarily on the Arab World and on Canada. At Simon Fraser and elsewhere he has taught classes on media, communications, democracy and power. Al-Rawi is currently an Associate Professor of News, Social Media & Public Communication in the School of Communication, Faculty of Communication, Art & Technology at Simon Fraser University in [[w:Vancouver|Vancouver]], British Columbia, Canada and a scientist with the [[w:International Panel on the Information Environment|International Panel on the Information Environment]]<ref><!--Ahmed Al-Rawi, IPIE-->{{cite Q|Q131349735}}</ref> He has previously taught at other universities in Canada as well as in the [[w:Netherlands|Netherlands]] and in [[w:Oman|Oman]]. Twenty years ago he worked as a freelance radio journalist for the Pacifica Radio Network and before that as a translator for Iraq National Television, [[w:Baghdad|Baghdad]], Iraq. == The threat == Internet company executives have knowingly increased political polarization and violence including the [[w:Rohingya genocide|Rohingya genocide]] in [[w:Myanmar|Myanmar]], because doing otherwise might have reduced their profits. Documentation of this is summarized in [[:Category:Media reform to improve democracy]]. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Ahmed Al-Rawi (2025, forthcoming) Mediated Racism and democracy in Canada: Interrogating the news industry, political systems, and public discourses, Routledge-->{{cite Q|Q131349901|date=2025b}} * <!-- Ahmed Al-Rawi (2025, forthcoming) Disruptive Information in Canada, Bloomsbury-->{{cite Q|Q131349920|date=2025a}} * <!-- Ahmed Al-Rawi et al. (2025) The Canadian Far-Right and Conspiracy Theories, Routledge-->{{cite Q|Q131349937}} * <!-- Ahmed Al-Rawi (2024) The Iraqi Spring: Social Media and Political Activism, Amsterdam U. Pr.-->{{cite Q|Q131350073}} * <!-- Ahmed Al-Rawi (2024) Online hate on social media, Palgrave Macmillan-->{{cite Q|Q131350104}} * <!-- Ahmed Al-Rawi (2024) ISIS' propaganda machine : global mediated terrorism, Routledge-->{{cite Q|Q131350154}} * <!-- Ahmed Al-Rawi (2023) Supernatural Creatures in Arabic Literary Tradition, Routledge-->{{cite Q|Q131350208}} * <!-- Ahmed Al-Rawi (2021) Cyberwars in the Middle East, Routledge-->{{cite Q|Q131350317}} * <!-- Ahmed Al-Rawi (2020) News 2.0: Journalists, Audiences and News on Social Media, Wiley-Blackwell-->{{cite Q|Q131350446}} * <!-- Ahmed Al-Rawi (2020) Women's Activism and New Media in the Arab World, SUNY Pr.-->{{cite Q|Q131350555}} * <!-- Ahmed Al-Rawi (2017) Islam on YouTube : Online Debates, Protests, and Extremism, Palgrave Macmillan-->{{cite Q|Q131350577}} * <!-- Ahmed Al-Rawi (2012) Media Practice in Iraq, Springer-->{{cite Q|Q131350656}} [[Category:Politics]] [[Category:Freedom and abundance]] [[Category:Media reform to improve democracy]] 9qyjzwyw8cjds6fnn5d7yda6bgax12n 0 316860 2690340 2689692 2024-12-05T04:01:57Z JackBot 238563 Bot: Fixing double redirect to [[The Varanasi Heritage Dossier/Khori (Sarveshvara) Ghat]] 2690340 wikitext text/x-wiki #REDIRECT [[The Varanasi Heritage Dossier/Khori (Sarveshvara) Ghat]] 8yx66vbsaovi9wad0smyzchp5go0gr5 0 316866 2690338 2689723 2024-12-05T04:01:56Z JackBot 238563 Bot: Fixing double redirect to [[The Varanasi Heritage Dossier/Darbhanga Ghat]] 2690338 wikitext text/x-wiki #REDIRECT [[The Varanasi Heritage Dossier/Darbhanga Ghat]] f592fabzj6hb9bfkmz3qcwxlyglzc6a 0 316871 2690339 2689746 2024-12-05T04:01:57Z JackBot 238563 Bot: Fixing double redirect to [[The Varanasi Heritage Dossier/Jalashayi (Jalasen) Ghat]] 2690339 wikitext text/x-wiki #REDIRECT [[The Varanasi Heritage Dossier/Jalashayi (Jalasen) Ghat]] 1e81dbjb2u39nvgog22f3qagy9hk02l 0 316888 2690337 2689846 2024-12-05T04:01:56Z JackBot 238563 Bot: Fixing double redirect to [[The Varanasi Heritage Dossier/Bundi Parkota Ghat]] 2690337 wikitext text/x-wiki #REDIRECT [[The Varanasi Heritage Dossier/Bundi Parkota Ghat]] 9u5aixcar176hu9hppzfa8voboz19ow 0 316906 2690302 2690201 2024-12-04T19:49:51Z Atcovi 276019 /* Module 14.3 - Organizational Development & Change (slide 28) */ 2690302 wikitext text/x-wiki '''Organizational Theory, Dynamics, and Change''' ==Module 14.1 - Conceptual & Theoretical Foundations of Organizations== '''Organization''': A group of people who band together for a common goal and follow a number of procedures to develop these products & services. They need ''strategic planning'' (buying from/selling to the unorganized). These organizations are a "way of life". Successful organizations are able to integrate several organizing forces (HR, finance, marketing, production) smoothly. === Classic Organizational Theory === '''Classic organizational theory''' proposes that organizations should emphasize architecture of organization rather than the processes of operation. Bureaucracy is the ideal way. The following are methods for describing an organization: * Division of labor * Delegation of authority * Structure * Span of control The theory assumes there was one best configuration for an organization, and assumes that organizations affect the behaviour of their members, but not the other way around. === Human Relations Theory === The '''human relations theory''' adds a human element to the study of organizations. '''McGregor's Theory X & Theory Y''' goes as the following: * '''Theory X:''' Managers believe subordinate behavior must be controlled. * '''Theory Y:''' Managers believe subordinates are active members of the organization, and responsible for their own behavior. The ''growth perspective of Argyris'' suggests there is natural developmental sequence in humans that can either be improved or destroyed by an organization. The proposed growth is a natural & healthy experience for an individual. === Contingency Theories === 3 organization types are described: * Small batch organization * Large batch & mass production organization * Continuous process organization The span of control varies systematically by the organization type. Introduces technology. ==== Lawrence & Lorsch ==== Lawrence & Lorsch came up with: # '''Mechanistic organizations''': Depend on formal rules & regulations, small span of control # '''Organic organizations''': Less formal procedures, larger span of control. They also identified the department as an "important level" for understanding organizations. ==== Mintzberg ==== 6 forms of coordination: # Mutual adjustments based in informal communication. # Direct supervision # Standardization of work processes # Standardization of KSAOs necessary for production # Standardization of outputs # Standardization of norms ==== Tavistock Institute's Sociotechnical approach ==== This uncovered a number of dramatic changes in social patterns of work that accompany technological change. This includes joint consideration of technology & social patterns. === Recent Approaches & Conclusion === # '''Pfeffer's resource theory''': An organization must be viewed in context of connections to other organizations. He states that the key to organizational survival is the ability to acquire and maintain resources. The conclusion of the theories of organization is that motivation metaphors can be applied to organizational theories. == Module 14.2 - Social Dynamics of Organizations == === Climate & Culture === Brief history of climate consists of [https://www.mindtools.com/aieezpa/lewins-leadership-styles-framework Lewin's autocratic vs. democratic climate]. Recent investigations have found that multiple climates may exist with any organization (service climate and safety climate, for example). The word '''culture''' was introduced to address issues regarding value & meaning of actions taken place in organizations that were not encompassed in climate. The ''Organizational Culture Inventory (OCI)'' measures culture in organizations. When looking at climate vs. culture, basically... * '''Climate''' is about context in which actions take place (created at lower levels of organization). * '''Culture''' is about the meaning intended by these actions (created and conveyed from higher levels of organization) ''What about when cultures clash?'' For multinational corporations, acknowledging the existence of different cultures is key. The following models operate sufficiently under ethnocentrism, polycentrism, regiocentrism, and geocentrism. === Socialization & Concept of Person-Organization (P-O) Fit === '''Organizational socialization''': the process by which a new employee becomes aware of values & organizational procedures. Recruitment is seen as socialization, as seen from research findings. '''Socialization & P-O fit models''': the extent to which skills, abilities, & interests of individual are compatible with the job demands (person-job fit). These are broadened to include fit between the person and the organization (P-O fit). See also the [https://marcr.net/marcr-for-career-professionals/career-theory/career-theories-and-theorists/work-adjustment-theory-dawis-lofquist/ work adjustment model]. === Three Stages of Socialization === # '''Anticipatory Socialization''': Getting in (learning about prospective organizations). # '''The Encounter Stage''': Breaking in (first encounters with the new organization). # '''The Metamorphosis Stage''': Settling in (making full entry into the organization) === Schneider's ASA model === '''ASA''' --> Attraction-Selection-Attrition (ASA) model: Organizations attempt to ''ATTRACT'' and ''SELECT'' particular types of people. ''ATTRITION'' occurs through direct/indirect actions. == Module 14.3 - Organizational Development & Change (slide 28) == === Lewin's 3-stage process === # '''Unfreezing''': Become aware of values & beliefs. # '''Changing''': Adopt new values, beliefs, and attitudes. # '''Refreezing''': Stabilization of new attitudes and values. '''Episodic changes''' do take place, but they are infrequent and discontinous. It is welcomed because it is focused, time-urgent, and reduces uncertainty amongst workers - but can be stressful at the same. These changes are usually managed. '''Continuous change''', on the other hand, is the opposite: ongoing and evolving. Usually improvised, rather than intentional like episodic changes. Lewin's model needs to be re-assessed in a continuous change environment. So it goes as follows: freeze → rebalance → unfreeze. '''Resistance to change''' could be caused by the following barriers: economic fear, fear of the unknown, threats to power balance, and (possibly) previous unsuccessful change efforts. === Large-Scale Organizational Change Initiatives === * '''Total quality management (TQM)''': emphasizes team-based behavior directed towards improving quality and meeting customer needs/demands. * '''Six Sigma systems''': provides training in statistical analysis, project management, and problem-solving methods to reduce defect rate of products. * '''Lean production manufacturing''': focus on reducing waste in every form. Consists of ''just-in-time production'' (detailed tracking of materials & production; draws both suppliers and customers into organizational circle). Often requires radical redesign of HRM systems to be successful. 602q72ug6zl8rpgwfclm21j53rjysn7 2690303 2690302 2024-12-04T19:50:06Z Atcovi 276019 removing slide notice 2690303 wikitext text/x-wiki '''Organizational Theory, Dynamics, and Change''' ==Module 14.1 - Conceptual & Theoretical Foundations of Organizations== '''Organization''': A group of people who band together for a common goal and follow a number of procedures to develop these products & services. They need ''strategic planning'' (buying from/selling to the unorganized). These organizations are a "way of life". Successful organizations are able to integrate several organizing forces (HR, finance, marketing, production) smoothly. === Classic Organizational Theory === '''Classic organizational theory''' proposes that organizations should emphasize architecture of organization rather than the processes of operation. Bureaucracy is the ideal way. The following are methods for describing an organization: * Division of labor * Delegation of authority * Structure * Span of control The theory assumes there was one best configuration for an organization, and assumes that organizations affect the behaviour of their members, but not the other way around. === Human Relations Theory === The '''human relations theory''' adds a human element to the study of organizations. '''McGregor's Theory X & Theory Y''' goes as the following: * '''Theory X:''' Managers believe subordinate behavior must be controlled. * '''Theory Y:''' Managers believe subordinates are active members of the organization, and responsible for their own behavior. The ''growth perspective of Argyris'' suggests there is natural developmental sequence in humans that can either be improved or destroyed by an organization. The proposed growth is a natural & healthy experience for an individual. === Contingency Theories === 3 organization types are described: * Small batch organization * Large batch & mass production organization * Continuous process organization The span of control varies systematically by the organization type. Introduces technology. ==== Lawrence & Lorsch ==== Lawrence & Lorsch came up with: # '''Mechanistic organizations''': Depend on formal rules & regulations, small span of control # '''Organic organizations''': Less formal procedures, larger span of control. They also identified the department as an "important level" for understanding organizations. ==== Mintzberg ==== 6 forms of coordination: # Mutual adjustments based in informal communication. # Direct supervision # Standardization of work processes # Standardization of KSAOs necessary for production # Standardization of outputs # Standardization of norms ==== Tavistock Institute's Sociotechnical approach ==== This uncovered a number of dramatic changes in social patterns of work that accompany technological change. This includes joint consideration of technology & social patterns. === Recent Approaches & Conclusion === # '''Pfeffer's resource theory''': An organization must be viewed in context of connections to other organizations. He states that the key to organizational survival is the ability to acquire and maintain resources. The conclusion of the theories of organization is that motivation metaphors can be applied to organizational theories. == Module 14.2 - Social Dynamics of Organizations == === Climate & Culture === Brief history of climate consists of [https://www.mindtools.com/aieezpa/lewins-leadership-styles-framework Lewin's autocratic vs. democratic climate]. Recent investigations have found that multiple climates may