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It would be a good introduction into how to care for plants. Of course, you don't want me in it- I've killed catusus before!--[[User:Rayc|Rayc]] 14:58, 19 August 2006 (UTC) :Nice idea, but I think that should probably be separate from the research project :). I do write wikibooks on gardening techniques, so wouldn't mind facilitating classes like that. ----[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 16:16, 19 August 2006 (UTC) == How to contribute == You should write up a section on how to contribute. I was thinking something like: # Find a flower that you pass by every day # If you own a digital camera, take a picture of it # Upload (steps to upload) # We (other users) will analys what type of plant it # Note which day you saw it first bloom # Note your location --[[User:Rayc|Rayc]] 04:48, 1 September 2006 (UTC) :That would actually be a great way to help people identify the flowers... good thinkin! --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 04:52, 1 September 2006 (UTC) ::I'm going to try and upload to see if it is workin. Never uploaded anything before!--[[User:Rayc|Rayc]] 05:00, 1 September 2006 (UTC) Can you tell what these are then?--[[User:Rayc|Rayc]] 05:04, 1 September 2006 (UTC) [[Image:Fieldofdlions.jpg|thumb|left]] :::LOL! Yup, those are [[w:Taraxacum officinale]] :). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 11:14, 1 September 2006 (UTC) == Mushrooms == Well, I was supposed to take a picture of some mushrooms while on vacation, but I can't find them on my PC anymore. Anyway, they looked orange and they were growing on a log with some white bark. Any idea what it was? --[[User:HappyCamper|HappyCamper]] 23:41, 6 October 2006 (UTC) == Historical data == Would participants in this project be interested in using herbarium specimens to record plant blooming dates? I'm running a distributed [http://herbariaunited.org/atHome/ project] to document museum herbarium specimens. I had already been considering including a [http://en.wikipedia.org/wiki/Phenology phenological] slant to the project, and having seen this project on wikiversity am wondering if collaboration is possible. As brief background, [http://herbariaunited.org/atHome/ herbaria@home] is a web based project to digitise university and museum [http://en.wikipedia.org/wiki/Herbarium herbarium] specimens. Currently we record the species, collector, site and date of collection, but I am thinking of also requesting data on the flowering state of the specimens. This might provide data which is directly relevant to the 'Bloom clock' project. It is worth noting that for taxonomic reasons herbarium specimens are usually flowering, so very many specimens would be relevant. Potentially this could add data from millions of specimens, with worldwide coverage. A good reason to look at historical data could be to study climate change, as flowering time and distribution can be a good way to track seasonal changes. e.g. http://www.bbc.co.uk/climate/evidence/phenology.shtml I'd be very interest to hear anyone's thoughts on this. -- [[User:Japonicus|Japonicus]] 17:07, 12 October 2006 (UTC) ^{[[User talk:Japonicus| talk ]]} ==Getting ready for the spring== Where should people look in the spring for blooming flowers? What does this project want besides the picture (temperature, location, soil quality, etc..)--[[User:Rayc|Rayc]] 06:25, 2 December 2006 (UTC) :I think location is probably the most important thing for now. Regions about the size of a US "County" would be best, since it's small enough to be fairly consistent in climate (though places like California with a million niches might need to be more specific). Temperature and rainfall might be helpful, soil quality shouldn't matter. Maybe the clock users should try to keep up a page on this? If so, maybe in their userspace... some sort of userbox which auto-linked to a certain subpage of the userspace might work. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 15:37, 18 December 2006 (UTC) == resetting the clock == A couple notes I've been thinking of the past few days: *I think it might be good to reset the clock on [[w:Solstice|Solstice]] (thus twice per year), as a not-so-arbitrary division on the year (plants that take their "cues" from day length would presumably bloom predictably in either hemisphere at a given latitude). This would be strange this first year because it means any southern hemisphere contributors would be starting at midsummer, but that will be fixed up 6 months from now (when the next clock is started, and the northern hemisphere would then have a clock starting in midsummer). *Limit the plants "watched" to weeds, wildflowers, and perennials (including trees and shrubs) that have been planted in the ground for at least one year. This would limit any skewed results from plants that might have been grown in nurseries of different climates. *Either tie this to other clocks, or rename the project as a "life clock". The reason for this is to also include things like fungi, insects, and perhaps migrating animals. In any case, I'm hoping to re-announce it on the colloquium and perhaps on some of the other wikis, though I'm wondering too whether there are web-forums that do this sort of thing where other interested people might be. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 15:33, 18 December 2006 (UTC) == Pink Blossoms in January! == Hmmm. I'll try to get pictures and more info on this beauty. For now, I'm just registering the time and place. I'm in [[Kentucky]] [[USA]] where you don't see many blossoms this time of year. Interesting project! [[User:CQ|CQ]] 14:44, 11 January 2007 (UTC) :Believe it or not, we've had Cherries blooming in Pennsylvania too! --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 12:56, 13 January 2007 (UTC) I noted a prunus blossoming monday, Jan 15th. Dont know exact species. Location: Gouda, Netherlands. To do anything with the data, we'd need a datastructure so data can be statistically analyzed. Ask for wikidata. You'll need entities for: * User (id, name) * species (id, bot. name,) * observation (species, observed by user, date observation, coordinates of observation, time of observation, identification confirmed by user, temperature, altitude) For while these are not availabl, I suggest a semicoln separated list is used: ''species''; Observed by User; , date observation; coordinate N; coordinate E; time of observation; identification confirmed by user; temperature; altitude;<br> ''vinca major''; [[:user:TeunSpaans|TeunSpaans]]; 2007-01-17; 52°05' N; 4°18' E; 18:00; ; 10C;0 [[User:TeunSpaans|TeunSpaans]] 07:29, 18 January 2007 (UTC) (funny, btw, the en: and nl: wikis have different coordinates for The Hague) :I think just having as much data as possible is the best way to go, as long as we have a fairly good idea of the location. As long as people gove some good information on the [[Bloom_clock_project/Contributors|contributors]] page, it should be reasonably useful (a bot can be used to organize the data at the end of the 6-month run). The important thing though is to make sure that if you're logging from a location other than your usual flower-spotting place, it needs to ba prominently noted (using a standard notation the bot will understand). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 11:14, 19 January 2007 (UTC) == data mining to discover from images? == Assuming that images of naturally-occurring flowers are available with time and date information, would it be possible to use what might be a huge number of timestamped bloom images to create the bloom clock? [[User:JPatrickBedell|JPatrickBedell]] 00:00, 15 January 2007 (UTC) :I'm not sure what you mean... your own photos? It would be a good idea, as long as we're sure ''where'' the photo was taken, as well as ''when''. Flowers will bloom at different times according to latitude (including hemisphere), altitude, and other local differences from year to year. For example: [[w:Crocus|Crocus]]es are going to bloom a lot earlier in Georgia than they do in Ontario. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 11:24, 15 January 2007 (UTC) == Drumb up some pictures == So it's to be spring soon. Hopefully. I'm going to see if I can get some people to start taking pictures by placing a notice on the community portal. Be ready for them.--[[User:Rayc|Rayc]] 18:26, 5 February 2007 (UTC) :Thanks Rayc!--[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 20:45, 5 February 2007 (UTC) ::And now I have a bunch of pictures for different dates. I'll upload them when I have time.--[[User:Rayc|Rayc]] 21:32, 29 March 2007 (UTC) == Making the clock pages prettier? == I'm wondering if it would be too much of an inconvenience for slow-connection folks to have pictures on the lists? If there's no picture, we could use [[:Image:No image available.svg]]... and maybe include that in the template. I was also thinking that maybe the logging for each particular plant should actually be a separate page, with the main page either transcluding or linking those pages. That way when plants are clearly out of season, they could be taken off the page to keep the list shorter, and could also allow people using common names to reach the same spot as those using binomials. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 13:54, 1 April 2007 (UTC) == Moving towards a transclusion model... == I feel like I'm talking to myself sometimes :). I think I've figured out a way to make the BCP better as both a teaching, learning, and data-collecting tool, but it's boggling my mind as to how I can explain it all. The strategy I have in mind involves knowledgeable people doing a lot of work, but it will help people learn, help people collect data, and help people teach. I've been ruminating for weeks -- and experimenting for a day -- on how to make the bloom clock more educational, more fun to use, more interesting to look at, and more useful as a data collection tool. The solution I've come up with is using page transclusion, and an additional template. I'm having a very hard time expressing my thoughts today (here and elsewhere), but here's what I have in mind: #use '''''several''''' lists to accomodate contributors with various levels of knowledge, including a "by scientific name" list (for people who know plants well), a "by common name" list (for people who know their plants, but don't speak botanical latin), and "by color and type" lists for those who really don't know much about plants, but want to learn. #keep the bcp template, with the understanding that that template will be replaced by transculded pages that use the bcp2 template. Still having expression problems, but maybe someone sees what I'm saying? --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 21:23, 8 April 2007 (UTC) :You're right SB Johnny, this is a hell-of-a-lotta work. I think the [[MediaWiki Project]] should get involved, as there a lot of things about MediaWiki as a database engine that lend themselves to a project like this. I posted below about "bloom clock dynamics" and I'm back-tracking your progress so far. I know what you mean by "talking to myself" and "boggling my mind" though, so take heart! I'll keep thinking about how to contribute to the BCP getting familiar with the structure. -- [[User:CQ|CQ]] 21:06, 23 May 2007 (UTC) == Documented approximate blooming times == Some Wikipedia articles have approximate blooming times from various sources (I encountered them when entering public domain tree info). For classifying those there are categories such as [http://en.wikipedia.org/wiki/Category:Mid_winter_flowers Category:Mid winter flowers]. See the category's parent cat for a summary. ([[User:207.195.192.45|207.195.192.45]] 02:15, 20 May 2007 (UTC)) == Bloom clock dynamics == This is a facinating project. I've started a course for '''[[generating dynamic content with MediaWiki]]''' and this would be an excellent content-rich application for getting some "hands-on" with dynamic content. Take a look and see if you get any ideas about different ways to implement "bloom clock dynamics" using CURRENT variables, parser functions and maybe javascript, perl or other scripting magic. I can envision an "Almanac" approach, but it's very scetchy at this point. -- [[User:CQ|CQ]] 20:51, 23 May 2007 (UTC) :At this point my primary interest is in collecting data, and finding ways to make data-collection fun. I'll hopefully have time this weekend to fill in some details, but I'm talking to a school and a conservation group about getting it more kid-friendly and interesting (my fellow wikibookian Whiteknight, who lives not so far from me, has volunteered to help me "train" people for this). Stay tuned... too busy and tired now to wax on (I'm a farmer, and it's spring)!--[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User talk:SB_Johnny|{{font|color=green|talk}}]]} 00:03, 24 May 2007 (UTC) == Collaborate with Wikispecies? == We could link each plant's log page to it's corresponding Wikispecies page. After all, they already link to Wikipedia, Wikimedia Commons, and Wikibooks. --[[User:Luai lashire|Luai lashire]] 18:22, 15 August 2007 (UTC) :If I understand correctly, isn't Wikispecies just a collection of metadata? Most Wikipedia articles have taxoboxes which achive pretty much the same purpose. I don't have any objection to adding links to Wikispecies in the templates, but I wonder how useful it can be to a bloom clock user. --[[User talk:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[meta:Wikimedia Pennsylvania|{{font|color=green|PA!}}]]} 12:55, 19 August 2007 (UTC) == How goes the bloom clock? == Looks like you've got a lot of data. Any analysis of it yet?--[[User:Rayc|Rayc]] 20:56, 6 September 2007 (UTC) :Not enough data for analysis yet, but we do have the [[Bloom Clock/Keys|Keys]] up and running for a few regions. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 10:23, 10 September 2007 (UTC) == Growing ornamental plants == I think we are automatically excluding growing ornamental plants and this stuff. Am I right?--[[User:Juan|Juan]] 15:03, 4 October 2007 (UTC) :Nope :). If it's growing outside, it's fair game. Probably about 1/2 of the plants on the clock are ornamentals and/or crops. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 17:48, 4 October 2007 (UTC) But do you know, that gardeners can actually offer flowering plants hole the year. Thats what they are doing. They are just replanting growing plants at the same plot.--[[User:Juan|Juan]] 18:16, 4 October 2007 (UTC) :I know (I'm a professional gardener :) ), but someone using the clock keys isn't necessarily going to know which flower is an annual and which is a perennial (or even a weed or a wildflower). While not necessarily helpful for the research aspect of the clock, logs of garden plants do help the educational aspects.--[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 20:21, 4 October 2007 (UTC) So what is the recomendation for me as a person who can differ between these plants? Anyway, I thing I will do just "wild" ones:-)--[[User:Juan|Juan]] 20:23, 4 October 2007 (UTC) :I've been using categories (soon to be replaced by templates) for this purpose in my region. See [[:Category:BCP/SEPA/NP]] (native plants), [[:Category:BCP/SEPA/IP]] (invasive plants), and [[:Category:BCP/SEPA/GP]] (garden plants). I haven't done anything with these yet, but will make an annotated flora next week (once the cats are replaced by templates... [[user:Mike's bot account]] will soon do the [[b:User_talk:Mike.lifeguard#Got_time_for_another_run_on_WV?|changes]]). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 12:31, 5 October 2007 (UTC) == Plants of special colection == Can I write down data also about plants from different regions, which cant be find in the wild. These plants from other regions, we have on our school fields.--[[User:Juan|Juan]] 11:55, 5 October 2007 (UTC) :''All'' flowering plants are good. The only trick is if you go to another region (a distinct one where the bloom times might be different). In that case, use another account. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 12:39, 5 October 2007 (UTC) Or, may I use different singnatures?--[[User:Juan|Juan]] 15:56, 5 October 2007 (UTC) == Categorising by month == I have just realized, that everithink is categorized in time by month. I would recomend to categorized it also by year. Of how well categorize flowering plants of the next October?--[[User:Juan|Juan]] 13:20, 15 October 2007 (UTC) ==Orphan Pages or Data== There are about 70 orphan Bloom Clock pages listed by the special pages orphan page utility. Someone from the proejct may wish to look at them, link them in or put them up for deletion if they are errors and not missing data. [[User:Mirwin|Mirwin]] 06:00, 23 December 2007 (UTC) :How can I find these pages? --[[User:Luai lashire|Luai lashire]] 18:44, 25 December 2007 (UTC) ::The pages can be found [http://en.wikiversity.org/w/index.php?title=Special:Lonelypages&limit=100&offset=50 here]. It looks like these are mostly the DPLs for the keys, which are categorized but not linked (they are linked via some newer templates, but the templates aren't on all the corresponding keys yet). In some other cases they are top level subbages, which really just serve to allow automatic linking from lower subpages, but again, they're categorized and therefore not realy orphans. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:05, 26 December 2007 (UTC) :Some of these pages seem a bit superfluous, such as [[Bloom Clock/Welcome]]- the information on this page is already on the main Bloom Clock page and as far as I can tell, the main Bloom Clock page doesn't even lead to the "Welcome" page. Couldn't this page just be deleted? --[[User:Luai lashire|Luai lashire]] 18:10, 26 December 2007 (UTC) ::A few of them probably could... I was experimenting around with different introductions. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 09:37, 31 December 2007 (UTC) == Global key experiment == I'm going to try over the next few days to start setting up a global key, based on the Southeast PA data. The idea is to have 8 "seasons": early winter, late winter, early spring, late spring, and so on. The hope is to have the data organized in a less "regional" manner for new contributors. For spring flowers that are reacting to temperature, we should probably be able to make adjusted calendars for various regions (for example, judging by cormaggio's latest entries, early spring in England would probably be in February-March, as opposed to March-April in Southeast PA. Using the SEPA data to make the standard has 3 advantages: #The largest data set (for now) is from that region. #The Winter in southeast PA follows the solstice-equinox calendar rather closely (i.e., hard freeze tends to set in in late december, and soil thaw tends to happen in late march). In that sense the data by season will be "typical". #In North America, this region sits on one of the major biome lines, meaning that it's on the southern limit of much of the northern flora, and on the northern limit of much of the southern flora. This makes it easy to "tie in" data from colder and warmer locations. I'm going to make the original keys by altering the region/month templates to add season categories and running DPL on the new categories. We can add the same categories to the region/month templates of other regions once we correlate the data. This will have to be rejiggered again each month, because I suspect that fall flowers will be reacting to daylight hours rather than growing degree days (summer flowers will probably use variable methods... let's see what the data tells us after a few years). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:02, 18 February 2008 (UTC) :OK, got a key started at [[Bloom Clock/Keys/Global]]. Hopefully looks ok. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:44, 19 February 2008 (UTC) == Questions about what to log == Question 1: I just bought a potted primrose which I intend to plant in my garden. It is blooming now, but it was raised in an artificial environment. Do I log it anyway, or wait until it's actually in my garden? (you can tell I'm a bit confused here!) :) Question 2: I saw some mosses today which had sent up stalks (not sure what the official term for those is); do we log those, or not? --[[User:Luai lashire|Luai lashire]] 23:08, 13 March 2008 (UTC) :Hi Luai! (Yes, I'm as excited about spring as you are :-)). For the primrose, it's a bit of a gray area, but I think you should probably log it after you plant it or start to leave it outside, because a lot of people plant them out as annuals in the spring, so they'll serve the function of an annual on the keys. Just add a comment about that when you sign the logs so future researchers are fully informed. :As for the '''[[w:sporophyte|sporophytes]]''', I really have no idea, because I really don't know anything about them other than that I like them and enjoy watching the little "stems" come up too. I'll ask the Wikipedians for advice on that (I need to give them an update anyway). :Well, I think, that it would be better to find different project for mosses, fungi, cyanophyta and these. Gardeners would not be probably interested in which period fungi are fruiting. But it would by interested for mycologists and these. All these data could be also usefull for phytogeographists. I hope for the future to make something for phytogeography, but firstly need to understand SB´s tepmlates a little bit more. But he might help right now, to found something simillar for mosses.--[[User:Juan|Juan]] 09:53, 20 March 2008 (UTC) == Regional Subdivisions == How deep I can go with region subdividing? Is it reasonable? How to devide? I think political divisions to department distrcits and so on might not be a good idea. E.g. now, I have tree different accounts for Southbohemian Region, Czech Republic. For me, its quite big. There are mountains, lowlands and so on.--[[User:Juan|Juan]] 09:49, 20 March 2008 (UTC) :Well, my 3 main accounts are used for a foothill region (where I live), a broad valley region (where I often work), and a low ridge area that's close enough to Philadelphia to have some heat island effect (each region is about 30-40 miles away from the others). Bloom time differences seem to be only a week or so, so for category purposes I've just been using "Southeastern Pennsylvania" for all three, but there's nothing to say I or someone else couldn't break them up later on if it seems appropriate (persumably this could be done by bots). Since the regional keys are broken down by months, 1-week differences won't show up at this point anyway. :Anticipating the next question :-)...: I don't think we can reasonably break down the keys to less than a month at this point unless we had a large number of people logging from any particular region, because it's unlikely that a single editor would have the time to log every plant blooming on a weekly basis. For example: during late spring and early summer in my region there are well over 200 species blooming (not including cultivars, varieties and grasses (I'm no good at identifying the latter)), and there's just no way I can promise to log all of them weekly. I'll probably be able to do a bit better at it this year because of the non-substituted templates. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 11:22, 20 March 2008 (UTC) == Global key update and problems == Well, I started templating for the global key, but I think I need to redefine the categories, because the "2-months-in-a-season" thing is causing inaccuracies. I'm going to try instead to directly tie global cats to a single month for SEPA, which I probably should have done in the first place but wasn't happy about how that breaks up the turning months (march/june/sept/dec). The issue is worst with things blooming here in April, which I had been catting as both early spring and mid spring, but I think it's better to just use it as early spring, and keep march to late winter (rather than late winter/early spring). SEPA is usually very cold through the end of March anyway (it's -0.9º C right now, and snowing), so should work out ok. One thing I'd like to work out on the global keys are some indicator species. Judging from some of Juan's contribs lately, these might need to be of 2 types: plants that bloom at the same time every year in a region (plants triggered by day length), and plants with variable bloom time (triggered by heat). We probably need 4 or 5 years of data to achieve this, but it's something to start watching out for. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 10:41, 22 March 2008 (UTC) == Bloom Tree (experimental) == ''Just for looking at occasionally.'' --[[User:McCormack|McCormack]] 16:39, 29 March 2008 (UTC) {{Robelbox|title=Bloom Tree|theme=9}} {{#categorytree:Bloom Clock|mode=all|depth=1}} {{Robelbox/close}} == Change in log pages == The newer log pages now have the recent logs '''below''' the archives, because of some problems caused with editing noinclude sections in the lates version of mediawiki. Eventually we should change them all over... the goal is to make it easier to see where new logs go ("on the last line", rather than "on a new line below the last signature under "Recent Logs" but above the noinclude tag"). I'm not sure whether this will be bot-fixable, but just don't be surprised when you see logs in a different place on newer pages and/or pages that have been updated. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 19:02, 13 April 2008 (UTC) :Ooohh... Oops. I already reverted a couple back to the way they used to be! I thought someone had made an error when they made the log page. Well, thanks for clarifying- I certainly won't make THAT mistake again. Although I must say, I fail to see how it makes it easier.... Having the archives of something at the very top of the page, even before the current text, seems counter-intuitive to me. --[[User:Luai lashire|Luai lashire]] 00:37, 14 April 2008 (UTC) ::Yeah, rewording might be better: maybe just having "Logs" as the first heading, and we can break the archived logs down by year as well over time. The trouble came about because now when you hit the [edit] button next to "recent logs", the edit window would also show the archived logs as well. With the recent logs at the bottom now, it shows only the recent log list rather than all the confusing wikicode. The mediawiki change also makes the archived logs a no-edit section on the old version :(. I tried asking our local techies about it, but unfortunately couldn't find the root of the issue. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 09:31, 14 April 2008 (UTC) == Global-temperate keys: progress and ways forward == I've been playing around with interpretations some more, and have come up with some pretty interesting results! I'm going to ask for advice on the WP ref desk sometime in the next day or two, but anyone interested can see the worksheets so far: *[[Bloom Clock/Keys/New Hampshire/Global Comparison Worksheet]] *[[Bloom Clock/Keys/Central Pennsylvania/Global Comparison Worksheet]] *[[Bloom Clock/Keys/Prague/Global Comparison Worksheet]] I'm trying to figure out when to incorporate regional data into the global data... not as easy as I would like. Stay tuned :). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 17:27, 14 April 2008 (UTC) ==Structural and other changes to the clock== I've been fiddling around as usual (I seem to be suffering from Michelangelo syndrome), and have come up with some improvements and fixes: ===Navigation=== I'm switching to linked headers for the monthly local keys. This will make links to the key pages appearing on the main regional page, which in turn can be followed to show the global-temperate infobox and the links. ===Data interpretations=== All "global-temperate" regions will soon have a data analysis and comparison system to match up to the global-temperate keys. Monthly keys will also have a template linking to the best match(es) on the g-t keys. ===Modifications to the templates=== ====Top==== *Add a "general information" field for discussing things not addressed by the other fields *Add a "cookbook link" field to make links to the wikibooks cookbook for edible plants with recipes *Remove the "quiz" link (the mediawiki for the quizzes has long been broken), and add a "fruit" link (for the soon-to-come fruit clock). *Add a "higher taxa" link (from species to genus, variety to species, etc.) ====Middle==== Add multiple "new region #" fields, since we have at least 3 new ones coming up. We'll name these starting at 3-12... the bot will be better able to keep track if we make sure each new field has a distinct name. ====Bottom==== Add several semi-defined fields to increase catagorizations, and get rid of "non-dichotomous" fields: *Flower shape: will use a "see instructions" clarifier... too many possible entries! *Inflorescence type: will use a "see instructions" clarifier... too many possible entries! *Veination: palmate/pinnate/linear *Species field: to keep species general records up to date for species with multiple cultivars *Separate color fields: embedded templates are too comfusing and hard to keep track of *multicolor field: mono/bi/multi/variable *Separate type fields: tree/shrub/vine/herbaceous *Separate pollination fields: far to complicated as it is now. **wind **animal *Nectar source quality: for logging plants that are often/sometimes/never covered in insects *Remove calander season field (already provided now by global cross-categorizations) *Toxic to humans *Toxic to livestock ===Summary=== It's a bunch of changes... I'll probably wait for more input and ideas about the bottom template for now, but will ask for a bot run to modify the middle and top templates (for one thing, we '''''desperately''''' need the new region fields!) within the next day or two. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:30, 21 April 2008 (UTC) ==Assigning New Region 1 and 2 == I'm going to modify these to accept templates from Juan (for West Bohemia), and HortMan (Victoria, Australia). More New regions will become available after the bot runs through. We really should have some sort of policy for assigning... thus far I'm just using the "10 or more in a month" approach. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:30, 21 April 2008 (UTC) == New region 3 assigned == New region 3 is now used for [[Bloom Clock/Keys/Moravia-Silesia|Moravia-Silesia]], Czech Republic, using the template series mo-si and categories MO-SI. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 17:45, 3 May 2008 (UTC) == New region 4 == I'm designating "new region 4" for the Czech name of plants (since we have 2 very active contributors from the Czech Republic). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 19:18, 8 May 2008 (UTC) == Changed log pages, change to bottom template in progress == Well, I went ahead and changed the template that makes new logs, and it's easy to redo the old ones using the script as well. The change in the mediawiki code that caused the earlier problems is still there, so it's still best to keep the recent logs at the bottom. The new change makes more sections for organizing the older logs, using heirarchichal headers so that whether you hit edit next to "Logs", "2008 logs" or "recent logs", you can always add new logs to the bottom. I replaced all the text at the top with a single template, which can easily be modified later on. I've noticed some confusion here and there about where logs should go, so that message should eventually be made clearer (for now, it at least works structurally). There's also been a lot of confusion lately about how to use the bottom template, so I'm hoping to change the non-dichotomous parts (where templates have been needed... e.g.: instead of adding <nowiki>{{bcp/white}}</nowiki> to the "flower color (...) =" field, you would now just add "y" or "yes" next to "white flowers = ". We'll be able to track the new and old versions after the bot run using a tracking categories. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 11:05, 12 May 2008 (UTC) ==Assigning New Region 5 to Aurora, Colorado== [[User:Ngravagna]] has added a lot of logs (and new pages) for this region... I haven't had much luck getting in touch with him to describe the region, but I do want to get the keys and cats up, so I'll add them myself for now and hope he'll get in touch :). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 11:05, 12 May 2008 (UTC) == Changes to BCP pages == I'll send talk page messages later, but '''new''' BCP pages will have a markedly different version of the bottom template. In brief, here's how to answer them: #for color options, just add "y" or "yes" after any colors that apply (for in-betweens like "greenish yellow", add "y" after both green and yellow) #for the other new options (formerly under "type" and "pollination method"), also just add "y" after whatever applies #"calendar season" is no longer on the template... the global-temperate keys will eventually serve this purpose much better (and also give us flexibility for monsoon seasons and other equatorial variants). #I'll explain the leaf veination and fruit present stuff later... got distracted on IRC and really have to run! The bot will at some point come through and add these fields as well... the old fields still work and all data will be preserved. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:45, 12 May 2008 (UTC) :By the way, as long as the bot is coming through, I asked him to update the middle template too. Please '''don't''' try to manually update any of the old versions... the bot follows a specific category. "New Region 3" and "New Region 4" still work as before, the bot will replace the text when it does the other updates. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:51, 12 May 2008 (UTC) == plant communities == what do you think about inserting a field for plant communities of occurrence (from literature) into the plant page template? like Fagion, Alno-Ulmion, ... can be useful to check species classification. but dont't know, whether this system is used internalionally. -- [[User:Turnvater Jahn|Turnvater Jahn]] 20:04, 21 May 2008 (UTC) == Change log dates == Is if possible (to handle for the bot) to log a plant and change the date of the signation? Last week i recorded a lot of blooming plants, but had no internet access to log them. Thanks. -- [[User:Turnvater Jahn|Turnvater Jahn]] 02:50, 5 June 2008 (UTC) :Well, one easy way is to just change your signature in preferences to include the date for the day you're logging for, then sign with <nowiki>*~~~</nowiki> rather than <nowiki>*~~~~</nowiki> so it doesn't show the date twice. I'm not sure a bot could do it, since it wouldn't have any way of knowing which logs to alter. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 08:57, 5 June 2008 (UTC) ::I just meant, whether the bot which summarizes the data to create the bloom clocks (isn't it a bot?) uses the date of signature to determine finding times. Not, wheater a bot could change the date. But my question is answered, since it should work. Thanks. ::And sorry for my bad edits. -- [[User:Turnvater Jahn|Turnvater Jahn]] 14:21, 5 June 2008 (UTC) :::Ah, I see. No, there is not bot for that yet. Adding the regional template adds the category which is picked up by the DPL. For example, when you (Turnvater Jahn) log a plant this month, you should also add {{tl|bcpm/leilo/6}} after the "| New Region 6 =" prompt on the middle template. For last month you use {{tl|bcpm/leilo/5}}. For ''both'' this month and last month, you add {{tl|bcpm/leilo/5}}, {{tl|bcpm/leilo/6}}. == Templates to automate data summarization == I had a similar problem like dicussed in the section above: how to summarize and visualize logs automatically. I thought, a bot would be needed to do so, but i was wrong. The templates i wrote for the [[de:Projekt:Atlas der Blütenpflanzen|plant atlas (de)]] were inspired (and partially copied) from BCP, so the problem is really similar. The solution i found consists of *using a template to log, in spite of the signature, e.g. [[de:Vorlage:PPA/Fund]] (finding entry), [[de:Vorlage:PPA/Kartierer]] (mapping contributor), [[de:Vorlage:PPA/Kartierer/Ort]] (contributor location(s)) *adding three parameters by "parameter = {{{xxx}}}" for **what you want the template to do (show on map, report coords, ...), **search field and **search for what, when using the template. If you use the log list as template then, you can specify "xxx" and use one list to draw a map, search for entries of a specified user, draw a visual bloom clock, categorize, ... Especially consider [[de:Projekt:Atlas der Blütenpflanzen/Quadranten]] (and click one of the green map entries), [[de:Projekt:Atlas der Blütenpflanzen/Kartierer]] and the species pages and subpages. E.g. ::[[de:Projekt:Atlas der Blütenpflanzen/Arten/Alliaria petiolata|Alliaria petiolata]] (i like the icon bot top left), ::[[de:Projekt:Atlas der Blütenpflanzen/Arten/Alliaria petiolata/Funde|its log page]], ::[[de:Projekt:Atlas der Blütenpflanzen/Arten/Alliaria petiolata/Karten|its map page]] So if templates with specification of user name and date of sight in spite of just signatures, and contributor/location templates at the contributors page (like [[de:Vorlage:PPA/KartiererListe]]) were used on PCB, automatic bloom time visualization would be possible. Perhaps on a world map? -- [[User:Turnvater Jahn|Turnvater Jahn]] 00:14, 10 June 2008 (UTC) :Well, we sort of have those templates already, but not ''in lieu'' of signatures. For one thing, <nowiki>"*~~~~"</nowiki> is a lot easier for new contributors to log rather than a template (we do use templates later to mark things). For another, it would create difficulties with archiving (e.g., [[BCP/Taraxacum officinale]] would be a rather unattractive page if the entire list from [[BCP/Taraxacum officinale/Logs]] was on the page. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 08:36, 10 June 2008 (UTC) ::It was just an idea. Of course, logging would be more time consuming and difficile, esp. for new contributors. Therfore i provide personalized log entry copy-and-paste templates on [[de:Projekt:Atlas der Blütenpflanzen/Kartierer]] (open the box). But the increased difficulty is an argument. ::Archiving is no problem, i think. You can also include two log pages, like <pre> {{Template:BCP/VisBloomClock |species = {{SUBPAGENAME}} |logs = {{{{FULLPAGENAME}}/Logs}} {{{{FULLPAGENAME}}/Logs/Archived}} |... }} </pre> ::But, as i said, just some idea. -- [[User:Turnvater Jahn|Turnvater Jahn]] 14:00, 10 June 2008 (UTC) :::Well, I had thought about using a separate archive page, but it ends up adding a lot of extra steps both in the profile-making process as well as simply making it more difficult to archive :-). A few other points: :::#Using the templates on the main profile page rather than on the log page allows multiple users from a single region to collaborate on a '''displayed''' result (see any profile for how that works, but the main thing is that it creates an easily understood list of when a plant has been seen blooming in a particular region, and also a link to a key). :::#Using the templates on the log page (let alone on '''both''' a log page and an archive log page) would cause serious issues when trying to apply DPL (which is needed both to semi-automatically create keys as well as compare bloom time data from region to region). :::#We actually do have some relatively simple-to use templates available (much simpler than the ones I think you're talking about), but even those are difficult to explain: (and (no offense, but) you yourself still have not gone back to add them... [[:Category:BCP/LEILO/5]]'s pages were all templated by me (and I don't think I got them all). :::Keep in mind that I'm not a programmer or anything, but the templates we have now are the results of almost 2 years of experimentation. A template that can create maps would be wonderful, but the primary thrust of the bloom clock is to serve as a research project, and the templates we have now serve that purpose quite well. Once you've familiarized yourself with how to use what we have, you'll be in a much better place to see how they can be improved upon :-). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:44, 10 June 2008 (UTC) ::::Ok, didnt't want to appear as smart ass, just referred to our conversation about bots. And sorry for not adding the templates, i already have a list of approx. 50 logs not done yet. i've spent a lot of time to PPA. -- [[User:Turnvater Jahn|Turnvater Jahn]] 19:30, 10 June 2008 (UTC) :::::No hurry :). Keep in mind though that the global-temperate comparison needs those categories to work (see [[Bloom Clock/Keys/Leipzig Lowlands, Saxony, Germany/Global Comparison Worksheet/May|worksheet]]). That's where the "research" side of things really kicks in.--[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 08:20, 11 June 2008 (UTC) ::::::Thanks for the link, from there i came to [[Template:Bcp-gtdpl]], which uses dynamicpagelist. I was looking for something like category intersection, which is not available in wp.de. But dynamicpagelist is. Schould be an alternative. -- [[User:Turnvater Jahn|<kbd><span style="color:#457345">'''Turnvater Jahn'''</span></kbd>]] ^{| 16:32 11.06.2008} ''(reset tabs)'' You mean wv.de, I think? Yeah, I have quite a few templates now using DPL (still experimenting), see [[:Category:Bloom clock project templates/DPL]]. Eventually all the keys will use these templates, since they're a bit more intuitive to use in comparison to just typing in the actual DPL scripts. --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 18:39, 11 June 2008 (UTC) == Template updates == It's time once again to get a bot run to modify the bottom template: First, we need "graminoid" to differentiate between herbaceous forbs and herbaceous grasses Second, some of the fields on the bottom are causing prblems for genus profiles, since they only allow one option when there might be several options that apply to a genus. For example, for Acer (maples), you need to be able to classify leaf complexity as pinnately compound (e.g., A. negundo), trifoliate (e.g., A. griseum), and simple (most other maples), and the same applies to lobes, petioles, and so on. I really want to start working on botanical keys during the winter, so it will have to be changed to allow more answers (switching them all to simple y/n/blank answers). This means a bunch of new fields on the bottom template, or perhaps just a new template for leaf descriptions only. Third, I want to add a 4th (or 5th?) template for hardiness zones. This would just list the usda hardiness zones from 1-12, and would be filled in to say if a plant is hardy, tender, annual or biennial in a given hardiness zone. The goal there is to gather horticultural data so that gardeners and designers could use the project to select plants for color and season when doing a design. I've also been thinking about adding height and spread, both for horticultural selection and for identification, but not uite sure how the fields would be defined, so that might wait until the next update. I'll start working up a scratch version over the next few days (will be raining here). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 09:41, 24 September 2008 (UTC) == Central Minnesota == I'd like to get more involved with the BCP, and to be perfectly honest I have no idea how the system works and would like to learn so I don't "mess anything up". :-D Anyway, now that it's almost October in good old (cold!) Minnesota where I live, I'm not going to be too much help in logging blooms, but I'd like to help set up the keys. Any data sorting, etc. that I can help out with? I'd also like to start setting up a key for central Minnesota, as I know a few nature centers around here that might like to bring the BCP into their program. What do I need to do to get started? [[User:Trinity507|Trinity507]] 17:43, 29 September 2009 (UTC) == Maintainers == Hey, are there still any maintainers of this project around?--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:43, 12 March 2018 (UTC) == Getting involved == I just recently discovered this page, even though I have been dabbling in wikiversity for the last couple of years. The topic of bloom time in different regions of the world is something I am interested in as an on-and-off gardner. I have a question regarding participation. The page says: * "To participate, read the page on How to Contribute, then go to the Contributors page and sign in" Since I am not interested enough to become a serious student, how else can I get involved? Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 22:32, 28 December 2023 (UTC) :You can create new pages for plants, work on analysis and create other related pages. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:09, 26 December 2024 (UTC) hh4c7akkvetrhvkbspxryhj3mme4o36 0 2347 2693455 2465922 2024-12-26T23:43:07Z MathXplore 2888076 Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]] 2693455 wikitext text/x-wiki {{TOCright}} {{nav3|Wikiversity|Wikiversity:School of Mathematics|School:Mathematics/Undergraduate/Pure Mathematics}} == Introduction to Graph Theory == An [[School:Mathematics/Courses by level|introductory course]] [[School:Mathematics|from the School of Mathematics]] This course aims to provide a thorough introduction to the subject of graph theory. == Course requirements == The following knowledge is required or desirable on commencement of study of this course: * knowledge of basic [[Introduction_to_Proofs|methods of proof]] * knowledge of basic [[Wikiversity:Statistics_101|probability]] == Course outline == This is an approximate depiction of the course: * Definitions * Bipartite Graphs * Hamilton Cycles and Eulerian Circuits * Planar Graphs ** Statement of Kuratowski's Theorem * Matchings in Bipartite Graphs ** Hall's Theorem * Connectivity ** Menger's Theorem * Extremal Graph Theory ** Hamilton and other cycles ** Turan's Theorem * Ramsey's Theorem * Graph Colourings ** Chromatic Polynomial ** Vizing's Theorem ** Four-Colour and Five-Colour Theorems *** Extensions to other surfaces * Eigenvalues ** Applications to Strongly Regular Graphs * The Probabilistic Method ** Lower bounds for Ramsey numbers ** Graphs with large girth and chromatic number == Lecture series == *[[/Lecture 1/]] Introduction and Definitions *[[/Lecture 2/]] Bipartite Graphs and Trees *[[/Lecture 3/]] Hamilton Cycles and Eulerian Circuits *[[/Lecture 4/]] Graph Traversal *[[/Lecture 5/]] Flows and Cuts *[[/Lecture 6/]] Planar Graphs [[/Glossary/|List of Definitions]] == Assignments == *[[/Problems 1/]] (on Lectures 1-5) *[[/Problems 2/]] (on Lectures 6-10) [[Category:Mathematics]] [[Category:Graph theory]] [[Category:Introductions]] {{BookCat}} pjfqvpsytjgsx3udjpfnkkoj0a0lm2f 0 2966 2693486 1823391 2024-12-27T00:25:50Z Tule-hog 2984180 Tule-hog moved page [[Network+/Architecture/Media/Introduction]] to [[Network+/Old guides/Network media]]: alter parent 1823391 wikitext text/x-wiki == Ethernet Overview == Ethernet has been with us since 1980. Early Ethernet was carried over stiff coax and later more malleable thin coax. Coax Ethernet installations require that the cable be snaked past each network device and that the device tap into the cable as it goes past. Although this wiring scheme reflects the way Ethernet uses a shared media to allow network communications, the station to station coax has serious real-world reliability problems. 10BASE-T was introduced in 1990 to address these problems. 10BASE-T uses durable and inexpensive twisted pair cable. Two pairs are required for each station: one pair for incoming traffic and one for outgoing. Home runs from each station to a central concentrator carry data to and from each station. With 10BASE-T, the vulnerable shared media portion of the network is now safely hidden in the closet and is not strewn all across the office. In a 10BASE-T system, a wiring fault typically takes out network service only to a single station. 10BASE-T includes diagnostic indicators that allow wiring faults to be easily identified. Because 10BASE-T offers both reliability and cost advantages over the original coax, most new Ethernet installations are of the twisted pair variety. The newer Fast Ethernet used by CobraNet is wired just like 10BASE-T except that it has ten times the bandwidth, requires a slightly higher grade cable and has some distance limitations not found in 10BASE-T. For more information on Ethernet, check out these network and Ethernet Ethernet is a family of frame-based computer networking technologies for local area networks (LANs). The name comes from the physical concept of the ether. It defines a number of wiring and signaling standards for the physical layer, through means of network access at the Media Access Control (MAC)/Data Link Layer, and a common addressing format. Ethernet is standardized as IEEE 802.3. The combination of the twisted pair versions of Ethernet for connecting end systems to the network, along with the fiber optic versions for site backbones, is the most widespread wired LAN technology. It has been in use from the 1990s to the present, largely replacing competing LAN standards such as token ring, FDDI, and ARCNET. In recent years, Wi-Fi, the wireless LAN standardized by IEEE 802.11, is prevalent in home and small office networks and augmenting Ethernet in larger installations. Ethernet is used in most, but not all, computer networking situations. It is a standard which defines rules that all computers follow to allow successful and efficient communication. ==Common Network Cabling== Cable is a medium which provides physical path for data transmission. Several types exist. Some networks use the same type, other use multiple types. A common early cable was the RS-232 D Serial cable. It transmitted one bit at a time, very slow connection. Parallel cables can talk at one bit per line at a time - an 8 pin cable can have 8 bits at once assuming one bit is one line and software handles all handshaking. [[Image:Vergleich 2von2 Crossoverkabel.gif|thumb|right]] RJ-11 is the terminator typically used on plugin phone cables. This cable type has been used for many years by TELCOs all over the world - six pin cable schmatics to follow at a later date. RJ-45 connectors now terminate most common cables used for hub based or star style TCP/IP networks. RJ-45 terminated cable ends are shown to right. Schematics to follow at a later date. Now that we understand what cables are used the next step is to learn signal processing. Early TCP/IP networks functioned with looped coax cables or very thick D connector style cables running from one computer to the next rather than via centralized star style hub components. ==Physical Network Communication Mediums== === Twisted-Pair Cable === Twisted-pair cabling is a copper wire that comes in two forms, shielded and unshielded. It is the most common form of wiring used in a network. It uses 8 wires twisted into pairs to cancel the effect of crosstalk (Noise from the adjacent wires). It is a relatively inexpensive form of LAN cabling. It can accommodate different topologies, but is mostly implemented in a star topology. === Unshielded Twisted-Pair Cable === UTP relies on the cancellation effect of twisting the wires to reduce ''Electromagnetic Interference'' (EMI). It is required to have a certain amount of twists per meter and it is connected using a ''Registered Jack 45 Connecter'' (RJ-45). UTP can run for 100 meters before the signal needs to be refreshed. UTP has advantages that make it ideal in some networks. * '''Easy to Install''' * '''Small, it does not take up much space in wiring ducts.''' * '''Cheapest type of cable.''' <br> UTP has 6 Categories. *'''Category 1:''' **Only reliable for transmitting telephone communications, not regular data transmissions. *'''Category 2:''' **Previously used in token rings. Speeds only up to 4kbit *'''Category 3:''' **Works in 10BASE-T networks.. Transfer rate of 10mbit *'''Category 4:''' **Used on 16mbit token ring networks. *'''Category 5:''' **Transfer rate of 100mbit. Unreliable for 1000BASE-T networks. *'''Category 5e:''' **Transfer rate of 1000mbit. Used in Gigabit Ethernet networks. *'''Category 6:''' **Same as Cat 5e but made to a higher standard === Shielded Twisted-Pair Cable === Shielded Twisted-Pair Cable uses twisted pairs along with a metallic foil shielding to reduce the crosstalk and EMI. It is usually connected using an STP connecter but can also be connected with an RJ-45. Although it reduces the interference better than UTP, STP has many drawbacks that keep it from having a mainstream use. * '''More Expensive''' * '''Must be grounded at both ends''' * '''Harder to install''' <br> Because of these drawbacks it is rarely implemented in Ethernet networks. It is more common in Europe. === Coaxial Cable === Coaxial Cable uses a copper wire for the conductor, on top of this is insulation for the wire. The third layer consists of a metallic foil or woven copper braid as a shielding, followed by a rubber jacket on the top. It is often referred to as Thicknet or Thinnet, depending on the specification. Coaxial cable was found in early Ethernet networks (~1980). Benefits: * '''It costs less to buy than Fiber Optic''' * '''It has speeds of 10mb/s to 100mb/s.''' * '''It costs more to install Coaxial cable.''' === Fiber Optic === Installation of fiber optic cabling is not the rocket science it once was. Great strides have been made in the durability cost and ease of termination of this media. Fiber optic cable offers two main advantages over twisted pair cable. First, data may be carried much further over fiber. Second, fiber is immune to electromagnetic interference. There are two basic types of fiber in use today: Multimode and Single Mode. Multimode fiber is used extensively in the data communications industry. Fast Ethernet carried over multimode fiber is known as 100BASE-FX. Ethernet may be carried up to 2 kilometers on this fiber. Single mode fiber is used extensively in the telecom industry. Single mode fiber allows much greater run lengths than multimode fiber. Although there is no official standard for carrying Ethernet over single mode fiber, numerous datacom products offer this capability. Two strands are required for each Ethernet link; one for transmit and one for receive. Both multimode and single mode fiber cables are available with varying numbers of strands. 4-strand cable costs about $0.80/meter. Transfer rates of up to 10GB/s and a distance of up to 1000 meters. This is an expensive type of medium and takes a special connector to terminate the signal. It has higher bandwidth possibilities and is best suited for backbone installations. It has 3 parts to the cable: Core, Cladding and Buffer. * '''Core:''' ** This is where the light is transmitted * '''Cladding:''' ** Just outside the core it traps the light inside the core and helps guide it around corners. * '''Buffer:''' ** The hard plastic coating on the outside of the cable that protects the core from moisture and physical damage. Fiber optic uses include grounding and bonding for: *LIghting protection systems *Grounding electrode systems *electrical bonding and grounding *Power protection *telecom bonding and grounding *Telecom circuit protector ==External links== *[http://yoda.uvi.edu/InfoTech/rj45.htm Wiring Standards at UVI] *[http://www.w2drz.ramcoinc.com/rj45.htm the RJ45 connector] *[http://www.dslreports.com/faq/bellsouth/11.0+Wiring+Diagrams+and+Schematics#5539 Other Diagrams] ==Bibliography== *Network Cabling fundamentals, by Beth Verity [[Category:Computer networks]] [[Category:Network administration]] 3p0c4tbftfhxf54jmahuqgqx9s0lrqj 2693526 2693486 2024-12-27T00:50:02Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693526 wikitext text/x-wiki == Ethernet Overview == Ethernet has been with us since 1980. Early Ethernet was carried over stiff coax and later more malleable thin coax. Coax Ethernet installations require that the cable be snaked past each network device and that the device tap into the cable as it goes past. Although this wiring scheme reflects the way Ethernet uses a shared media to allow network communications, the station to station coax has serious real-world reliability problems. 