exist with any organization (service climate and safety climate, for example). The word '''culture''' was introduced to address issues regarding value & meaning of actions taken place in organizations that were not encompassed in climate. The ''Organizational Culture Inventory (OCI)'' measures culture in organizations. When looking at climate vs. culture, basically... * '''Climate''' is about context in which actions take place (created at lower levels of organization). * '''Culture''' is about the meaning intended by these actions (created and conveyed from higher levels of organization) ''What about when cultures clash?'' For multinational corporations, acknowledging the existence of different cultures is key. The following models operate sufficiently under ethnocentrism, polycentrism, regiocentrism, and geocentrism. === Socialization & Concept of Person-Organization (P-O) Fit === '''Organizational socialization''': the process by which a new employee becomes aware of values & organizational procedures. Recruitment is seen as socialization, as seen from research findings. '''Socialization & P-O fit models''': the extent to which skills, abilities, & interests of individual are compatible with the job demands (person-job fit). These are broadened to include fit between the person and the organization (P-O fit). See also the [https://marcr.net/marcr-for-career-professionals/career-theory/career-theories-and-theorists/work-adjustment-theory-dawis-lofquist/ work adjustment model]. === Three Stages of Socialization === # '''Anticipatory Socialization''': Getting in (learning about prospective organizations). # '''The Encounter Stage''': Breaking in (first encounters with the new organization). # '''The Metamorphosis Stage''': Settling in (making full entry into the organization) === Schneider's ASA model === '''ASA''' --> Attraction-Selection-Attrition (ASA) model: Organizations attempt to ''ATTRACT'' and ''SELECT'' particular types of people. ''ATTRITION'' occurs through direct/indirect actions. == Module 14.3 - Organizational Development & Change == === Lewin's 3-stage process === # '''Unfreezing''': Become aware of values & beliefs. # '''Changing''': Adopt new values, beliefs, and attitudes. # '''Refreezing''': Stabilization of new attitudes and values. '''Episodic changes''' do take place, but they are infrequent and discontinous. It is welcomed because it is focused, time-urgent, and reduces uncertainty amongst workers - but can be stressful at the same. These changes are usually managed. '''Continuous change''', on the other hand, is the opposite: ongoing and evolving. Usually improvised, rather than intentional like episodic changes. Lewin's model needs to be re-assessed in a continuous change environment. So it goes as follows: freeze → rebalance → unfreeze. '''Resistance to change''' could be caused by the following barriers: economic fear, fear of the unknown, threats to power balance, and (possibly) previous unsuccessful change efforts. === Large-Scale Organizational Change Initiatives === * '''Total quality management (TQM)''': emphasizes team-based behavior directed towards improving quality and meeting customer needs/demands. * '''Six Sigma systems''': provides training in statistical analysis, project management, and problem-solving methods to reduce defect rate of products. * '''Lean production manufacturing''': focus on reducing waste in every form. Consists of ''just-in-time production'' (detailed tracking of materials & production; draws both suppliers and customers into organizational circle). Often requires radical redesign of HRM systems to be successful. t3zea18eogfwegcr7pzbnzk09gdev7m 0 316951 2690313 2690223 2024-12-04T23:06:53Z MathXplore 2888076 Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]] 2690313 wikitext text/x-wiki The '''electrical resistance''' of an object is a measure of its opposition to the flow of electric current. The reciprocal/opposite electrical property is Conductance. Resistance is measured in ohms, symbolized by the Greek letter omega (Ω), named after Georg Simon Ohm. There is a material science behind what makes a good electrical conductor (things like aluminum and copper, which are good at transmitting electricity), and resistors (things like rubber, wood and plastics, which do not like to transmit electricity). There are items like heating elements and items like [[Electronics/Resistors]] that have a fixed resistance value. Resistance is used to ensure the current and voltage after the item causing the resistance is at a specified value. If the resistor is the only path of electrical current flow (series circuit), there will be a voltage drop after the resistance item. If there are multiple path of electrical current flow (parallel circuit), there will be a current drop where the parallel path meets. === Ohms Law: === With some of the other electrical properties, the resistance of an object can be calculated. <math>Ohms = Volts^2/Watts</math> <math>Ohms = Watts/Amps^2 </math> <math>Ohms=Volts/Amps</math> {{BookCat}} jtsgmhzp2iwmvrfyq0axdxnl956bo62 0 316954 2690253 2024-12-04T13:20:57Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Exercise{{{opt|}}} |Text= Let {{mat|term= M |}} denote an {{ Definitionlink |invertible| |Context=matrix| }} {{ Definitionlink |upper triangular matrix| |pm=. }} Show that the {{ Definitionlink |inverse matrix| }} {{mat|term= M^{-1} |}} is also an upper triangular matrix. |Textform=Exercise |Category= }}" 2690253 wikitext text/x-wiki {{ Mathematical text/Exercise{{{opt|}}} |Text= Let {{mat|term= M |}} denote an {{ Definitionlink |invertible| |Context=matrix| }} {{ Definitionlink |upper triangular matrix| |pm=. }} Show that the {{ Definitionlink |inverse matrix| }} {{mat|term= M^{-1} |}} is also an upper triangular matrix. |Textform=Exercise |Category= }} i76bk9os57tu3welrhz4wj43nizsxk0 0 316955 2690254 2024-12-04T13:21:48Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Exercise{{{opt|}}} |Text= Let {{mat|term= M |}} denote an {{ Definitionlink |invertible| |Context=matrix| }} {{ Definitionlink |upper triangular matrix| |pm=. }} Show that the diagonal elements of {{mat|term= M |pm=}} are not {{mat|term= 0 |pm=.}} |Textform=Exercise |Category= }}" 2690254 wikitext text/x-wiki {{ Mathematical text/Exercise{{{opt|}}} |Text= Let {{mat|term= M |}} denote an {{ Definitionlink |invertible| |Context=matrix| }} {{ Definitionlink |upper triangular matrix| |pm=. }} Show that the diagonal elements of {{mat|term= M |pm=}} are not {{mat|term= 0 |pm=.