10BASE-T was introduced in 1990 to address these problems. 10BASE-T uses durable and inexpensive twisted pair cable. Two pairs are required for each station: one pair for incoming traffic and one for outgoing. Home runs from each station to a central concentrator carry data to and from each station. With 10BASE-T, the vulnerable shared media portion of the network is now safely hidden in the closet and is not strewn all across the office. In a 10BASE-T system, a wiring fault typically takes out network service only to a single station. 10BASE-T includes diagnostic indicators that allow wiring faults to be easily identified. Because 10BASE-T offers both reliability and cost advantages over the original coax, most new Ethernet installations are of the twisted pair variety. The newer Fast Ethernet used by CobraNet is wired just like 10BASE-T except that it has ten times the bandwidth, requires a slightly higher grade cable and has some distance limitations not found in 10BASE-T. For more information on Ethernet, check out these network and Ethernet Ethernet is a family of frame-based computer networking technologies for local area networks (LANs). The name comes from the physical concept of the ether. It defines a number of wiring and signaling standards for the physical layer, through means of network access at the Media Access Control (MAC)/Data Link Layer, and a common addressing format. Ethernet is standardized as IEEE 802.3. The combination of the twisted pair versions of Ethernet for connecting end systems to the network, along with the fiber optic versions for site backbones, is the most widespread wired LAN technology. It has been in use from the 1990s to the present, largely replacing competing LAN standards such as token ring, FDDI, and ARCNET. In recent years, Wi-Fi, the wireless LAN standardized by IEEE 802.11, is prevalent in home and small office networks and augmenting Ethernet in larger installations. Ethernet is used in most, but not all, computer networking situations. It is a standard which defines rules that all computers follow to allow successful and efficient communication. ==Common Network Cabling== Cable is a medium which provides physical path for data transmission. Several types exist. Some networks use the same type, other use multiple types. A common early cable was the RS-232 D Serial cable. It transmitted one bit at a time, very slow connection. Parallel cables can talk at one bit per line at a time - an 8 pin cable can have 8 bits at once assuming one bit is one line and software handles all handshaking. [[Image:Vergleich 2von2 Crossoverkabel.gif|thumb|right]] RJ-11 is the terminator typically used on plugin phone cables. This cable type has been used for many years by TELCOs all over the world - six pin cable schmatics to follow at a later date. RJ-45 connectors now terminate most common cables used for hub based or star style TCP/IP networks. RJ-45 terminated cable ends are shown to right. Schematics to follow at a later date. Now that we understand what cables are used the next step is to learn signal processing. Early TCP/IP networks functioned with looped coax cables or very thick D connector style cables running from one computer to the next rather than via centralized star style hub components. ==Physical Network Communication Mediums== === Twisted-Pair Cable === Twisted-pair cabling is a copper wire that comes in two forms, shielded and unshielded. It is the most common form of wiring used in a network. It uses 8 wires twisted into pairs to cancel the effect of crosstalk (Noise from the adjacent wires). It is a relatively inexpensive form of LAN cabling. It can accommodate different topologies, but is mostly implemented in a star topology. === Unshielded Twisted-Pair Cable === UTP relies on the cancellation effect of twisting the wires to reduce ''Electromagnetic Interference'' (EMI). It is required to have a certain amount of twists per meter and it is connected using a ''Registered Jack 45 Connecter'' (RJ-45). UTP can run for 100 meters before the signal needs to be refreshed. UTP has advantages that make it ideal in some networks. * '''Easy to Install''' * '''Small, it does not take up much space in wiring ducts.''' * '''Cheapest type of cable.''' <br> UTP has 6 Categories. *'''Category 1:''' **Only reliable for transmitting telephone communications, not regular data transmissions. *'''Category 2:''' **Previously used in token rings. Speeds only up to 4kbit *'''Category 3:''' **Works in 10BASE-T networks.. Transfer rate of 10mbit *'''Category 4:''' **Used on 16mbit token ring networks. *'''Category 5:''' **Transfer rate of 100mbit. Unreliable for 1000BASE-T networks. *'''Category 5e:''' **Transfer rate of 1000mbit. Used in Gigabit Ethernet networks. *'''Category 6:''' **Same as Cat 5e but made to a higher standard === Shielded Twisted-Pair Cable === Shielded Twisted-Pair Cable uses twisted pairs along with a metallic foil shielding to reduce the crosstalk and EMI. It is usually connected using an STP connecter but can also be connected with an RJ-45. Although it reduces the interference better than UTP, STP has many drawbacks that keep it from having a mainstream use. * '''More Expensive''' * '''Must be grounded at both ends''' * '''Harder to install''' <br> Because of these drawbacks it is rarely implemented in Ethernet networks. It is more common in Europe. === Coaxial Cable === Coaxial Cable uses a copper wire for the conductor, on top of this is insulation for the wire. The third layer consists of a metallic foil or woven copper braid as a shielding, followed by a rubber jacket on the top. It is often referred to as Thicknet or Thinnet, depending on the specification. Coaxial cable was found in early Ethernet networks (~1980). Benefits: * '''It costs less to buy than Fiber Optic''' * '''It has speeds of 10mb/s to 100mb/s.''' * '''It costs more to install Coaxial cable.''' === Fiber Optic === Installation of fiber optic cabling is not the rocket science it once was. Great strides have been made in the durability cost and ease of termination of this media. Fiber optic cable offers two main advantages over twisted pair cable. First, data may be carried much further over fiber. Second, fiber is immune to electromagnetic interference. There are two basic types of fiber in use today: Multimode and Single Mode. Multimode fiber is used extensively in the data communications industry. Fast Ethernet carried over multimode fiber is known as 100BASE-FX. Ethernet may be carried up to 2 kilometers on this fiber. Single mode fiber is used extensively in the telecom industry. Single mode fiber allows much greater run lengths than multimode fiber. Although there is no official standard for carrying Ethernet over single mode fiber, numerous datacom products offer this capability. Two strands are required for each Ethernet link; one for transmit and one for receive. Both multimode and single mode fiber cables are available with varying numbers of strands. 4-strand cable costs about $0.80/meter. Transfer rates of up to 10GB/s and a distance of up to 1000 meters. This is an expensive type of medium and takes a special connector to terminate the signal. It has higher bandwidth possibilities and is best suited for backbone installations. It has 3 parts to the cable: Core, Cladding and Buffer. * '''Core:''' ** This is where the light is transmitted * '''Cladding:''' ** Just outside the core it traps the light inside the core and helps guide it around corners. * '''Buffer:''' ** The hard plastic coating on the outside of the cable that protects the core from moisture and physical damage. Fiber optic uses include grounding and bonding for: *LIghting protection systems *Grounding electrode systems *electrical bonding and grounding *Power protection *telecom bonding and grounding *Telecom circuit protector ==External links== *[http://yoda.uvi.edu/InfoTech/rj45.htm Wiring Standards at UVI] *[http://www.w2drz.ramcoinc.com/rj45.htm the RJ45 connector] *[http://www.dslreports.com/faq/bellsouth/11.0+Wiring+Diagrams+and+Schematics#5539 Other Diagrams] ==Bibliography== *Network Cabling fundamentals, by Beth Verity [[Category:Networking]] [[Category:Network administration]] r9rkiebq9067etxuiyurhaabgcqe51t 0 9295 2693521 2218281 2024-12-27T00:49:12Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693521 wikitext text/x-wiki ==Introduction== Computer network is a collection of nodes, where the computing stations (resources) reside. The nodes communicate to each other via [[w:communication network|communication network]]. Transmission, and switching accomplishes the course of communications. ===Network Types=== Networks can be classified by switching methods, topology, functions and representation. ====[[w:Circuit switching|Circuit Switching]]==== is good for long continuous and time-constrained communications. *Before the data transmission began, station to station (end-to-end) physical path must be established. All channels in the path are used simultaneously ruding the transmission. The entire path must be used for communication, until the release of the circuit. *The best way to memorize operation, consists of three phases: **Circuit Establishment (Resources are allocated) **Data Transfering (Resource are used) **Circuit Termination (Resources are deallocated) ====Message Switching==== A message is a logical unit of information that one station sends to another. It does not need a path to be established. Instead, the message travels over existing channels. It hops (travels) through the network in a node-to-node manner, in store-and-forward fashion, until it reaches its destination. ====Packet Switching==== Transmits data in a bursty nature. Small users get fast service in presence of large users. Messages are decomposed into packets. Packets are the smaller units of data. *Many packets of the same message can be transmitted at the same time. In other words, the message is cut is small pieces which are transmitted simultaneously, yet by different channels. *The message needs to be reassembled after arriving at the destination. ====Datagram Packet Switching==== *Data packets are treated independently of each other. *Each packet can travel according to the route it found on the network. *Sequential order is not required to take place. *The connection operation consists only of data transfer phase. ====Virtual Circuit packet switching==== Virtual circuit is a logical connection, which is established prior to the start of communications. Logical connection is a route, over which all packets travel between the stations. Packets are delivered along the circuits in sequential order. *Connection-Oriented Operation: **Connection Established **Data Transferred **Connection Terminated ===Protocol Concepts=== [[Image:FileTransferExample.JPG|thumb|200px|File Transfer Example]] Consider a simple example of file transfer, which happens almost every day with us. How does it happen: *Source either activates direct data communication path, or informs the network about desired destination. *Source system must be sure that the destination system is ready to receive data. *File transfer application of the source system must be sure that the file management program on the destination system is ready to receive files. Conversion must be performed if two systems are incompatible. ====Protocols==== Protocol is a set of rules that governs the exchange of the information. It deals with *Syntax, which is the format, coding, signals magnituded *Semantics, which takes care of the implementation of control information and error handling. *Timing the sequence in which everything occurs. Anything, capable of transmitting and receiving information is called an entity. A system is a physically distinct collection of entities. Protocols are responsible for: *Delivering user data *Fragmentation and reassambly *Encapsulation *Connection Control, which consists of **Establishment of the connection **Data transfer **Connection Termination *Controling the data flow between two protocol entities *Error Control *Sequencing, which is maintaining the proper order of Protocol data Unit Delivery *Addressing, which is a proper manner of naming and referencing protocol entities. *Multiplexing, allowing single connection to multiple users *Expedited data delivery, security, which comes under a frame of Transmision Service. ====Layered Approach==== Entity implements the functions of a given layer, and the protocols for communicating with peer entities. An entity communicates with entities on *Its layer *on layers above, or below it, acting as an interface The entity on the layers N communicated with the entities on lower layers, by invokations of N-1 primitives. Primitives: *''''Request'''': used to invoke requested service, passing parameters for full specification *In order to show that a procedure has been invoked by a peer service, '''indication''' migh be useful. *Response is issued to complete or acknowledge some procedure, invoked by an indication to user. *Confirmation, used by a service provider to complete or acknowledge some procedure. [[Image:Multilayer.JPG]] ===OSI MOdel=== Suggested reading: [[w:OSI model|OSI model]] ===TCP/IP protocol architecture=== Suggested Reading, from WIKIPedia: *[[w:Internet protocol suite|Internet protocol suite]] ==Data Link: Issues and Requirements== Data link control is respoinsible of data communications between tow transmitting-receving station, connected directly. The most important aspects of the Data Link COntrol are *Frame synchronization **'''character oriented''', where the '''character''' is the basic unit (atom) of information. **'''bit oriented''', where the "atom" of information is a '''bit'''. *Line Configuration, is a criteria that differs the way of data links. [[Image:FrameSynchSimple.JPG]] **Topology (point to point, multipont) **Duplexity **Simplex, one connection at a time, in one directeion over a single line **Half Duplex, One connection at time, during the talk. **Full duplex 2 way connection during the talk. *Line discipline **contention **polling **selection ===Bit Oriented Data Link Protocols=== To study *[[w:High-Level Data Link Control|High Level Data Link COntrol]] (HDLC) - ISO * [[w:LAPB|LAPB]] - ITU-T (for [[w:x.25|x.25]]) * [[w:Point-to-Point Protocol|Point-to-Point Protocol]] used on the internet to connect '''home users to ISPs''', as well as '''router to router'''. While working, it relies on: **[[w:Line Control protocol|Line Control protocol]] for '''establish'''ing connection **In order to check the validity of a password, PPP uses either [[w:Password authentication protocol|PAP]] or [[w:Challenge-handshake authentication protocol|CHAP]]. It comes between '''Establish''', ''and'' '''Authenticate''' states. **[[w:Network Control Protocol|Network Control Protocol]] for negotiating network layer option. Comes between '''Network''' and '''Open''' states. **And for providing IP service over the established connection, [[w:Internet Protocol Control Protocol|IPCP]] is in use. It comes between the states of '''Authenticate''' and '''Network'''. Just for reference *Advanced Communications Control Procedures (ADCCP) *Synchronous data Link COntrol *[[w:Link Access Procedures, D channel|Link Access Procedures, D channel]]LAPD - ITU-T (for ISDN) *Link Access Procedure for Frame-mode Bearer Sercies (LAPF) *Logic Link COntrol(LLC) - IEEE 802.2 ==Network Layer== This layer and its protocols are responsible for the terminal to terminal data packet delivery. One of its main design issues are: ===Subnet to host interface=== {| class="wikitable" |- !Connection Type !Methode Description !Diagram |- |Virtual Circuit service |Network layer excercises flow and error control over the established circuit. Network tries its best to deliver packets in sequence. |[[Image:VirtualCircuitServiceNWK.JPG|thumb|200px]] |- |Datagram Service |Network layer accepts packets from users, and tries to deliver them as isolated units. Packets may arrive not in sequence, and not in full order. |[[Image:VirtualCircuitServiceNWK.JPG|thumb|200px]] |- |Internal Operation. Internal virtual circuit operation. Virtual Circuit Switching. |A virtual circuit determines a route between two terminals in a network. Then all the packets follow the same way. |[[Image:VirtualCircuit878.jpg|thumb|200px]] |- |Datagram Switching (Internal datagram Operation) |Each packet is treated independently, and finds its own route through the network. |[[Image:InternaldatagramOperation.jpg|thumb|200px]] |} ===External vs. Internal operation=== In case of external.internal virtual circuit, whenever the user requests VC, a dedicated route is set through the network. All packets will follow it. *External Virtual Circuit with internal datagrams. **Different packets of the same message may follow different routes through the network. **The original sequence and the integrity of message is achieved by resassemlby of packets at the destication node. *External/Internal datagram, features independent treatment of each packet. General COmparison {| class="wikitable" |- !Ext/Int !Virtual circuits !Datagrams |- |Internal Operations |MInimizes per-packet overhead *routing decisions are made only on at-call set ups *Sequenced Delivery |Robust with respect to the outages of links and nodes. | |- |External Operation |End-to-End sequencing with flow control. |No call set-up.No need to hold packets in case of an error packet. |} ===Packet Routing=== ===Congestion management=== ===X.25 Standard=== The protocol specifies three level interface between [[w:Data Terminal Equipment|Data Terminal Equipment]] and [[w:Data Control Equipment|Data Control Equipment]] *From TOP to BOTTOM, the levels of the