}} |Textform=Exercise |Category= }} dqgqlkv2dheoci6g13f3k43cdbyvk3s 0 316956 2690256 2024-12-04T13:26:40Z Bocardodarapti 289675 New resource with "{{Number in course{{{opt|}}}|Exercise|12|6}}" 2690256 wikitext text/x-wiki {{Number in course{{{opt|}}}|Exercise|12|6}} 3qhpxvrcxdyk0w8nxhm8w9ytnbwgjdi 6 316958 2690266 2024-12-04T14:42:30Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6B (20241204 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690266 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6B (20241204 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} g7we6mm69yoda24u5kcx4054n6lvtj8 2690273 2690266 2024-12-04T15:00:49Z Young1lim 21186 /* Summary */ 2690273 wikitext text/x-wiki == Summary == {{Information |Description=VLSI.Arith: Carry Skip Adders 1A (20241204 - 20241203 wrong files) |Source={{own|Young1lim}} |Date=2024-12-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} k5t7rim5k8cvdv31tt93wxt5p7w0haa 0 316959 2690267 2024-12-04T14:47:34Z Lbeaumont 278565 Created the adventure 2690267 wikitext text/x-wiki [[File:Bundesarchiv Bild 101I-635-3966-27, Panzerfabrik in Deutschland.jpg|thumb|During World War II, production of German tanks such as the [[w:Panther tank|Panther]] was accurately estimated by Allied intelligence using statistical methods.]] The [[w:German_tank_problem|German tank problem]] is a set of statistics problems named for an important historical application. During the course of the Second World War, the Western Allies made sustained efforts to determine the extent of German production and approached this in two major ways: conventional intelligence gathering and statistical estimation. In many cases, statistical analysis substantially improved on conventional intelligence. In some cases, conventional intelligence was used in conjunction with statistical methods, as was the case in estimation of Panther tank production just prior to D-Day. How can a sample of serial numbers, obtained from equipment (tanks in the historical case) observed in the field, be used to estimate the number of units being manufactured? Describe your proposed solution before reading about [[w:German_tank_problem|various expert solutions]]. {{CourseCat}} [[Category:Mathematics/Activities]] ddxxyiw993l0woonsshsbjbth7pb5ko 6 316960 2690271 2024-12-04T14:54:55Z Young1lim 21186 {{Information |Description=C04.SA0: Address-of and de-reference operators 1A (20241204 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690271 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address-of and de-reference operators 1A (20241204 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 841zbzr32q34dm3g7etcitafwtrv8iq 6 316961 2690274 2024-12-04T15:04:44Z Young1lim 21186 {{Information |Description=VLSI.Arith: Carry Skip Adders 1A (20241204 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690274 wikitext text/x-wiki == Summary == {{Information |Description=VLSI.Arith: Carry Skip Adders 1A (20241204 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 4hkbd0xtk12rq6kl2m9fyr2ndnlso4r 6 316962 2690276 2024-12-04T15:07:01Z Young1lim 21186 {{Information |Description=Laurent.5: Permutations 6B (20241204-1 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690276 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutations 6B (20241204-1 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 5y9x42ces8qz6pqiauhvuax9jexljaz 0 316963 2690280 2024-12-04T16:14:52Z 82.219.7.128 New resource with "Food tests are tests for different biological molecules (food groups). Examples of food groups include protein, carbohydrates and starch. == Testing for Starch (experiment) == Iodine is used to test for starch in food products. This test involves different types of foods, which include pasta, biscuits, crisps, cereal hoops, salt, mustard powder and sugar, though any types of food will work in this experiment. === Safety Precautions: === Iodine is very toxic to aquatic..." 2690280 wikitext text/x-wiki Food tests are tests for different biological molecules (food groups). Examples of food groups include protein, carbohydrates and starch. == Testing for Starch (experiment) == Iodine is used to test for starch in food products. This test involves different types of foods, which include pasta, biscuits, crisps, cereal hoops, salt, mustard powder and sugar, though any types of food will work in this experiment. === Safety Precautions: === Iodine is very toxic to aquatic life and it is harmful to human skin in large quantities., so take precautions before conducting this experiment. Make sure you do not eat the food while it is being experimented on. === Conducting the Experiment: === Make sure you have prepared all of the required foods and iodine before conducting this experiment. Start by taking the food and adding iodine to one of the foods. Notice how one of the foods change colours. What does this show? Continue the experiment with the other foods. === Conclusion: === The findings show that if foods have starch in them, they will go black, while if they do not have starch in them, they will go orange. e9xdb9r62oroqsciqrkyyf3nlbfjwg9 2690284 2690280 2024-12-04T16:55:57Z Atcovi 276019 PROD 2690284 wikitext text/x-wiki <!-- TO CONTEST THIS PROPOSED DELETION, remove the following template, including this comment, up to the CLOSING COMMENT --> {{Ombox | type = delete | image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]] | text = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[WV:WIW|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|{{REVISIONTIMESTAMP}} +90 days}}''' may allow it to be [[WV:PROD|kept]]. <br> You may remove {{tl|proposed deletion}} from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]]. {{#if:|---- <div style="text-align:center; margin-bottom:0em;"> ''The Nominator gave the following reason for their nomination'': {{cquote|{{{1}}}}} </div> }}}}{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}/|''Link to any subpages this page might have'']]}} {{event trigger|date=December 4, 2024|when=90 days|[[Category:Pending deletions|{{PAGENAME}}]]}} {{event trigger|date=December 4, 2024|when=60 days|[[Category:60-day proposed deletions|{{PAGENAME}}]]}} [[Category:Proposed deletions|{{PAGENAME}}]] <!-- CLOSING COMMENT, remove up to and including this comment --> Food tests are tests for different biological molecules (food groups). Examples of food groups include protein, carbohydrates and starch. == Testing for Starch (experiment) == Iodine is used to test for starch in food products. This test involves different types of foods, which include pasta, biscuits, crisps, cereal hoops, salt, mustard powder and sugar, though any types of food will work in this experiment. === Safety Precautions: === Iodine is very toxic to aquatic life and it is harmful to human skin in large quantities., so take precautions before conducting this experiment. Make sure you do not eat the food while it is being experimented on. === Conducting the Experiment: === Make sure you have prepared all of the required foods and iodine before conducting this experiment. Start by taking the food and adding iodine to one of the foods. Notice how one of the foods change colours. What does this show? Continue the experiment with the other foods. === Conclusion: === The findings show that if foods have starch in them, they will go black, while if they do not have starch in them, they will go orange. p9txs4vrvksmhip8ww0heqatjv8gj9f 2690349 2690284 2024-12-05T08:33:56Z 82.219.7.128 2690349 wikitext text/x-wiki Food tests are tests for different biological molecules (food groups). Examples of food groups include protein, carbohydrates and starch. == Testing for Starch (experiment) == Iodine is used to test for starch in food products. This test involves different types of foods, which include pasta, biscuits, crisps, cereal hoops, salt, mustard powder and sugar, though any types of food will work in this experiment. === Safety Precautions: === Iodine is very toxic to aquatic life and it is harmful to human skin in large quantities., so take precautions before conducting this experiment. Make sure you do not eat the food while it is being experimented on. === Conducting the Experiment: === Make sure you have prepared all of the required foods and iodine before conducting this experiment. Start by taking the food and adding iodine to one of the foods. Notice how one of the foods change colours. What does this show? Continue the experiment with the other foods. === Conclusion: === The findings show that if foods have starch in them, they will go black, while if they do not have starch in them, they will go orange. nw6h04cxmnnsqc1wbj1brtoedzilwh8 2690350 2690349 2024-12-05T08:41:54Z 82.219.7.128 2690350 wikitext text/x-wiki Food tests are tests for different biological molecules (food groups). Examples of food groups include protein, carbohydrates and starch. Different types of solutions, such as iodine, benedict's solution, and biuret are used to test for products in certain food groups. == Testing for Starch (experiment) == Iodine is used to test for starch in food products. This test involves different types of foods, which include pasta, biscuits, crisps, cereal hoops, salt, mustard powder and sugar, though any types of food (such as sweet sprinkles and condiments) will work in this experiment to test for starch. === Preparations: === Before conducting '''any''' biology experiment, make sure to define your dependent, independent and control variables. === Safety Precautions: === Iodine is very toxic to aquatic life and it is harmful to humans in large quantities, so take these precautions into account before conducting this experiment. Make sure you do not eat the food while it is being experimented on, because this may ruin the results of the experiment and may be harmful. === Conducting the Experiment: === Make sure you have prepared all of the required foods and iodine before conducting this experiment. Start by taking the food and adding iodine to one of the foods. Notice how one of the foods change colours. What does this show? Continue the experiment with the other foods. What do these findings show? === Conclusion: === The findings show that if foods have starch in them, the iodine turns black, while if they do not have starch in them, they will go orange. What are you able to learn from this experiment. sy5p3axmdhzvevq1zr21qwen5sjuuly 0 316964 2690289 2024-12-04T18:52:34Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Example |Text= The {{ Definitionlink |subgroups| |pm= }} of the integers are of the form{{{extra1|}}} {{ Mathcor|term1= \Z n |with|term2= n \geq 0 |pm=, }} due to {{ Factlink |Factname= Subgroups of Z/One generator/Fact |Nr= |pm=. }} The {{ Definitionlink |factor group| |pm= }} are denoted by {{ Math/display|term= {{op:Zmod|n|}} |pm= }} {{ Extra/Bracket |text={{Quotation2|{{mat|term= \Z |pm=}} modulo {{mat|term= n |pm=}}|}}| |pm=. }} For {{ Relat..." 2690289 wikitext text/x-wiki {{ Mathematical text/Example |Text= The {{ Definitionlink |subgroups| |pm= }} of the integers are of the form{{{extra1|}}} {{ Mathcor|term1= \Z n |with|term2= n \geq 0 |pm=, }} due to {{ Factlink |Factname= Subgroups of Z/One generator/Fact |Nr= |pm=. }} The {{ Definitionlink |factor group| |pm= }} are denoted by {{ Math/display|term= {{op:Zmod|n|}} |pm= }} {{ Extra/Bracket |text={{Quotation2|{{mat|term= \Z |pm=}} modulo {{mat|term= n |pm=}}|}}| |pm=. }} For {{ Relationchain |n || 0 || || || |pm=, }} this is just {{mat|term= \Z |pm=}} itself; for {{ Relationchain |n ||1 || || || |pm=, }} this is the {{ Definitionlink |trivial group| |pm=. }} In general, the equivalence relation on {{mat|term= \Z |pm=}} defined by the subgroup {{mat|term= \Z n |pm=}} is given in the way that {{ Mathcor|term1= a |and|term2= b |pm= }} are equivalent if and only if their difference {{mathl|term= a-b |pm=}} belongs to {{mat|term= \Z n |pm=,}} that is, if it is a multiple of {{mat|term= n |pm=.}} Therefore, {{ Extra/Bracket |text= {{ Relationchain/b | n |\geq| 1 || || || |pm= }}| |pm=, }} every integer number is equivalent to exactly one of the {{mat|term= n |pm=}} numbers {{ Math/display|term= 0,1,2 {{commadots|}} n-1 |pm= }} {{ Extra/Bracket |text=or, as we also say, {{Keyword|congruent modulo {{mat|term= n |pm=}}|pm=}}| |pm=, }} namely to the remainder upon division through {{mat|term= n |pm=.}} These remainders form a system of representatives for the factor group, and contains {{mat|term= n |pm=}} elements. The fact that the quotient mapping {{ Mapping/display |name= | \Z | {{op:Zmod|n|}} | a | [a] {{=|}} a {{modulo}} n |pm=, }} is a homomorphism might be expressed by saying that the remainder of a sum of two integers depends only on their remainders, not on the numbers themselves{{ Extra/{{{extra2|}}} |text=This holds also for the product of two numbers, meaning that this mapping is a ring homomorphism| |Ipm=.|Epm=. }} As an image of the {{ Definitionlink |cyclic group| |pm={{ Extra/{{{extra3|}}} |text={{:Group theory/Cyclic group/Definition|}} | |Ipm=|Epm= }} }} {{mat|term= \Z |pm=,}} the group {{mathl|term= {{op:Zmod|n|}} |pm=}} is also cyclic; {{mat|term= 1 |pm=}} {{ Extra/Bracket |text=but also {{mat|term= -1 |pm=}}| |pm= }} is always a generator. |Textform=Example |Category= }} l6pzcnoegas9w8t8ct4jtx8otqwsnax 0 316965 2690290 2024-12-04T18:57:10Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Definition |Text= Let {{mat|term= G |pm=}} be a {{ Definitionlink |group| |pm=, }} and let {{ Relationchain |H |\subseteq|G || || || |pm= }} denote a {{ Definitionlink |subgroup| |pm=. }} {{mat|term= H |pm=}} is called a {{Word of definition|normal subgroup|pm=}} if {{ Relationchain/display | xH ||Hx || || || |pm= }} holds for all {{ Relationchain | x |\in| G || || || |pm=, }} that is, if every {{ Definitionlink |left coset| |pm= }} of {{mat|term=..." 2690290 wikitext text/x-wiki {{ Mathematical text/Definition |Text= Let {{mat|term= G |pm=}} be a {{ Definitionlink |group| |pm=, }} and let {{ Relationchain |H |\subseteq|G || || || |pm= }} denote a {{ Definitionlink |subgroup| |pm=. }} {{mat|term= H |pm=}} is called a {{Word of definition|normal subgroup|pm=}} if {{ Relationchain/display | xH ||Hx || || || |pm= }} holds for all {{ Relationchain | x |\in| G || || || |pm=, }} that is, if every {{ Definitionlink |left coset| |pm= }} of {{mat|term= x |pm=}} coincides with the right coset of {{mat|term= x |pm=.}} |Textform=Definition |Category= |Word of definition=Normal subgroup }} 7fh35wdov64knifvb94hf6zmrg9nc4h 0 316966 2690291 2024-12-04T19:00:42Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Fact |Text= {{ Factstructure |Situation={{:Subgroup/Situation|pm=.}} |Condition= |Segue= |Conclusion= Then the following statements are equivalent. {{ Enumeration3 |{{mat|term= H |pm=}} is a {{ Definitionlink |normal subgroup| |pm= }} of {{mat|term= G |pm=.}} |We have {{ Relationchain | xhx^{-1} |\in| H || || || |pm= }} for all {{ Mathcor|term1= x \in G |and|term2= h \in H |pm=. }} |{{mat|term= H |pm=}} is invariant under every {{ Definitionlink..." 2690291 wikitext text/x-wiki {{ Mathematical text/Fact |Text= {{ Factstructure |Situation={{:Subgroup/Situation|pm=.}} |Condition= |Segue= |Conclusion= Then the following statements are equivalent. {{ Enumeration3 |{{mat|term= H |pm=}} is a {{ Definitionlink |normal subgroup| |pm= }} of {{mat|term= G |pm=.