interface are: ==='''Packet Levels'''=== enables subscribers of the network to set up logical connections to the subscribers. These logical connections, form a pre-planed routes and are called, [[w:Virtutal circuits|Virtutal circuits]] THis level relies on [[w:Packet Layer Protocol|Packet Layer Protocol]] *Virtual Circuit Service of X.25 *Dynamically established Virtual Call. *Network Assigned, permanet virtual circuit. *Packet types {|class="wikitable" name="X.25 packet types" |- !Packet Type !DCE -> DTE !DTE -> DCE !Service !VC !PVC |- |Call setup and Cleaning |Incoming Call |Call Request | |X | |- | |Call Connected |Call Accepted | |X | |- | |Clear Indication |Clear Request | |X | |- | |Clear Confirmation |Clear Confirmation | |X | |- |Data and Interrupt |Data |Data | |X |X |- | |Interrupt |Interrupt | |X |X |- | |Interrupt Confirmation |Interrupt Confirmation | |X |X |- |Flow COntrol and Reset |RR |RR | |X |X |- | |RNR |RNR | |X |X |- | | |REJ | |X |X |- | |Reset Indication |Reset Request | |X |X |- | |Reset Confirmation |Reset Confirmation | |X |X |- |Restart |Restart Indication |Restart Request | |X |X |- |Restart |Restart Confirmation |Restart Confirmation | |X |X |- |Diagnostic |Diagnostic | | |X |X |- |Registration |Registration Confirmation |Registration Request | |X |X |} ====[[Multilink Procedure]]==== *Multilink procedure allows multiline DTE-DCE connections. *Each link is governed by [[w:SLP|SLP]] [[w:LAPB|LAPB]] *MLP frame format **Multilink '''Control Field''' (MLC) (2 octets) ***12 bit sequence number accross al links is needed, to reorder frames sent, which may arrive in different order. ***Restination reorders the packets according to MLP sequence numbers ***The X.25 packet. ===Physical Level=== deals with the physical interface between the link that attaches computer to the network, and the computer. *[[w:X.21|X.21]] *X.21 bis, similar to EIA-232-D ===X25 details=== ====Error control and recovery==== *Reset packet: This operation requires to reinitialize a virtual circuit. *Restart packet: This operation requires to reset all active virtual circuits. ====Multiplexing==== *DTE is allowed to establish up to 4095 [[w:Virtual Circuit|Virtual Circuits]] simultaneously. *each packet contains a 12bit Virtual Circuit number. (0 is reserved for "restart" and diagnostic packets. ====Flow Control==== *Sliding Window protocol with default size of 2 (max is 7 or 127) ====Acknowledgements==== They may have either local or end to end significance. *when D=0, the acknowledgement is made between DTE and the network. *when D=1, the acknowledgement is made from Remote DTE ==[[w:Frame Relay|Frame Relay Networks]]== *This technology is designed to provide lower delays, and higher throughput, in comparison with traditional switching networks. *F.R. originated as a part of ISDN standardization, yet can be provided for bothISDN and non-ISDN networks. *In x.25 the control packets are carried on the same channel, and the same virtual circuit as the data packet. In fact in-band signaling takes place there. Simply speaking control and data packets travel in series, not in parallel. Multiplexing and switching takes place at level 3, whicle both layers 2 and 3 include flow control. *Frame Relay provides separate channels/virtual circuits for data and control packets. In other words, data and control packets travel in parallel ways. ===FrameRelay protocol architecture=== Consider two separate planes of operations, COntrol (C) plane and user (U) plane. *C plane is responsible for establishment and termination of logical connections. *U plane is responsible to the transfer of user data between the subscribers. *Multiplexing, and logical connection switching takes place at level 2 (instead of 3) leaving level 3 for processing. *Flow control is not end to end. No the hop to hop. ===Call Control=== Call controls signalling is carried on a separate logical connection. Thus the state table maintenance is not required for intermediate nodes. *It used LAPD (Q.921) protocol, for relizable data link control *Q.931/Q.933 control signalling mesages are exchanged atop of Q.921 protocol. ====Frame relay call control functions==== *Exchange of a Q.931/Q.933 message over a logical connection dedicated call - '''establishes and release''' a frame relay connection. *Sending the SETUP message, either side can request '''the establishment'''. The side which sends setup message may chose DLCI option by choosing the unused value. Otherwise the accepting side will assign DLCI with the conenct message. Other side can answer with *CONNECT message to confirm the connection, or the *RELEASE COMPLETE message to refus the connection. After the connection is established, data transfer can proceed. Clearing a connection is accomplished by the exchange of DISCONNECT, RELEASE, and RELEASE COMPLETE messages. ====Q.931/Q.933 repertoire==== {|class="wikitable" name="X.25 packet types" |- !Call establishment messages !Call clearing messages !Miscalleneous messages |- | *Alerting *call proceeding *connect *connect acknowledgement *progress *setup | *Disconnect *Release *Release complete | *Status *Status enquiry |} ===User data Transer=== User plane provides LAP-F protocol for data transfer (Link Access Procedure for Frame Mode bearer Services) It is defined in Q.922. Only the code function which are used in frame relay are defined here: *delimiting, alignment, and transparancy *multiplexing/demultiplexing, using the address field *inspecting the frame for integral number of oxtets **prior to zero bit insertion **after zero bit extraction Core functions of theLAPF constitue a subwayler in a data link layer. They provide bare data transfer service without any flow and error control. Besides this, the user can chose end-to-end function. Network can relay the frames with the following properties *Small probability of frame loss. *Preservation of the order of frame. ===Network Function=== ===ITU-T Recommendations=== ===Q.921/Q.922=== ==Routing protocols== Computer communication protocols can be divided on *nonroutable *routed *routing protocols Routing protocls in turn can be divided on by methode on *[[w:distance vector routing|distance vector routing]] *[[w:link state routing|link state routing]] ===Network address translation=== (NAT) is a very important issue of routing. NAT can use either one-to one mapping or one to many mapping. NAT has several advantages. *It conserves public [[w:IP address|IP addresses]] *NAT hides internal IP scheme from the outside world *allows easy renumbering of IP address translation. For example: when a user decides to change a provider, along with external IP Addresses, NAT will presedve all the internal IP addresses with the internal to external translation. However NAT has following disadvantages *Delay comes froma NAT router to perform address translation. *End to end IP traceablity is lost due to IP translation *It is hard to find the originator of the message. ====Static NAT==== This kind of translation is ONE TO ONE. The configuration is simple, because NAT Routes will be a default gateway for all clients ====Dynamic NAT==== *Group-to-Group mapping, where a group of valid outside IP addresses are mapped to a group of private IP addresses. *Network administrator doesnt care of specific one-to one mapping. *Any private IP address will be automatically translated to any IP address. [[Category:Networking]] bxe1ixwqkhy1yc4qbdxi8rb2tg4x22f 102 10405 2693517 2183669 2024-12-27T00:48:32Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693517 wikitext text/x-wiki {{center top}}<big>'''Welcome to the Department of Computer Networks!'''</big>{{center bottom}} {{center top}}''Also Part of [[:Category:Engineering and Technology|Engineering and Technology]]''{{center bottom}} {{tertiary}} {{bag}} {{yawn}} {{it}} {{cleanup|Reorganization/Standardization}} '''[[w:Computer networking|Computer networking]]''' is the scientific and engineering discipline concerned with communication between computer systems. Such networks involve at least two devices capable of being networked with at least one usually being a computer. The devices can be separated by a few meters (e.g. via Bluetooth) or thousands of kilometers (e.g. via the Internet). Computer networking is sometimes considered a sub-discipline of [[Telecommunications engineering|telecommunications]]. Computer networking can be examined from either the 7 layer [[Network+/Standards/OSI Model|OSI Model]], or the 4 layer [[Network+/Standards/Theory and Concepts|TCP/IP Model]] <!---Begin Dump from Topic:Networking----> ==Introduction== Networking is the practice of enabling and harnessing the transmission of data between computer systems. A simple analogy of a data network is two tin cans connected by a simple string. As in the analogy, the following holds true for actual data network implementations as well: * The network exists merely as a medium for communications across it * A protocol of some form is needed to initiate and carry on conversations (this is not intrinsic to the network itself) * The same protocol (a spoken language) can also be used with different media for the same purpose; different networks have different advantages and uses A network may require one engineer to design it, another to build it, and yet another to administer it. The skills needed by each are related, but not necessarily dependent; hence, Networking is interdisciplinary. Networking developed within the discipline of Computer Science, but an extensive background in Computer Science is not necessary to study or even practice Networking. A ''Network Engineer'' works with data networks in some form, but the scope of that work and the skills required may be as diverse - even from one job to the next - as those of any scientist. == Topics == Networking is the practice of enabling and harnessing the transmission of data from one computer system to another. The TCP/IP model is used in presenting the following topics. # [[Introduction to Networking]] # [[Network+/Standards/TCP/IP Model/Introduction|TCP/IP Fundamentals]] # [[Application Layer]] # [[Transport Layer]] # [[Internet Layer]] # [[Link Layer]] # [[Mobile Networks]] # [[IT Security/Network|Network Security]] # [[Network Administration]] # [[Portal:Pursuing network certification|Pursuing network certification]] ==[[w:OSI model|OSI Seven Layer Model]]== The alternative OSI layer can be studied [[Network+/Standards/OSI Model|here]]. *Layer 7, the '''Application layer''', provides service directly related to the applications. Those services vary on applications. *Layer 6, the '''Presentation layer''', format the data to look like common for all applications *Layer 5, the '''Session Layer''', establishes a connection between two nodes. Deals with whether the connection is [[w:Full duplex|full duplex]], [[w:Half Duplex|half duplex]] etc. *Layer 4, '''Transport layer''', handles and delivers data, whenever it is connection-oriented or connectionless. It includes some [[w:flow control|flow control]] issues. *Layer 3, '''Network Layer''', establishes the connection between two nodes, using the IP addressing. *Layer 2, the '''Data Link Layer''', frames data and provides low-level flow control *Layer 1, '''Physical layer''', transmits low bitstreams (data), deals with electrical signalling, [[w:Network cabling|cabling]] and hardware interface Each topic includes an outline, suggested activities and links to useful resources. They are constructed for self-study, but should be adaptable to a group environment. <!--- End Dump from Topic: Computer NetworkING ----> == Courses == * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Windows Server Administration]] ==See also== * [[School:Computer Science]] * [[w:NetSim | NetSim]] - Useful network simulator for lab experimentation and teaching on networks ==External Resources== {{wikibooks}} [[Category:Networking]] [[Category:Engineering and Technology|{{PAGENAME}}]] [[fr:Département:Réseau]] [[ru:Компьютерные сети]] 7frv0tb1es8xfl6cstq75i9axysnb9z 2693544 2693517 2024-12-27T01:00:20Z Tule-hog 2984180 fix lks 2693544 wikitext text/x-wiki {{center top}}<big>'''Welcome to the Department of Computer Networks!'''</big>{{center bottom}} {{center top}}''Also Part of [[:Category:Engineering and Technology|Engineering and Technology]]''{{center bottom}} {{tertiary}} {{bag}} {{yawn}} {{it}} {{cleanup|Reorganization/Standardization}} '''[[w:Computer networking|Computer networking]]''' is the scientific and engineering discipline concerned with communication between computer systems. Such networks involve at least two devices capable of being networked with at least one usually being a computer. The devices can be separated by a few meters (e.g. via Bluetooth) or thousands of kilometers (e.g. via the Internet). Computer networking is sometimes considered a sub-discipline of [[Telecommunications engineering|telecommunications]]. Computer networking can be examined from either the 7 layer [[Network+/Old Guides/OSI Model|OSI Model]], or the 4 layer [[Network+/Standards/Theory and Concepts|TCP/IP Model]] <!---Begin Dump from Topic:Networking----> ==Introduction== Networking is the practice of enabling and harnessing the transmission of data between computer systems. A simple analogy of a data network is two tin cans connected by a simple string. As in the analogy, the following holds true for actual data network implementations as well: * The network exists merely as a medium for communications across it * A protocol of some form is needed to initiate and carry on conversations (this is not intrinsic to the network itself) * The same protocol (a spoken language) can also be used with different media for the same purpose; different networks have different advantages and uses A network may require one engineer to design it, another to build it, and yet another to administer it. The skills needed by each are related, but not necessarily dependent; hence, Networking is interdisciplinary. Networking developed within the discipline of Computer Science, but an extensive background in Computer Science is not necessary to study or even practice Networking. A ''Network Engineer'' works with data networks in some form, but the scope of that work and the skills required may be as diverse - even from one job to the next - as those of any scientist. == Topics == Networking is the practice of enabling and harnessing the transmission of data from one computer system to another. The TCP/IP model is used in presenting the following topics. # [[Introduction to Networking]] # [[TCP/IP Fundamentals]] # [[Application Layer]] # [[Transport Layer]] # [[Internet Layer]] # [[Link Layer]] # [[Mobile Networks]] # [[IT Security/Network|Network Security]] # [[Network Administration]] # [[Portal:Pursuing network certification|Pursuing network certification]] ==[[w:OSI model|OSI Seven Layer Model]]== The alternative OSI layer can be studied [[Network+/Old guides/OSI Model|here]]. *Layer 7, the '''Application layer''', provides service directly related to the applications. Those services vary on applications. *Layer 6, the '''Presentation layer''', format the data to look like common for all applications *Layer 5, the '''Session Layer''', establishes a connection between two nodes. Deals with whether the connection is [[w:Full duplex|full duplex]], [[w:Half Duplex|half duplex]] etc. *Layer 4, '''Transport layer''', handles and delivers data, whenever it is connection-oriented or connectionless. It includes some [[w:flow control|flow control]] issues. *Layer 3, '''Network Layer''', establishes the connection between two nodes, using the IP addressing. *Layer 2, the '''Data Link Layer''', frames data and provides low-level flow control *Layer 1, '''Physical layer''', transmits low bitstreams (data), deals with electrical signalling, [[w:Network cabling|cabling]] and hardware interface Each topic includes an outline, suggested activities and links to useful resources. They are constructed for self-study, but should be adaptable to a group environment. <!--- End Dump from Topic: Computer NetworkING ----> == Courses == * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Windows Server Administration]] ==See also== * [[School:Computer Science]] * [[w:NetSim | NetSim]] - Useful network simulator for lab experimentation and teaching on networks ==External Resources== {{wikibooks}} [[Category:Networking]] [[Category:Engineering and Technology|{{PAGENAME}}]] [[fr:Département:Réseau]] [[ru:Компьютерные сети]] faq7ircfpb8stpt2okkavrnyl73zbfj 2693545 2693544 2024-12-27T01:00:41Z Tule-hog 2984180 fix lk 2693545 wikitext text/x-wiki {{center top}}<big>'''Welcome to the Department of Computer Networks!'''</big>{{center bottom}} {{center top}}''Also Part of [[:Category:Engineering and Technology|Engineering and Technology]]''{{center bottom}} {{tertiary}} {{bag}} {{yawn}} {{it}} {{cleanup|Reorganization/Standardization}} '''[[w:Computer networking|Computer networking]]''' is the scientific and engineering discipline concerned with communication between computer systems. Such networks involve at least two devices capable of being networked with at least one usually being a computer. The devices can be separated by a few meters (e.g. via Bluetooth) or thousands of kilometers (e.g. via the Internet). Computer networking is sometimes considered a sub-discipline of [[Telecommunications engineering|telecommunications]]. Computer networking can be examined from either the 7 layer [[Network+/Old guides/OSI Model|OSI Model]], or the 4 layer [[Network+/Standards/Theory and Concepts|TCP/IP Model]] <!