}} |We have {{ Relationchain | xhx^{-1} |\in| H || || || |pm= }} for all {{ Mathcor|term1= x \in G |and|term2= h \in H |pm=. }} |{{mat|term= H |pm=}} is invariant under every {{ Definitionlink |inner automorphism| |pm= }} of {{mat|term= G |pm=.}} }} |Extra= }} |Textform=Fact |Category= |Factname= |Request= }} h5z06jz8c22brfhu1su4jna9g8riqu1 0 316967 2690292 2024-12-04T19:03:41Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= (1) means for given {{ Relationchain | h |\in| H || || || |pm= }} that we can write {{ Relationchain | xh ||\tilde{h}x || || || |pm= }} with some {{ Relationchain | \tilde{h} |\in| H || || || |pm= }}. Multiplication by {{mat|term= x^{-1} |pm=}} from the right yields {{ Relationchain |xhx^{-1} ||\tilde{h} |\in|H || || |pm=; }} therefore, {{mat|term= (2) |pm=}} holds. Reading this argume..." 2690292 wikitext text/x-wiki {{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= (1) means for given {{ Relationchain | h |\in| H || || || |pm= }} that we can write {{ Relationchain | xh ||\tilde{h}x || || || |pm= }} with some {{ Relationchain | \tilde{h} |\in| H || || || |pm= }}. Multiplication by {{mat|term= x^{-1} |pm=}} from the right yields {{ Relationchain |xhx^{-1} ||\tilde{h} |\in|H || || |pm=; }} therefore, {{mat|term= (2) |pm=}} holds. Reading this argument backwards gives the implication {{mat|term= (2) \Rightarrow (1) |pm=.}} Moreover, {{mat|term= (2) |pm=}} is an explicit reformulation of {{mat|term= (3) |pm=.}} |Closure= }} |Textform=Proof |Category=See }} so05uf7zg91mk3k0i0dvqea0cdk9w52 0 316968 2690293 2024-12-04T19:09:39Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Example |Text= We consider the {{ Definitionlink |permutation group| |Context=| |pm= }} {{ Relationchain |G ||S_3 || || || |pm= }} for a set with three elements, that is, {{mat|term= S_3 |pm=}} consists of all bijective mappings of the set {{mathl|term= \{1,2,3\} |pm=}} to itself. The trivial group {{mat|term= \{ \operatorname{id} \} |pm=}} and the whole group are {{ Definitionlink |normal subgroups| |pm=. }} The subset {{ Relationchain |H || \{ \..." 2690293 wikitext text/x-wiki {{ Mathematical text/Example |Text= We consider the {{ Definitionlink |permutation group| |Context=| |pm= }} {{ Relationchain |G ||S_3 || || || |pm= }} for a set with three elements, that is, {{mat|term= S_3 |pm=}} consists of all bijective mappings of the set {{mathl|term= \{1,2,3\} |pm=}} to itself. The trivial group {{mat|term= \{ \operatorname{id} \} |pm=}} and the whole group are {{ Definitionlink |normal subgroups| |pm=. }} The subset {{ Relationchain |H || \{ \operatorname{id} \, , \varphi \} || || || |pm=, }} where {{mat|term= \varphi|pm=}} is the element that swops {{ Mathcor|term1= 1 |and|term2= 2 |pm= }} and fixes {{mat|term= 3 |pm=,}} is a {{ Definitionlink |subgroup| |pm=. }} However, it is not a normal subgroup. To show this, let {{mat|term= \psi|pm=}} denote the bijection that fixes {{mat|term= 1 |pm=}} and swops {{ Mathcor|term1= 2 |and|term2= 3 |pm=. }} The inverse of {{mat|term= \psi|pm=}} is {{mat|term= \psi|pm=}} itself. The {{ Definitionlink |conjugation| |Context=| |pm= }} {{ Relationchain | \psi \varphi \psi^{-1} || \psi \varphi \psi || || || |pm= }} is the mapping that sends {{ Mathcor|term1= 1 |to|term2= 3 |pm=, }} {{ Mathcor|term1= 2 |to|term2= 2 |pm=, }} and {{ Mathcor|term1= 3 |to|term2= 1 |pm=. }} This bijection does not belong to {{mat|term= H |pm=.}} |Textform=Example |Category= }} tjbi528egt6di3le77enlv3u89og81h 0 316969 2690294 2024-12-04T19:11:36Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Fact |Text= {{ Factstructure |Situation={{:Group homomorphism/Situation|pm=.}} |Condition= |Segue= |Conclusion= Then the {{ Definitionlink |kernel| |Context=group |pm= }} {{mathl|term= {{op:Kern| \varphi |}} |pm=}} is a {{ Definitionlink |normal subgroup| |pm= }} in {{mat|term= G |pm=.}} |Extra= }} |Textform=Fact |Category= |Factname= |Request=Kernel of a group homomorphism is ... }}" 2690294 wikitext text/x-wiki {{ Mathematical text/Fact |Text= {{ Factstructure |Situation={{:Group homomorphism/Situation|pm=.}} |Condition= |Segue= |Conclusion= Then the {{ Definitionlink |kernel| |Context=group |pm= }} {{mathl|term= {{op:Kern| \varphi |}} |pm=}} is a {{ Definitionlink |normal subgroup| |pm= }} in {{mat|term= G |pm=.}} |Extra= }} |Textform=Fact |Category= |Factname= |Request=Kernel of a group homomorphism is ... }} polewk8i6kfghqhj48vcjlnvot8azhe 0 316970 2690296 2024-12-04T19:15:25Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= By {{ Factlink |Factname= Group homomorphism/Kernel/Subgroup/Fact |Nr= |pm=, }} we know that the kernel is a {{ Definitionlink |Premath= |subgroup| |Context=| |pm=. }} We use {{ Factlink |Factname= Normal subgroup/Characterization/Fact |pm=. }} Hence, let {{ Relationchain | x |\in| G || || || |pm= }} be arbitrary, and {{ Relationchain | h |\in| {{op:Kern|\varphi|}} || || || |pm=. }} The..." 2690296 wikitext text/x-wiki {{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= By {{ Factlink |Factname= Group homomorphism/Kernel/Subgroup/Fact |Nr= |pm=, }} we know that the kernel is a {{ Definitionlink |Premath= |subgroup| |Context=| |pm=. }} We use {{ Factlink |Factname= Normal subgroup/Characterization/Fact |pm=. }} Hence, let {{ Relationchain | x |\in| G || || || |pm= }} be arbitrary, and {{ Relationchain | h |\in| {{op:Kern|\varphi|}} || || || |pm=. }} Then {{ Relationchain/display | \varphi {{mabr| xh x^{-1} |}} || \varphi(x) \varphi(h) \varphi {{mabr| x^{-1} |}} || \varphi(x) e_H\varphi {{mabr| x^{-1} |}} || \varphi(x) \varphi(x)^{-1} || e_H |pm=, }} therefore, {{mathl|term= xh x^{-1} |pm=}} belongs to the kernel. |Closure= }} |Textform=Proof |Category=See }} psn6haas1py961ez5lrr4pcejlfx9fc 0 316971 2690297 2024-12-04T19:22:03Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Fact |Text= {{ Factstructure |Situation={{:Normal subgroup/Situation|pm=.|}} Let {{mathl|term= G/H |pm=}} be the set of all {{ Definitionlink |cosets| |pm= }} {{ Extra/Bracket |text=the quotient set| |pm=, }} and let {{Mapping/display |name=q | G | G/H | g | [g] |pm=, }} denote the {{ Definitionlink |canonical projection| |pm=. }} |Condition= |Segue= |Conclusion= Then there exists a uniquely determined group structure on {{mathl|term= G/H |pm=}} su..." 2690297 wikitext text/x-wiki {{ Mathematical text/Fact |Text= {{ Factstructure |Situation={{:Normal subgroup/Situation|pm=.|}} Let {{mathl|term= G/H |pm=}} be the set of all {{ Definitionlink |cosets| |pm= }} {{ Extra/Bracket |text=the quotient set| |pm=, }} and let {{Mapping/display |name=q | G | G/H | g | [g] |pm=, }} denote the {{ Definitionlink |canonical projection| |pm=. }} |Condition= |Segue= |Conclusion= Then there exists a uniquely determined group structure on {{mathl|term= G/H |pm=}} such that {{mat|term= q |pm=}} is a {{ Definitionlink |group homomorphism| |pm=. }} |Extra= }} |Textform=Fact |Category= |Factname= |Request=Normal subgroup and factor group }} gkrgeme64k6qvwvytdpn0cvwtaf7oab 0 316972 2690298 2024-12-04T19:23:35Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Situation{{{opt|}}} |Text= Let {{mat|term={{{G|G}}} |}} be a {{ Definitionlink |group| |pm=, }} and let {{ Vergleichskette | {{{H|H}}} |\subseteq| {{{G|G}}} || || || }} be a {{ Definitionslink |normal subgroup| |pm={{{pm|}}}}} |Textform=Situation |}}" 2690298 wikitext text/x-wiki {{ Mathematical text/Situation{{{opt|}}} |Text= Let {{mat|term={{{G|G}}} |}} be a {{ Definitionlink |group| |pm=, }} and let {{ Vergleichskette | {{{H|H}}} |\subseteq| {{{G|G}}} || || || }} be a {{ Definitionslink |normal subgroup| |pm={{{pm|}}}}} |Textform=Situation |}} q0mr1pxhh6dhxxe6kuaajbvyzxoxwsz 2690299 2690298 2024-12-04T19:23:51Z Bocardodarapti 289675 2690299 wikitext text/x-wiki {{ Mathematical text/Situation{{{opt|}}} |Text= Let {{mat|term={{{G|G}}} |}} be a {{ Definitionlink |group| |pm=, }} and let {{ Relationchain | {{{H|H}}} |\subseteq| {{{G|G}}} || || || }} be a {{ Definitionlink |normal subgroup| |pm={{{pm|}}}}} |Textform=Situation |}} s84r0aqcu200m8nz1ihc4hy6o057191 0 316973 2690300 2024-12-04T19:33:13Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= Since the canonical projection shall be a group homomorphism, the operation must fulfill {{ Relationchain/display | [x] [y] || [xy] || || || |pm=. }} We have to show that this rule gives a well-defined operation on {{mathl|term= G/H|pm=,}} that is, is independent of the choice of representatives. Hence, we have to show for {{ mathcor|term1= [x] {{=|}} [x'] |and|term2= [y]{{=|}}[y'] |pm= }}..." 2690300 wikitext text/x-wiki {{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= Since the canonical projection shall be a group homomorphism, the operation must fulfill {{ Relationchain/display | [x] [y] || [xy] || || || |pm=. }} We have to show that this rule gives a well-defined operation on {{mathl|term= G/H|pm=,}} that is, is independent of the choice of representatives. Hence, we have to show for {{ mathcor|term1= [x] {{=|}} [x'] |and|term2= [y]{{=|}}[y'] |pm= }} that {{ Relationchain | [xy] || [x'y'] || || || |pm= }} holds. Due to the condition, we can write {{ mathcor|term1= x'{{=}}xh |and|term2= hy'{{=}} \tilde{h} y{{=}}yh' |pm= }} with {{ Relationchain | h, \tilde{h}, h' |\in| H || || || |pm=. }} Therefore, {{ Relationchain/display | x'y' || (xh)y' || x(hy') || x(yh') || xyh' |pm=. }} This means {{ Relationchain | [xy] || [x'y'] || || || |pm=. }} From this, the group property, the homomorphism property of the projection and the uniqueness follows. |Closure= }} |Textform=Proof |Category=See }} mzsofnqwx2cb1xathtjqy139kscrpg9 0 316974 2690301 2024-12-04T19:39:14Z Bocardodarapti 289675 New resource with "{{ Mathematical text/Definition |Text= {{:Normal subgroup/Situation|pm=.}} The {{ Definitionlink |quotient set| |pm= }} {{ Math/display|term= G/H |pm=, }} endowed with the {{ Extra/Bracket |text=according to {{ Factlink |Factname= Group/Normal subgroup/Residue class group/Fact |pm= }}| |Ipm=|Epm= }} uniquely determined group structure, is called the {{Word of definition|factor group of |pm=}} {{mat|term= G |pm=}} {{Word of definition|modulo|pm=}} {{mat|term= H |pm=.}} T..." 2690301 wikitext text/x-wiki {{ Mathematical text/Definition |Text= {{:Normal subgroup/Situation|pm=.}} The {{ Definitionlink |quotient set| |pm= }} {{ Math/display|term= G/H |pm=, }} endowed with the {{ Extra/Bracket |text=according to {{ Factlink |Factname= Group/Normal subgroup/Residue class group/Fact |pm= }}| |Ipm=|Epm= }} uniquely determined group structure, is called the {{Word of definition|factor group of |pm=}} {{mat|term= G |pm=}} {{Word of definition|modulo|pm=}} {{mat|term= H |pm=.}} The elements {{ Relationchain | [g] |\in| G/H || || || |pm= }} are called {{Word of definition|residue classes|pm=.}} For a residue class {{mat|term=[g] |pm=,}} every element {{ Mathcor|term1= g' \in G |with|term2= [g'] = [g] |pm= }} is called a {{Word of definition|representative|pm=}} of {{mat|term=[g] |pm=.}} |Textform=Definition |Category= |Word of definition=Factor group }} sw86roqu0kzz64w484p8ywlwlyl3mf3 2 316975 2690345 2024-12-05T07:09:45Z RockTransport 2992610 New resource with "{{User British Citizen}}" 2690345 wikitext text/x-wiki {{User British Citizen}} oygivamrnxw6sjx7m55gaufhi7jd1yq 2690346 2690345 2024-12-05T07:14:24Z RockTransport 2992610 2690346 wikitext text/x-wiki {{User British Citizen}} {{user language|en|N}} {{user language|id|4}} {{user language|de|2}} {{user language|fr|2}} {{user language|es|1}} {{user language|zh|1}} Hello there! I am a student currently living in the United Kingdom. I have helped out on another Wikimedia projects such as Wikivoyage and Commons, and I am making great contributions. a9busnrsg63afuvtfh5dwm3rcnn9so7 0 316976 2690347 2024-12-05T08:24:32Z Chair-x86 2994439 New resource with " Just like how C is compatible with C++, Assembly is compatible with C. Though unlike the C-C++ relation, C actually almost IS assembly. Whenever you compile C code the compiler first turns it into a intermediate language and then turns that intermediate language into assembly based on you system details. The assembly code later gets assembled and linked to become the actual binary file. So this mainly means two things: # you can generate assembly code from C code. # C..." 2690347 wikitext text/x-wiki Just like how C is compatible with C++, Assembly is compatible with C. Though unlike the C-C++ relation, C actually almost IS assembly. Whenever you compile C code the compiler first turns it into a intermediate language and then turns that intermediate language into assembly based on you system details. The assembly code later gets assembled and linked to become the actual binary file. So this mainly means two things: # you can generate assembly code from C code. # C functions and variables are actually assembly variables and you can call C functions in assembly and vise versa (in most cases) == Generating assembly code from C code == The gcc command:<syntaxhighlight lang="bash"> gcc -S FILE.c -o FILE.s </syntaxhighlight>(Where 'FILE' is the name of you file) This command will take your C file and compile it as normal, but will stop when it has an assembly file that then becomes the output. == Using C and Assembly Functions Interchangeably == Because C libraries and programs get compiled to assembly / machine-code, assembly programs can access those compiled C functions. The same way C functions can, after they are compiled access assembly functions. == Inline Assembly == C has the 'asm' function that takes an assembly code block as a parameter, the compiler will preserve this code during compilation for it to end up as part of the resulting assembly code. The "Hello, World!" program written this way would look like this:<syntaxhighlight lang="c" line="1" start="3">int main(){ asm(" .data foo: .string "Hello, World!