---Begin Dump from Topic:Networking----> ==Introduction== Networking is the practice of enabling and harnessing the transmission of data between computer systems. A simple analogy of a data network is two tin cans connected by a simple string. As in the analogy, the following holds true for actual data network implementations as well: * The network exists merely as a medium for communications across it * A protocol of some form is needed to initiate and carry on conversations (this is not intrinsic to the network itself) * The same protocol (a spoken language) can also be used with different media for the same purpose; different networks have different advantages and uses A network may require one engineer to design it, another to build it, and yet another to administer it. The skills needed by each are related, but not necessarily dependent; hence, Networking is interdisciplinary. Networking developed within the discipline of Computer Science, but an extensive background in Computer Science is not necessary to study or even practice Networking. A ''Network Engineer'' works with data networks in some form, but the scope of that work and the skills required may be as diverse - even from one job to the next - as those of any scientist. == Topics == Networking is the practice of enabling and harnessing the transmission of data from one computer system to another. The TCP/IP model is used in presenting the following topics. # [[Introduction to Networking]] # [[TCP/IP Fundamentals]] # [[Application Layer]] # [[Transport Layer]] # [[Internet Layer]] # [[Link Layer]] # [[Mobile Networks]] # [[IT Security/Network|Network Security]] # [[Network Administration]] # [[Portal:Pursuing network certification|Pursuing network certification]] ==[[w:OSI model|OSI Seven Layer Model]]== The alternative OSI layer can be studied [[Network+/Old guides/OSI Model|here]]. *Layer 7, the '''Application layer''', provides service directly related to the applications. Those services vary on applications. *Layer 6, the '''Presentation layer''', format the data to look like common for all applications *Layer 5, the '''Session Layer''', establishes a connection between two nodes. Deals with whether the connection is [[w:Full duplex|full duplex]], [[w:Half Duplex|half duplex]] etc. *Layer 4, '''Transport layer''', handles and delivers data, whenever it is connection-oriented or connectionless. It includes some [[w:flow control|flow control]] issues. *Layer 3, '''Network Layer''', establishes the connection between two nodes, using the IP addressing. *Layer 2, the '''Data Link Layer''', frames data and provides low-level flow control *Layer 1, '''Physical layer''', transmits low bitstreams (data), deals with electrical signalling, [[w:Network cabling|cabling]] and hardware interface Each topic includes an outline, suggested activities and links to useful resources. They are constructed for self-study, but should be adaptable to a group environment. <!--- End Dump from Topic: Computer NetworkING ----> == Courses == * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Windows Server Administration]] ==See also== * [[School:Computer Science]] * [[w:NetSim | NetSim]] - Useful network simulator for lab experimentation and teaching on networks ==External Resources== {{wikibooks}} [[Category:Networking]] [[Category:Engineering and Technology|{{PAGENAME}}]] [[fr:Département:Réseau]] [[ru:Компьютерные сети]] 0olrova0xwt1p5skrokbhj71iz815i0 14 14989 2693442 2389525 2024-12-26T23:30:10Z Tule-hog 2984180 rm merge tag, see [[Category talk:Computer networks#Merge proposal|talk]] 2693442 wikitext text/x-wiki [[Category:Information technology]] [[Category: Networking]] 61vel73nqdgszkgn9m335ljzopcf3ag 2693443 2693442 2024-12-26T23:32:09Z Tule-hog 2984180 clarify cat 2693443 wikitext text/x-wiki Computer Networks is an information technology [[WV:LR|resource]] that covers computer network standards, media and topologies, devices, services, security, management, and troubleshooting. The course also assists learners in preparing for CompTIA [[:Category:Network+|Network+]] Certification (a separate resource on Wikiversity). [[Category:Information technology]] [[Category: Networking]] dlfn0g1bkfs4qur0ay0plyxsyujzt86 0 28230 2693477 1180519 2024-12-27T00:11:31Z Tule-hog 2984180 add cat 2693477 wikitext text/x-wiki <big>'''The Teletraffic Hyperlinked Textbook'''</big> This is the combined work of the ELEN5007 class of 2007 (now ELEN7015). == Modules == # What is [[/Blocking/]]? # What are the types of Congestions? # [[/What is Congestion control?/]] # [[/What is the Engset calculation?/]] # [[/Flow Control Telecoms/ | What is Flow control?]] # [[/Teletraffic_grading/ | What are Gradings?]] # [[/What_is_the_High_loss_calculation/]]? # [[/How does Internet telephony traffic differ/]]? # [[/Limited_availability/ | What is the limited availability calculation?]] # What is Long-tailed traffic? (6 modules=1course)* # What is Mobile Quality of Service? (1 module)* # What is Multi-media traffic? (1 module)* # What is Quality of Service? (1 module)* # [[/What is queueing/]]? # [[/Routing/ | What is Routing?]] # [[/What is tariffing?/]] # [[/What is Trade in Services?/]] # How do we Manage Teletraffic data? # [[/Forecasting_Telephony_Traffic/ | How is Telephony Traffic Forecast?]] # [[/How is Telephony Traffic Modelled?/]] # [[/How is telephony traffic simulated/]]? # [[/Trunking/ | What is Trunking?]] * Existing modules (ca. 2006), to be imported from cnx.org == Contributors == {| cellpadding=5 cellspacing=3 |- | '''Name''' || '''email''' || '''modules''' |- | Ian Kennedy || dr.iankennedy at gmail.com || ''Editor in Chief'' |- | Jorenjeye Arubayi 0705588N || lasborn at yahoo.co.uk || Module 15. [[/Routing/ | What is routing?]], Module 20. [[/How is Telephony Traffic Modelled?/]] |- | Chilekwa Bwalya 0701344F || chilekwab at hotmail.com || Module 8. [[/How_does_Internet_telephony_traffic_differ/|How does Internet telephony traffic differ?]] Module 14. [[/What_is_queueing/ | What is Queueing?]] |- | Nosipho Dhladhla 0705577P || mnorid at yahoo.com || Module 4. [[/What is the Engset calculation?/]] Module 17. [[/What is Trade in Services?/]] |- | Nomfundo Dlamini 0607335G || dlamini at volt.ee.wits.ac.za || Module 1. [[/Blocking/ | What is Blocking?]] Module 21. [[/How_is_telephony_traffic_simulated/ | How is telephony traffic simulated?]] |- | Daniel Enslin 0008750T || bestbuddy69 at hotmail.com || Module 6. [[/Teletraffic_grading/|What is Grading?]] Module 19. [[/Forecasting_Telephony_Traffic/|How is Telephony Traffic Forecast?]] |- | Chisala Moses 0700386W || moseslsa at yahoo.com || Module 3. [[/What is Congestion control?/]] Module 16. [[/What is tariffing?/]] |- | Michel Le Vieux 9408490D || michel at mtnns.net || Module 5. [[/Flow Control Telecoms/ | What is Flow control?]] Module 22. [[/Trunking/ | What is Trunking?]] |- | Stephen kyalo Musango Makonge 0710458J || steve at integritylimited.ws || Module 7. [[/What_is_the_High_loss_calculation/ | What is the High loss calculation?]] Module 9. [[/Limited_availability/ | What is the Limited Availability Calculation?]] |} [[Category:Teletraffic engineering|*]] [[Category:Computer networking]] k4l2818844d1cigv5vj4xrw9i4pgxoh 2693485 2693477 2024-12-27T00:23:15Z Tule-hog 2984180 fix cat 2693485 wikitext text/x-wiki <big>'''The Teletraffic Hyperlinked Textbook'''</big> This is the combined work of the ELEN5007 class of 2007 (now ELEN7015). == Modules == # What is [[/Blocking/]]? # What are the types of Congestions? # [[/What is Congestion control?/]] # [[/What is the Engset calculation?/]] # [[/Flow Control Telecoms/ | What is Flow control?]] # [[/Teletraffic_grading/ | What are Gradings?]] # [[/What_is_the_High_loss_calculation/]]? # [[/How does Internet telephony traffic differ/]]? # [[/Limited_availability/ | What is the limited availability calculation?]] # What is Long-tailed traffic? (6 modules=1course)* # What is Mobile Quality of Service? (1 module)* # What is Multi-media traffic? (1 module)* # What is Quality of Service? (1 module)* # [[/What is queueing/]]? # [[/Routing/ | What is Routing?]] # [[/What is tariffing?/]] # [[/What is Trade in Services?/]] # How do we Manage Teletraffic data? # [[/Forecasting_Telephony_Traffic/ | How is Telephony Traffic Forecast?]] # [[/How is Telephony Traffic Modelled?/]] # [[/How is telephony traffic simulated/]]? # [[/Trunking/ | What is Trunking?]] * Existing modules (ca. 2006), to be imported from cnx.org == Contributors == {| cellpadding=5 cellspacing=3 |- | '''Name''' || '''email''' || '''modules''' |- | Ian Kennedy || dr.iankennedy at gmail.com || ''Editor in Chief'' |- | Jorenjeye Arubayi 0705588N || lasborn at yahoo.co.uk || Module 15. [[/Routing/ | What is routing?]], Module 20. [[/How is Telephony Traffic Modelled?/]] |- | Chilekwa Bwalya 0701344F || chilekwab at hotmail.com || Module 8. [[/How_does_Internet_telephony_traffic_differ/|How does Internet telephony traffic differ?]] Module 14. [[/What_is_queueing/ | What is Queueing?]] |- | Nosipho Dhladhla 0705577P || mnorid at yahoo.com || Module 4. [[/What is the Engset calculation?/]] Module 17. [[/What is Trade in Services?/]] |- | Nomfundo Dlamini 0607335G || dlamini at volt.ee.wits.ac.za || Module 1. [[/Blocking/ | What is Blocking?]] Module 21. [[/How_is_telephony_traffic_simulated/ | How is telephony traffic simulated?]] |- | Daniel Enslin 0008750T || bestbuddy69 at hotmail.com || Module 6. [[/Teletraffic_grading/|What is Grading?]] Module 19. [[/Forecasting_Telephony_Traffic/|How is Telephony Traffic Forecast?]] |- | Chisala Moses 0700386W || moseslsa at yahoo.com || Module 3. [[/What is Congestion control?/]] Module 16. [[/What is tariffing?/]] |- | Michel Le Vieux 9408490D || michel at mtnns.net || Module 5. [[/Flow Control Telecoms/ | What is Flow control?]] Module 22. [[/Trunking/ | What is Trunking?]] |- | Stephen kyalo Musango Makonge 0710458J || steve at integritylimited.ws || Module 7. [[/What_is_the_High_loss_calculation/ | What is the High loss calculation?]] Module 9. [[/Limited_availability/ | What is the Limited Availability Calculation?]] |} [[Category:Teletraffic engineering|*]] [[Category:Computer networks]] 8vqrmctjbt815cnz5ri4nsd9ba2cxrs 2693527 2693485 2024-12-27T00:50:12Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693527 wikitext text/x-wiki <big>'''The Teletraffic Hyperlinked Textbook'''</big> This is the combined work of the ELEN5007 class of 2007 (now ELEN7015). == Modules == # What is [[/Blocking/]]? # What are the types of Congestions? # [[/What is Congestion control?/]] # [[/What is the Engset calculation?/]] # [[/Flow Control Telecoms/ | What is Flow control?]] # [[/Teletraffic_grading/ | What are Gradings?]] # [[/What_is_the_High_loss_calculation/]]? # [[/How does Internet telephony traffic differ/]]? # [[/Limited_availability/ | What is the limited availability calculation?]] # What is Long-tailed traffic? (6 modules=1course)* # What is Mobile Quality of Service? (1 module)* # What is Multi-media traffic? (1 module)* # What is Quality of Service? (1 module)* # [[/What is queueing/]]? # [[/Routing/ | What is Routing?]] # [[/What is tariffing?/]] # [[/What is Trade in Services?/]] # How do we Manage Teletraffic data? # [[/Forecasting_Telephony_Traffic/ | How is Telephony Traffic Forecast?]] # [[/How is Telephony Traffic Modelled?/]] # [[/How is telephony traffic simulated/]]? # [[/Trunking/ | What is Trunking?]] * Existing modules (ca. 2006), to be imported from cnx.org == Contributors == {| cellpadding=5 cellspacing=3 |- | '''Name''' || '''email''' || '''modules''' |- | Ian Kennedy || dr.iankennedy at gmail.com || ''Editor in Chief'' |- | Jorenjeye Arubayi 0705588N || lasborn at yahoo.co.uk || Module 15. [[/Routing/ | What is routing?]], Module 20. [[/How is Telephony Traffic Modelled?/]] |- | Chilekwa Bwalya 0701344F || chilekwab at hotmail.com || Module 8. [[/How_does_Internet_telephony_traffic_differ/|How does Internet telephony traffic differ?]] Module 14. [[/What_is_queueing/ | What is Queueing?]] |- | Nosipho Dhladhla 0705577P || mnorid at yahoo.com || Module 4. [[/What is the Engset calculation?/]] Module 17. [[/What is Trade in Services?/]] |- | Nomfundo Dlamini 0607335G || dlamini at volt.ee.wits.ac.za || Module 1. [[/Blocking/ | What is Blocking?]] Module 21. [[/How_is_telephony_traffic_simulated/ | How is telephony traffic simulated?]] |- | Daniel Enslin 0008750T || bestbuddy69 at hotmail.com || Module 6. [[/Teletraffic_grading/|What is Grading?]] Module 19. [[/Forecasting_Telephony_Traffic/|How is Telephony Traffic Forecast?]] |- | Chisala Moses 0700386W || moseslsa at yahoo.com || Module 3. [[/What is Congestion control?/]] Module 16. [[/What is tariffing?/]] |- | Michel Le Vieux 9408490D || michel at mtnns.net || Module 5. [[/Flow Control Telecoms/ | What is Flow control?]] Module 22. [[/Trunking/ | What is Trunking?]] |- | Stephen kyalo Musango Makonge 0710458J || steve at integritylimited.ws || Module 7. [[/What_is_the_High_loss_calculation/ | What is the High loss calculation?]] Module 9. [[/Limited_availability/ | What is the Limited Availability Calculation?]] |} [[Category:Teletraffic engineering|*]] [[Category:Networking]] dy3zs1my4dx727vjtt3025rsbbt33fu 0 29085 2693478 2194686 2024-12-27T00:11:44Z Tule-hog 2984180 rm cat 2693478 wikitext text/x-wiki ''originally written by Moses Chisala'' == Summary == The document defines congestion control and the five algorithms found in the TCP protocols: slow start Exponential backoff, sliding window, Fast retransmit and Fast recovery. In addition, it also gives an example and an exercise basing on simple “steady-state model” of TCP technique. == Definition == Congestion control is a method used for monitoring the process of regulating the total amount of data entering the network so as to keep traffic levels at an acceptable value. This is done in order to avoid the telecommunication network reaching what is termed [[w:congestive collapse]]. Congestion control mostly applies to packet-switching network. A wide variety of approaches have been proposed, however the "objective is to maintain the number of packets within the network below the level at which performance falls off dramatically." There are two transport layer protocols where congestion control is implemented; * [[w:Transmission Control Protocol|Transmission Control Protocol]] * [[w:User Datagram Protocol|User Datagram Protocol]] In the TCP there are several congestion control algorithms strateges used: * [[w:Slow start|slow start]] * [[w:Exponential backoff|exponential backoff]] == Model: The simple “steady-state model” of TCP == * Fixed roundtrip time <var>R</var> in seconds. * A packet is dropped each time the window reaches <var>W</var> packets. * TCP’s congestion window: <var>W</var>, <var>W</var> / 2, <var>W</var> / 2 + 1,..., <var>W</var> - 1, <var>W</var>, <var>W</var> / 2 The average sending rate <var>T</var> pkts per sec: <var>T</var> = 3<var>W</var> / 4<var>R</var> The packet drop rate <var>p</var>: <var>p</var> = 1 / (⅜ * <var>W</var>²) <var>T</var> in pkts per sec: <var>T</var> = &radic;(3 / 2) / (<var>R</var> * &radic;<var>p</var>) or in bytes per sec, given <var>B</var> bytes per pkt: <var>T</var> = &radic;(3 / 2) * <var>B</var> / (R * &radic;<var>p</var>) The improved “steady-state model” of TCP: An improved steady-state model of TCP includes a fixed packet drop rate,retranmit timeouts, and the exponential backoff of the retransmit timer. <!-- TeX would fit here. --> <var>T</var> = <var>B</var> / [<var>R</var> * &radic;(2 * <var>p</var> / 3) + 2 * <var>R</var> * {3 * &radic;(3 * <var>p</var> / 8)} * <var>p</var> * (1 + 32 * <var>p</var>²)] ; <var>T</var> : sending rate in bytes/sec ; <var>B</var> : packet size in bytes ; <var>R</var> : roundtrip time ; <var>p</var> : packet drop rate[1] * [[w:Sliding window|sliding window]] * [[w:Fast retransmit|fast retransmit]] * [[w:Fast recovery|fast recovery]] === Example === The date rate between two routers is 230.4 Kbps, and the TCP congestion window is 30 packets. calculate the round trip time <var>R</var>. (Assuming 1 packet = 512 - bytes) ==== Solution ==== <!-- better ideas to mark up these equations? ('''no tables''' of course) I mean especially the third - it becomes unreadable if written in one line. --> <var>T</var> = 230.3 Kbps <var>W</var> = 30 packets = 30 * 512 * 8 bits <var>R</var> = 3 * <var>W</var> / 4 * <var>T</var> = 3 * 30 * 512 * 8 / (4 * 230300) = 0.4 s == Exercises == An internet cafe connects to an ISP with bandwidth of 10 Mbps. The TCP’s congestion window is 64 KB. (1 packet = 512 bytes) # Calculate the round trip time at network equilibrum # What is the packet drop rate? # What is the capacity of the TCP’s congestion window packets? [[/Solutions/]] == References == <!-- there are no reference marks in this text, so I left reference list (almost) untouched... --> <div class="references-small"> # [http://www.icir.org/floyd/talks/ATT_Nov99.pdf Sally Floyd and Mark Handley, AT&T Center for Internet Research at ICSI, December 1999] # Wikipedia article about [[w:Congestion_control|congestion control]] # Data and Computer Communication,Fifth Edition, William Stallings # Iverson, V.B., Teletraffic Engineering and Network Planning, COM Course 34340, Technical University of Denmark, May 2006. # Dr. Kennedy, ELEN7015 lecture notes, Department of Electrical Engineering, University of the Witwatersrand, 2007. # Network Working Group, NASA Glenn/Sterling Software, M. Allman, V. Paxson, W. Stevens </div> [[Category:Teletraffic engineering]] ln7n7zjwr63ml1fcfp8rbs5dlsda6s7 0 31391 2693479 2224686 2024-12-27T00:12:05Z Tule-hog 2984180 rm cat 2693479 wikitext text/x-wiki Author: Moses Chisala == What is Tariffing? == === Summary === This document is about [[w:Tariffing|'''tariffing''']] as applied in the Telecommunications industrial. Several factors have been looked at; tariffing policy entities looking at the customer and what should be priced. Also components of tariffing or tariffs such as; Why are tariffs charged?, Components of tariffs, Special tariffs and Impact of tariffs on traffic. === Definition === The word '''tariffing''' comes from the word [http://www.answers.com/topic/tariff ''tariff''] which means; tax, duty, due, fee, excise, levy or toll paid towards the use of a specific service. In the telecommunication environment it applies to the charging of telecommunication services that have been already used or prior to. Therefore, '''tariffing''' can be defined as the process of fixing a duty, fee or a price on the telecommunication services provided by the service provider and utilized by the end user (consumer) and the '''public policy regulating body''' acting as an overseer. The regulating body provides standards and guidelines to both the service provider and the end user. The standards and regulations imposed differ from one country to another. Some of the elements affecting tariffing are:- Monopoly or competition; Pricing and Tariffs; Universal Service; Network interconnections and Abuse of Dominance. '''Tariffing policy entities:''' '''''Customer''''' An end user customer uses one telecommunications network to initiate a communication to another customer of the same network or another. An ''''''interconnector'''''' is a network operator that terminates a communication from a customer of another network operator to a customer of its network. '''''What Should Be Priced?''''' a. Rate Elements and Rate Structure "Price elements and the structure of prices are intended to address two related issues: on the supply side, to