\n" .text leaq foo(%rip), %rax movq %rax, %rdi call puts@PLT movl $0, %eax ret ");</syntaxhighlight> logilnslprjlfalyugoo8zjau6q7fa8 2690348 2690347 2024-12-05T08:25:49Z Chair-x86 2994439 2690348 wikitext text/x-wiki Just like how C is compatible with C++, Assembly is compatible with C. Though unlike the C-C++ relation, C actually almost IS assembly. Whenever you compile C code the compiler first turns it into a intermediate language and then turns that intermediate language into assembly based on you system details. The assembly code later gets assembled and linked to become the actual binary file. So this mainly means two things: # you can generate assembly code from C code. # C functions and variables are actually assembly variables and you can call C functions in assembly and vise versa (in most cases) == Generating assembly code from C code == The gcc command:<syntaxhighlight lang="bash"> gcc -S FILE.c -o FILE.s </syntaxhighlight>(Where 'FILE' is the name of you file) This command will take your C file and compile it as normal, but will stop when it has an assembly file that then becomes the output. == Using C and Assembly Functions Interchangeably == Because C libraries and programs get compiled to assembly / machine-code, assembly programs can access those compiled C functions. The same way C functions can, after they are compiled access assembly functions. == Inline Assembly == C has the 'asm' function that takes an assembly code block as a parameter, the compiler will preserve this code during compilation for it to end up as part of the resulting assembly code. The "Hello, World!" program written this way would look like this:<syntaxhighlight lang="c" line="1" start="3">int main(){ asm(" .data foo: .string "Hello, World!\n" .text leaq foo(%rip), %rax movq %rax, %rdi call puts@PLT movl $0, %eax ret ");</syntaxhighlight>Next: [[AT&T-X86: Registers & exit codes|Registers & exit codes]] mr4yvba5b4uqbjjcp14gum4rbbq9epn 6 316977 2690353 2024-12-05T11:09:03Z Young1lim 21186 {{Information |Description=Work2.1A: Packages (20241203 - 20241202) |Source={{own|Young1lim}} |Date=2024-12-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690353 wikitext text/x-wiki == Summary == {{Information |Description=Work2.1A: Packages (20241203 - 20241202) |Source={{own|Young1lim}} |Date=2024-12-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} d5iu2fkctq1p0grnyv6oubq57meg098 6 316978 2690355 2024-12-05T11:09:44Z Young1lim 21186 {{Information |Description=Work2.1A: Packages (20241204 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690355 wikitext text/x-wiki == Summary == {{Information |Description=Work2.1A: Packages (20241204 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 45ggxv8c5zrndqd11yawglyfwc0pbex 2690356 2690355 2024-12-05T11:18:55Z Young1lim 21186 Young1lim uploaded a new version of [[File:Python.Work2.Package.1A.20241204.pdf]] 2690355 wikitext text/x-wiki == Summary == {{Information |Description=Work2.1A: Packages (20241204 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 45ggxv8c5zrndqd11yawglyfwc0pbex 2690357 2690356 2024-12-05T11:19:47Z Young1lim 21186 Young1lim uploaded a new version of [[File:Python.Work2.Package.1A.20241204.pdf]] 2690355 wikitext text/x-wiki == Summary == {{Information |Description=Work2.1A: Packages (20241204 - 20241203) |Source={{own|Young1lim}} |Date=2024-12-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 45ggxv8c5zrndqd11yawglyfwc0pbex 6 316979 2690359 2024-12-05T11:20:30Z Young1lim 21186 {{Information |Description=Work2.1A: Packages (20241205 - 20241204) |Source={{own|Young1lim}} |Date=2024-12-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690359 wikitext text/x-wiki == Summary == {{Information |Description=Work2.1A: Packages (20241205 - 20241204) |Source={{own|Young1lim}} |Date=2024-12-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} qne3h8rlnvierfvke0jgxp8tt50s9d9 0 316980 2690360 2024-12-05T11:38:15Z Bocardodarapti 289675 New resource with " {{Subtitle|Homomorphism theorems}} {{:Group theory/Homomorphism theorems/Examples/Section}} {{ inputfactproof |Group theory/Isomorphy theorem for residue class groups/Fact|Theorem|| }} In short, we have {{ Relationchain/display | G/H || (G/N)/(H/N) || || || |pm=. }} {{Subtitle|Residue class rings}} On the residue class group for a normal subgroup in a group, there are quite often additional structures available, if the group and the normal subgroup fulfill certain..." 2690360 wikitext text/x-wiki {{Subtitle|Homomorphism theorems}} {{:Group theory/Homomorphism theorems/Examples/Section}} {{ inputfactproof |Group theory/Isomorphy theorem for residue class groups/Fact|Theorem|| }} In short, we have {{ Relationchain/display | G/H || (G/N)/(H/N) || || || |pm=. }} {{Subtitle|Residue class rings}} On the residue class group for a normal subgroup in a group, there are quite often additional structures available, if the group and the normal subgroup fulfill certain properties. In the next lecture, we will look at residue class spaces for a linear subspace. Here, we discuss briefly residue class rings for an ideal in a commutative ring. We recall the definition of a ring homomorphism. {{ inputdefinition |Ring theory/Ring homomorphism/Definition|| }} {{ inputfactproofexercise |Commutative ring/Ring homomorphism/Kernel/Ideal/Fact|Lemma|| || }} {{:Commutative ring/Residue class ring/Group known/Introduction/Section}} {{Subtitle|The residue class rings of {{mat|term= \Z |pm=}}}} {{ inputimage |Anillo cíclico|png | 300px {{!}} {{!}} |epsname=Anillo_cíclico Romero Schmidtke |User=FrancoGG |Domain=es.wikipedia.org |License=CC-BY-SA-3.0 }} We know already the residue class groups {{mathl|term= {{op:Zmod|d}} |pm=;}} they are cyclic groups of order {{mat|term= d |pm=.}} Moreover, these groups get now also a ring structure. {{ inputfactproofhere |Residue class rings of Z/Ring homomorphism/Fact|Corollary|| |Proof text=This is a spezial case of the considerations above. }} {{ inputfactproofexercise |Residue class rings of Z/Field/Integer/Prime number/Fact|Theorem|| }} The residue class rings {{ Relationchain | S || K[X]/(P) || || || |pm= }} are also quite easy to understand. If {{mat|term= P |pm=}} has degree {{mat|term= d |pm=,}} then every residue class in {{mat|term= S |pm=}} is represented by a unique polynomial of degree {{mathl|term= <d |pm=.}} This polynomial is the remainder that we get by dividing through {{mat|term= P |pm=.}} bgv194448258tooowxiob6l8fnokezw 0 316981 2690361 2024-12-05T11:52:21Z Bocardodarapti 289675 New resource with "{{ Mathematical section{{{opt|}}} |Content= {{ inputfactproof |Group homomorphism/Homomorphism theorem/Surjective and kernel/Fact|Theorem|| }} {{ inputexample |Group homomorphism/Homomorphism theorem/Residue class groups of Z/Example|| }} The mapping constructed in this theorem is called {{Word of definition|induced mapping|pm=}} or {{Word of definition|induced homomorphism|pm=,}} and the theorem is called the {{Keyword|theorem about the induced homomorphism|pm=.}} {{..." 2690361 wikitext text/x-wiki {{ Mathematical section{{{opt|}}} |Content= {{ inputfactproof |Group homomorphism/Homomorphism theorem/Surjective and kernel/Fact|Theorem|| }} {{ inputexample |Group homomorphism/Homomorphism theorem/Residue class groups of Z/Example|| }} The mapping constructed in this theorem is called {{Word of definition|induced mapping|pm=}} or {{Word of definition|induced homomorphism|pm=,}} and the theorem is called the {{Keyword|theorem about the induced homomorphism|pm=.}} {{ inputfactproof |Group homomorphism/Surjective and residue class group/Fact|Corollary|| }} {{ inputexample |Cyclic group/Z mod n/Example|| }} {{ inputexample |R mod Z2 pi/Isomorphism theorem/Example|| }} {{ inputexample |C/Cx/Exponential mapping/Isomorphism theorem/Example|| }} {{ inputexample |General and special linear group/Determinant/Isomorphism theorem/Example|| }} {{ inputfactproof |Group homomorphism/Factorization/Fact|Theorem|| }} This statement is often briefly expressed by saying: {{Emphasize|term=image {{mat|term= =|pm=}} preimage modulo kernel|pm=.}} |Textform=Section |Category= |}} l27fyaalucgqa78d1cgjfprz0hsbyxv