ration scarce resources, and on the demand side, to change consumption behavior of end users. The decision to make the next telephone call depends on the price of that incremental call, not the average price of all calls. The decision to talk for the next minute, once a call is placed, is based on the perceived price of that incremental minute. The decision to subscribe to a service is based on the perceived price of that incremental service. Each of these marginal decisions is performed by comparing the perceived price to the perceived benefit to be obtained. Customers often undertake these decisions with poor information, incomplete understanding, and only a vague notion of the actual price that will be charged. Time-of-day pricing is a crude form of peak load pricing, intended to cause users to change their behavior by shifting some of their calling to off-peak periods. Multi-part tariffs (fixed charge + usage-sensitive charge) can be used to segment the market according to user characteristics." [http://www.pegasus.or.id/Reports/89)%20Telecom%20Pricing.pdf [2]] The figure below shows a simple telecommunication network indicating the routing of the local call and long-distance call. Simple Telecommunication Network b. “Product” Definitions "Fundamentally, the circuit switched telephone network is a time-sharing network. Telephone companies set different prices for different minutes of use, depending on the identity of the user, the distance of the call, and the time of day." [http://www.pegasus.or.id/Reports/89)%20Telecom%20Pricing.pdf [2]] The following componets also should be considered when looking at tariffing; [[w:Tariffing#Why_are_tariffs_charged.3F|Why are tariffs charged?]] [[w:Tariffing#Components_of_tariffs|Components of tariffs]] [[w:Tariffing#Special_tariffs|Special tariffs]] [[w:Tariffing#Impact_of_tariffs_on_traffic|Impact of tariffs on traffic]] '''Price-elasticity of demand''' Elasticity of demand for new installations may be estimated taking into account the subjective price perception of potential customers. Price elasticity expresses the sensitivity of customers to the cost of the service. The elasticity parameter is calculated as the ratio of percentage change in demand (quantity sold per period) caused by a percentage change in price. ==== Example ==== '''Example 1''' If the average price for the new service that has been introduced in the network is R15 and the elasticity of revenue is 0.7. Asuming 4% drop in quantity demand. Calculate I. elasticity of quantity demand II. the new price '''Solutions''' '''I.''' <math>E_{RP} = 1 + E_{qp}</math> 0.7 = 1 + Eqp Eqp = '''- 0.3''' '''II.''' Eqp = [(Q1/Qo) - 1]/[(P1/Po-1] - 0.3 = [-0.04]/[15/Po-1] Po = '''R13.24''' ====Exercises ==== '''Question 1''' A mobile network provider charging R2 per 1Mb data download has a subscriber base of 1.2 million for a period of six months. The subscriber base increases in five months to 1.6 million after the 64% reduction per download. Calculate I. the initial quantity demand II. the relative Change in quantity demand III. the relative change in price IV. elasticity of quantity demand V. elasticity of revenues VI. comment on the answer in v. {{collapse top|Solutions to Module 16}} {{collapse top|I}} '''Qo = 200,000'''{{collapse bottom}} {{collapse top|II}} '''(Q1-Qo)/Qo = 0.6'''{{collapse bottom}} {{collapse top|III}}'''(P1-Po)/Po = -0.64'''{{collapse bottom}} {{collapse top|IV}} '''Eqp = -0.9375'''{{collapse bottom}} {{collapse top|V}} ''' ERP = 0.0625'''{{collapse bottom}} {{collapse bottom}} === References === [1] INTERNATIONAL TELECOMMUNICATION UNION SERIES D SUPPLEMENT 3 (03/93) [2] http://www.ingrimayne.com/econ/elasticity/Elastic1.html [3] Steensrtup M., Routing in Communication Networks. Prentice Hall Inc, New Jersey, 1995 [4] Hanharan H., Integrated Digital Communications. School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg, 2006. [5] [[[w:Tariffing|http://en.wikipedia.org/wiki/Tariffing]]] [6] Kennedy I.G., Why is Network Planning Important?, Lecture Notes, ELEN5007 - Teletraffic Engineering, School of Electrical and Information Engineering, University of the Witwatersrand, 2005. [[Category:Teletraffic engineering]] [[Category:Tariffing]] 30ikoz1ez2e489new7clu1es3j2ezis 2693483 2693479 2024-12-27T00:21:08Z Tule-hog 2984180 /* Exercises */ mv from [[What is Tariffing?/VI]] 2693483 wikitext text/x-wiki Author: Moses Chisala == What is Tariffing? == === Summary === This document is about [[w:Tariffing|'''tariffing''']] as applied in the Telecommunications industrial. Several factors have been looked at; tariffing policy entities looking at the customer and what should be priced. Also components of tariffing or tariffs such as; Why are tariffs charged?, Components of tariffs, Special tariffs and Impact of tariffs on traffic. === Definition === The word '''tariffing''' comes from the word [http://www.answers.com/topic/tariff ''tariff''] which means; tax, duty, due, fee, excise, levy or toll paid towards the use of a specific service. In the telecommunication environment it applies to the charging of telecommunication services that have been already used or prior to. Therefore, '''tariffing''' can be defined as the process of fixing a duty, fee or a price on the telecommunication services provided by the service provider and utilized by the end user (consumer) and the '''public policy regulating body''' acting as an overseer. The regulating body provides standards and guidelines to both the service provider and the end user. The standards and regulations imposed differ from one country to another. Some of the elements affecting tariffing are:- Monopoly or competition; Pricing and Tariffs; Universal Service; Network interconnections and Abuse of Dominance. '''Tariffing policy entities:''' '''''Customer''''' An end user customer uses one telecommunications network to initiate a communication to another customer of the same network or another. An ''''''interconnector'''''' is a network operator that terminates a communication from a customer of another network operator to a customer of its network. '''''What Should Be Priced?''''' a. Rate Elements and Rate Structure "Price elements and the structure of prices are intended to address two related issues: on the supply side, to ration scarce resources, and on the demand side, to change consumption behavior of end users. The decision to make the next telephone call depends on the price of that incremental call, not the average price of all calls. The decision to talk for the next minute, once a call is placed, is based on the perceived price of that incremental minute. The decision to subscribe to a service is based on the perceived price of that incremental service. Each of these marginal decisions is performed by comparing the perceived price to the perceived benefit to be obtained. Customers often undertake these decisions with poor information, incomplete understanding, and only a vague notion of the actual price that will be charged. Time-of-day pricing is a crude form of peak load pricing, intended to cause users to change their behavior by shifting some of their calling to off-peak periods. Multi-part tariffs (fixed charge + usage-sensitive charge) can be used to segment the market according to user characteristics." [http://www.pegasus.or.id/Reports/89)%20Telecom%20Pricing.pdf [2]] The figure below shows a simple telecommunication network indicating the routing of the local call and long-distance call. Simple Telecommunication Network b. “Product” Definitions "Fundamentally, the circuit switched telephone network is a time-sharing network. Telephone companies set different prices for different minutes of use, depending on the identity of the user, the distance of the call, and the time of day." [http://www.pegasus.or.id/Reports/89)%20Telecom%20Pricing.pdf [2]] The following componets also should be considered when looking at tariffing; [[w:Tariffing#Why_are_tariffs_charged.3F|Why are tariffs charged?]] [[w:Tariffing#Components_of_tariffs|Components of tariffs]] [[w:Tariffing#Special_tariffs|Special tariffs]] [[w:Tariffing#Impact_of_tariffs_on_traffic|Impact of tariffs on traffic]] '''Price-elasticity of demand''' Elasticity of demand for new installations may be estimated taking into account the subjective price perception of potential customers. Price elasticity expresses the sensitivity of customers to the cost of the service. The elasticity parameter is calculated as the ratio of percentage change in demand (quantity sold per period) caused by a percentage change in price. ==== Example ==== '''Example 1''' If the average price for the new service that has been introduced in the network is R15 and the elasticity of revenue is 0.7. Asuming 4% drop in quantity demand. Calculate I. elasticity of quantity demand II. the new price '''Solutions''' '''I.''' <math>E_{RP} = 1 + E_{qp}</math> 0.7 = 1 + Eqp Eqp = '''- 0.3''' '''II.''' Eqp = [(Q1/Qo) - 1]/[(P1/Po-1] - 0.3 = [-0.04]/[15/Po-1] Po = '''R13.24''' ====Exercises ==== '''Question 1''' A mobile network provider charging R2 per 1Mb data download has a subscriber base of 1.2 million for a period of six months. The subscriber base increases in five months to 1.6 million after the 64% reduction per download. Calculate I. the initial quantity demand II. the relative Change in quantity demand III. the relative change in price IV. elasticity of quantity demand V. elasticity of revenues VI. comment on the answer in v. {{collapse top|Solutions to Module 16}} {{collapse top|I}} '''Qo = 200,000''' {{collapse top|II}} '''(Q1-Qo)/Qo = 0.6''' {{collapse top|III}}'''(P1-Po)/Po = -0.64''' {{collapse top|IV}} '''Eqp = -0.9375''' {{collapse top|V}} ''' ERP = 0.0625''' {{collapse top|VI}} Demand is [http://www.ingrimayne.com/econ/elasticity/Elastic1.html '''inelastic'''] whenever the elasticity coefficient is less than modulars one. When it is greater than then the demand is called [http://www.ingrimayne.com/econ/elasticity/Elastic1.html elastic]. {{collapse bottom}} === References === [1] INTERNATIONAL TELECOMMUNICATION UNION SERIES D SUPPLEMENT 3 (03/93) [2] http://www.ingrimayne.com/econ/elasticity/Elastic1.html [3] Steensrtup M., Routing in Communication Networks. Prentice Hall Inc, New Jersey, 1995 [4] Hanharan H., Integrated Digital Communications. School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg, 2006. [5] [[[w:Tariffing|http://en.wikipedia.org/wiki/Tariffing]]] [6] Kennedy I.G., Why is Network Planning Important?, Lecture Notes, ELEN5007 - Teletraffic Engineering, School of Electrical and Information Engineering, University of the Witwatersrand, 2005. [[Category:Teletraffic engineering]] [[Category:Tariffing]] l72dhqap594mfgttjwsuvj8wyf0nidq 0 31468 2693484 477111 2024-12-27T00:22:23Z Tule-hog 2984180 nominate speedy (previously abandoned resource, integrated into main) 2693484 wikitext text/x-wiki {{speedy|Moved directly into [[Teletraffic engineering/What is tariffing?]]}} '''Q1. VI.''' '''Ans''' Demand is [http://www.ingrimayne.com/econ/elasticity/Elastic1.html '''inelastic'''] whenever the elasticity coefficient is less than modulars one. When it is greater than then the demand is called [http://www.ingrimayne.com/econ/elasticity/Elastic1.html elastic]. [[Category:Answers]] [[Category:Tariffing]] l4bh6qep2tf7sxum4u0xsjbw5fa4xtz 14 33835 2693439 2389526 2024-12-26T23:28:18Z Tule-hog 2984180 rm merge tag, see [[Category talk:Computer networks#Merge proposal|talk]] 2693439 wikitext text/x-wiki [[Category:Computer science]] [[Category:Networking]] 9w63yt52eg9ktnix761y90tazv935gb 2693516 2693439 2024-12-27T00:48:19Z Tule-hog 2984180 [[wv:bold|boldly]] tag {[[template:category redirect|category redirect]]} 2693516 wikitext text/x-wiki {{category redirect|Networking}} dfru00ja0ptjjbsfd7txd4m4jcvsx1f 10 34399 2693423 2650111 2024-12-26T23:20:22Z Tule-hog 2984180 rm usurped tsh 2693423 wikitext text/x-wiki <noinclude> ==Usage== Type <code><nowiki>{{subst:welcome}}</nowiki></code> on the user's discussion page. No signature is required; it will be placed automatically. </noinclude>==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]]<includeonly>, {{{{{|safesubst:}}}PAGENAME}}</includeonly>!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:{{{{{|safesubst:}}}REVISIONUSER}}|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. 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See you around Wikiversity! --~~<includeonly></includeonly>~~</div> <!-- Template:Welcome --> {{Robelbox/close}}<noinclude> [[Category:Welcome templates]] [[de:Vorlage:Willkommen]] [[el:Πρότυπο:Καλωσόρισμα]] [[es:Plantilla:Bienvenido usuario]] [[fr:Modèle:Bienvenue]] [[it:Template:Benvenuto]] </noinclude> 30zmuv0j714javvpqh37x315c6vev7y 2 37604 2693464 188408 2024-12-26T23:58:23Z Tule-hog 2984180 rm [[:Category:Networks]] 2693464 wikitext text/x-wiki The generative model of feedback networks <ref>Cited by Wei, Wang, Qiuping, Nivanen, Lauret et al (2006-01-12) [http://www.citebase.org/abstract?id=oai%3AarXiv.org%3Aphysics%2F0601091 How to fit the degree distribution of the air network?] </ref>, <ref>Cited by de Meneses, Marcela, da Cunha, Sharon, and Soares et al (2006-01-12) [http://www.citebase.org/abstract?id=oai%3AarXiv.org%3Acond-mat%2F0601273 Preferential attachment scale-free growth model with random fitness]</ref> studied by [[Douglas R. White|White]], [[Nataša Kejžar|Kejžar]], [[Constantino Tsallis|Tsallis]], [[J. Doyne Farmer|Farmer]], and White, or '''social-circles network model''', defines a class of random graphs generated by simple processes that are common to edge formation and feedback loops in social circles. This class is distinct from the [[small-world network]] and the [[scale-free network]] models in [[Network analysis|network analysis]] but also captures many of the characteristics of real-world [[social network]]s. == Highlights == * A feedback network is one where active nodes or agents send tokens or communications through their network to help locate potential partners for new ties. * When a previously unconnected partner is located given constraints of distance a new tie is formed, creating a feedback loop. * Failing to locate a partner within the network, the active node engaged in partner selection recruits a new partner from outside the existing network and forms a tie. * Like many real-world networks, feedback networks modeled by this process evolve large networks with cohesive ties (social circles). In this way, [[Watts and Strogatz model|small worlds]] with cohesive subgroups and relatively short distances may be generated, [[scale-free networks]] may be generated, or a variety of other network topologies may be generated. * The generative feedback network model simulates these processes and their network outcomes as new nodes and edges are added to an initial network. * Only three parameters are used to govern network formation in this probabilistic generative model. These govern levels of node activity, distance decay in how soon network traversal fails to find a partner, and the extent to which local hubs are used in network traversal. * The model provides an explanation for how hubs are formed in networks, how well-traveled edges are formed, how cycles and [[Structural cohesion|cohesion]] are formed, and how other features of real-world networks may result from different combinations of three basic network parameters. * Empirical network data can be fitted to the parameters of the general model. == Rationale == A social-circles network models by a graph-generating process the formation of [[structural cohesion|cohesion]] and [[Star network|dispersion]] in connected networks as a function of acitvity and feedback. The process begins with a single node. The next events in the evolution of the network are that: (1) a node is chosen randomly (proportional to the number of current links of each node raised to power ''alpha'') to emit a feedback token, (2) the token attempts to travel by a non-circular path through the current network by choosing its next neighbor randomly, proportional to the number of edges of the neighbor minus the node already traversed, raised to power ''gamma''; (3) the token travels through the network until it reaches a random distance ''d'', proportional to 1 over ''d'' raised to a power ''beta'', and failing that, the source node that originated the token gains an edge to a node that is added to the network (dispersion of edges), but if the token succeeds in reaching a target at distance d, a new 'feedback' edge is added between the source and the target. The three parameters of the model are ''alpha, beta, gamma.'' The notion of 'feedback' in this model is very general and may involve intentionality on the part of source nodes as agents ''looking for a partner'' or simply nodes in a network emitting signals randomly that contain information that may enable the source and target to lock into a feedback relation, represented by the formation of a new edge. Whether searches or signals have the capacity to locate a target in an existing network is affected by the current size and topology of the network, the [[distance decay]] (''beta'') in traversal, and the intelligence (''gamma'') exhibited in the method of search. Failure to locate targets results in a generic substitution (new node and an edge connecting to it), like looking in the phone book for a doctor rather that asking for a referral. == The generative model == At each step three stochastic processes are run. A node ''i'' is selected, a distance ''d'' (an imaginary number of steps to get to a searched node ''j'') is chosen, and the search path of distance ''d'' is generated. :A node ''i'' is selected with a probability proportional to its degree ::: <math>p_\alpha(i)=\frac{[deg(i)]^\alpha}{\sum_{m=1}^N [deg(m)]^\alpha}</math> :A distance ''d'' is chosen with the probability ::: <math>p_\beta(d)=\frac{d^{-\beta}}{\sum_{m=1}^\infty m^{-\beta}}</math> :where ''d'' represents the number of steps necessary to create a new cycle. If the number of steps cannot be concluded, a new node with a new edge is addded. :A search path of length ''d'' is generated. At a given moment a node ''r'' is a current node on the path. The next node is selected among the neighbors of ''r'' that were not yet visited. The probability of selecting a certain neighbor (say ''l'') is proportional to their 'unused degree'. The 'unused degree', ''u(l)'', is considered a degree of the node from which all the already visited neighbors are subtracted. The probability is given by ::: <math>p_\gamma(l)=\frac {[1+u(l)^{\gamma}]} {\sum_{m=1}^N[1+u(m)^{\gamma}]}</math> == Results == Social-circles network simulation with varying model parameters results in a variety of networks whose topology in terms of [[Social network#Density|network density]], [[Clustering coefficient|clustering]], [[structural cohesion|cohesion]], and [[degree distribution]] varies in ways that resemble the variety of real-world networks. Study of the degree distributions and edge-traversal frequencies shows fit to the statistical distributions of [[Tsallis entropy]], familiar to [[Constantino Tsallis|nonextensive physics]]. This provides a baseline test of the model to real-world networks, one that contrasts with that of the [[scale-free network]], for which one baseline test is whether the degree distribution follows a [[power law]]. The [[Tsallis entropy]] distribution asymptotes to a power law for large degree but bends toward the [[exponential distribution]] in the lower part of the degree distribution. ==Degree Distributions== Parameters <math>\alpha, \beta, \gamma</math> in the social circles model are shown to generate networks with degree distributions that fit a general family of curves that combine power-law with exponential tendencies. The probability density function for degree <math>k</math> in this family, with parameters <math>\delta, \kappa, q </math>, and <math>p_0</math>, is given by: :<math>p(k)=p_0k^\delta e_q^{-k/\kappa}</math> where the [[q-exponential|''q''-exponential]] function <math>e^{-x}_q</math> with parameter <math>q</math> is defined as :<math>e^{-x}_q = (1+(1-q)x)^{1/(1-q)} (e_1^x = e^x) </math> The ''q''-exponential function arises naturally as the solution of the equation <math>dk/dt=k^q</math>, which is <math>k=e^{-x}_q = [1+(1-q) a t]^{1/(1-q)}</math>. For the case where <math>q=1</math>, <math>k=e^{at}</math>. The function is related to the stationary solution of a nonlinear Fokker-Planck equation known as the ''porous medium equation.'' It describes correlated diffusion known as [[Darcy's law]], the scientific basis of fluid permeability used in the earth sciences. This is a constitutive equation that describes the flow of a fluid through a porous medium as formulated by Henry Darcy to describe results of experiments on the flow of water through beds of sand (see references [2,3,4]). In the density function, <math>p_0</math> coincides with <math>p(0)</math> if and only if <math>\delta=0</math>; <math>\kappa</math> is a characteristic degree. In a generalization of the water-flow analogy, the density function for degree would reflect a ''q''-exponential component for which ''q''=1 is omnidirectional flow, and as ''q'' increases there is initially a steep low-dimensional gradient that changes to flat and high dimensional as ''q'' grows to infinity. The other component corresponding to <math>p_0k^\delta</math> would reflect a characteristic magnitude of flow also affected by ''q''. A loose analogy is to traders whose activities etch the routes of their commercial networks, amplified by gradient intensities and the heterogeneity of terrain. ==Retrofitting== The parameters <math>\delta, \kappa, q</math>, and <math>p_0</math> of the fitted models, which may also be applied to the degree distributions of empirical networks, were found to closely approximate 1-to-1 functions of the generating parameters, as given by the approximations : <math>\delta = -1.5 \alpha </math> : <math>q = 1+.64\alpha + 6\alpha\gamma(\beta-1)^2</math> : <math>\kappa = 235\gamma(1-\alpha)(\beta-1)</math> : <math>p_0 = 2 + \alpha/4 - \beta -1.3\gamma</math> These allow approximate solutions for the generating parameters that fit observed empirical degree of networks: : <math>\alpha = - 2\delta/3</math> : <math>\beta = 3/2 + p_o/2 + q/9</math> : <math>\gamma = 5|(q/k)-.5|^{1/2} + 5p0+ 11q/k</math> The last of these approximations (for <math>\gamma</math>) is the least accurate (more precise estimates may yet be found). The authors of the paper have provided a more exact optimal matching between the four parameters fitted from empirical degree distributions of actual networks (assuming they follow the <math>q</math>-exponential) in spreadsheet form.<ref> White et al. [http://eclectic.ss.uci.edu/~drwhite/0/parameter_values1.html Retrofitting the Generative Feedback Network (Social Circles) Models ]</ref> This makes the social circles model a prime candidate for accounting for the degree distributions of empirical networks in terms of a generative network model with the three parameters <math>\alpha, \beta, \gamma</math>: activity bias of agents, distance decay, and navigability of the network. [[Maximum likelihood]] estimates (MLE) of the four <math>\delta, \kappa, q</math>, and <math>p_0</math> parameters for empirical networks and [[goodness-of-fit]] tests can establish the statistical relationship between model and data. Shalizi (2007) provides an [http://www.cscs.umich.edu/~crshalizi/research/tsallis-MLE/ MLE] procedure for the Pareto II or ''q''-exponential parameter fitting.<ref>Shalizi, Cosma. 2007. [http://www.cscs.umich.edu/~crshalizi/research/tsallis-MLE/ Maximum Likelihood Estimation for ''q''-Exponential (Tsallis) Distributions.]</ref> ==Replication and extensions== Kejžar in 2007 replicated these results by simulating feedback networks up to N=5000 nodes, extended the findings to: #Degree distributions for directed edges (indegree, outdetree). These were similar to those of the undirected model, especially for <math>\gamma=0</math> but differed toward higher q when <math>\gamma \rightarrow 1</math> #Densification. Growth of edges relative to nodes is initially linear, but soon changees to approximate a slightly superlinear power-law (for <math>\alpha=0, \beta=1.6, \gamma=0.5, e\approx 0.08+n^{1.08}</math>, for example, where ''e'' are edge numbers and ''n'' numbers of node). Even with 1 billion nodes, however, this is scarcely more than an average degree of 5. Up to 6000 nodes, the average degree is less than two. For large degrees, however, the degree distribution approaches a power law with an exponent <math>1/(q-1) -\delta</math>. Densification is a natural consequence. #Average shortest paths. If ''l'' is the average shortest path and ''n'' the number of nodes, ''l'' shrinks with ''n'' if <math>\gamma=1</math> but rises if <math>\gamma=0</math> and a weighted sum of <math>\alpha, \gamma</math> determines the crossover (low <math>\alpha</math>, high <math>\gamma</math> has the most rapid shrinkage). In every case there tends to be a "shrinking limit" as <math>n \to \infty</math> #Variance of shortest paths. Most of the shortest paths between nodes approach the "shrinking limit." In short, the generative feedback model has realistic and nicely continuous characteristics, and these simulations show no evidence that the density function for degree ''k'' is not the correct function. == References == # [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PLEEE8000073000001016119000001&idtype=cvips&gifs=yes "Generative Model for Feedback Networks"] in ''Physical Review E'', 016119 (2006, [[Douglas R. White]], [[Nataša Kejžar]], [[Constantino Tsallis]], [[J. Doyne Farmer]], [h3ttp://intersci.ss.uci.edu/wiki/index.php/Scott_D._White Scott D. White]. [http://www.santafe.edu/research/publications/wpabstract/200508034|Final paper in pdf as SFI working paper]. [http://www.europhysicsnews.com/us/html/EPNsommaire36.html Reviewed 2005 in Europhysicsnews 36(6):218-220] by [http://www.europhysicsnews.com/full/36/article12.pdf Stefan Thurner]. #[http://www.ingentaconnect.com/content/els/03784371/1995/00000222/00000001/art00211 "Non-extensive statistical mechanics and generalized Fokker-Planck equation." 1995. A. R. Plastino and A. Plastino]. Physica A 222, 347-354 #[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ000046000012123303000001&idtype=cvips&gifs=yes “Nonlinear diffusion equation, Tsallis formalism and exact solutions.’ 2005 P. C. Assis, Jr, L. R. da Silva, E. K. Lenzi, L. C. Malacarne, and R. S. Mendes] Journal of Mathematical Physics 46, 123303 [http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=JMAPAQ000046000012123303000001&idtype=cvips&prog=normal pdf] #[http://www.springerlink.com/content/90e0lbqhycnwleke/ "Nonextensive statistical mechanics: A brief introduction", 2004. C. Tsallis and E. Brigatti.] Continuum Mechanics and Thermodynamics 16(3):223-235. [http://www.springerlink.com/content/90e0lbqhycnwleke/fulltext.pdf pdf] #Nataša Kejžar. 2007. Evolution of Networks. Doctoral thesis. Department of Mathematics. University of Ljubljana. == Links == * [http://eclectic.ss.uci.edu/~drwhite/pub/paj/1250-0-2-0.paj Simulated data in a Pajek Network Project *.paj file for 1250 nodes paramaters 0-2-0] * [http://eclectic.ss.uci.edu/~drwhite/EBMsim/500-0-1d9-0x002.htm Walk-through of a network simulation, edge by edge with the same parameters] * [http://www.insna.org] [[INSNA]] International Network for Social Network Analysis -- professional association of network analysts == Notes == {{reflist}} [[Category:Social networks]] j4e76ej7jheejp6jjtp38e0hwx8oz5c 0 40973 2693539 2113709 2024-12-27T00:58:23Z Tule-hog 2984180 Tule-hog moved page [[Network+/Standards/TCP/IP Model/Introduction]] to [[Network+/Old guides/IP Model]]: alter parent 2113709 wikitext text/x-wiki The Transmission Control Protocol/Internet protocol model refers to a set of rules governing communication between devices across IP networks like the Ethernet. It is implemented as a series of layers. Why layers? Layers allow us to deal with complex systems by separating the problem into its individual components, then dealing with it in a modular manner. This modularization allows the system to be maintained and updated without causing trouble to other parts of the system. == Introduction == The TCP/IP Model is a Department of Defense created system for network transmission. Similar to the OSI Model, the TCP/IP Model uses layers to describe the process of data transmission across a network. Instead of using a seven layers, the TCP/IP Model breaks down the transmission into four layers which are: 4. The Application Layer<br> 3. The Transport Layer<br> 2. The Internet Layer<br> 1. The Link Layer<br> You may see some layers called by different names in different places but the concept of operation remains the same. == Application Layer == The [[Application Layer]], as the name implies, is where the applications or high level protocols reside. Protocols such as FTP, SMTP, [[Domain Name System|DNS]], SNMP, and HTTP all function at this layer. Some programs or applications that are built upon these protocols include the humble internet browser, email, and bittorent. The application layer has many important protocols including HTTP for web-browsing, FTP for file transfer, IMAPv4 for instant messaging, and SMTP for e-mail delivery. {{RoundBoxTop}} The application layer hosts high level protocols such as HTTP and FTP, which enable us to surf the web and transfer files {{RoundBoxBottom}} == Transport Layer == The [[Transport Layer]] handles session management between host computers. TCP, UDP, IGMP and ICMP all function at this layer. Raw data from the application layer is broken down and encapsulated into '''segments''', which are then passed into the lower layer. {{RoundBoxTop}} The Transport Layer attempts to enable the successful transfer of information by maintaining an end to end perspective, ensuring data that exits one end reaches the other. {{RoundBoxBottom}} The Logical Link Control was handled to this layer from a sub layer of Data Link Layer at OSI Model, TCP/IP Model do not implement Logical Link Control at Network Interface Layer. == Internet Layer == The [[Internet Layer]] packages information from the higher layers, determines the path the information is to take across the network, and then encapsulates the segments from the Transport layer into "envelopes" called packets. Path selection happens here, therefore IP and IPSec both operate at this level. Routers and some switches operate at this level. The internet layer has quite a number of protocols like ICMP (Internet control message protocol), IPv4, IPv6 (v4 and v6 refer to version numbers) The internet layer defines the set of rules used to transmit datagrams. In order for both the transmitter and the receiver to understand each other a structure needs to be defined for the datagram. Currently, a transition is occurring between IP protocol version 4 (IPv4), and the more flexible and scalable version 6 ([[IPv6]]). It is not uncommon to see a combination of the two protocols in similar settings. The IP datagram is composed of 20 bytes for the header plus a variable data field that varies based on optimization. == Link Layer == The [[Link Layer]] breaks down the packets from the Internet Layer into frames and then eventually into bits for transmission across the physical network medium. Signaling and network medium standards such as Ethernet, Token Ring, FDDI, X .25, Frame Relay, RS-232, and v.35 are defined in this layer. Network Interface Controller (NIC) Cards, network hubs, repeaters, bridges, and switches operate at this level. Also included in the link layer of the TCP/IP model is the physical mediums (this is a separate layer in the OSI model). The physical mediums consist of twisted copper cables and also wireless transmission mediums that allow bits to travel from machine to machine. The physical layer is how the signal is physically transported from location A to location B across a network. == Differences from the OSI Model == The Application Layer (Layer 4) of the TCP/IP Model combines the Application (Layer 7), Presentation (Layer 6), and Session (Layer 5) layers of the OSI Model. The Link Layer of the TCP/IP Model (Layer 1) combines the Data Link (Layer 2) and Physical (Layer 1) layers of the OSI Model. == See also == * [[Internet Protocol Analysis]] [[Category:Computer Networks]] [[Category:TCP/IP Fundamentals|*]] nruyy9dqkwb7yv6w0obbhvcudoh4ydz 2693548 2693539 2024-12-27T01:03:14Z Tule-hog 2984180 rm cat (tag redir instead) 2693548 wikitext text/x-wiki The Transmission Control Protocol/Internet protocol model refers to a set of rules governing communication between devices across IP networks like the Ethernet. It is implemented as a series of layers. Why layers? Layers allow us to deal with complex systems by separating the problem into its individual components, then dealing with it in a modular manner. This modularization allows the system to be maintained and updated without causing trouble to other parts of the system. == Introduction == The TCP/IP Model is a Department of Defense created system for network transmission. Similar to the OSI Model, the TCP/IP Model uses layers to describe the process of data transmission across a network. Instead of using a seven layers, the TCP/IP Model breaks down the transmission into four layers which are: 4. The Application Layer<br> 3. The Transport Layer<br> 2. The Internet Layer<br> 1. The Link Layer<br> You may see some layers called by different names in different places but the concept of operation remains the same. == Application Layer == The [[Application Layer]], as the name implies, is where the applications or high level protocols reside. Protocols such as FTP, SMTP, [[Domain Name System|DNS]], SNMP, and HTTP all function at this layer. Some programs or applications that are built upon these protocols include the humble internet browser, email, and bittorent. The application layer has many important protocols including HTTP for web-browsing, FTP for file transfer, IMAPv4 for instant messaging, and SMTP for e-mail delivery. {{RoundBoxTop}} The application layer hosts high level protocols such as HTTP and FTP, which enable us to surf the web and transfer files {{RoundBoxBottom}} == Transport Layer == The [[Transport Layer]] handles session management between host computers. TCP, UDP, IGMP and ICMP all function at this layer. Raw data from the application layer is broken down and encapsulated into '''segments''', which are then passed into the lower layer. {{RoundBoxTop}} The Transport Layer attempts to enable the successful transfer of information by maintaining an end to end perspective, ensuring data that exits one end reaches the other. {{RoundBoxBottom}} The Logical Link Control was handled to this layer from a sub layer of Data Link Layer at OSI Model, TCP/IP Model do not implement Logical Link Control at Network Interface Layer. == Internet Layer == The [[Internet Layer]] packages information from the higher layers, determines the path the information is to take across the network, and then encapsulates the segments from the Transport layer into "envelopes" called packets. Path selection happens here, therefore IP and IPSec both operate at this level. Routers and some switches operate at this level. The internet layer has quite a number of protocols like ICMP (Internet control message protocol), IPv4, IPv6 (v4 and v6 refer to version numbers) The internet layer defines the set of rules used to transmit datagrams. In order for both the transmitter and the receiver to understand each other a structure needs to be defined for the datagram. Currently, a transition is occurring between IP protocol version 4 (IPv4), and the more flexible and scalable version 6 ([[IPv6]]). It is not uncommon to see a combination of the two protocols in similar settings. The IP datagram is composed of 20 bytes for the header plus a variable data field that varies based on optimization. == Link Layer == The [[Link Layer]] breaks down the packets from the Internet Layer into frames and then eventually into bits for transmission across the physical network medium. Signaling and network medium standards such as Ethernet, Token Ring, FDDI, X .25, Frame Relay, RS-232, and v.35 are defined in this layer. Network Interface Controller (NIC) Cards, network hubs, repeaters, bridges, and switches operate at this level. Also included in the link layer of the TCP/IP model is the physical mediums (this is a separate layer in the OSI model). The physical mediums consist of twisted copper cables and also wireless transmission mediums that allow bits to travel from machine to machine. The physical layer is how the signal is physically transported from location A to location B across a network. == Differences from the OSI Model == The Application Layer (Layer 4) of the TCP/IP Model combines the Application (Layer 7), Presentation (Layer 6), and Session (Layer 5) layers of the OSI Model. The Link Layer of the TCP/IP Model (Layer 1) combines the Data Link (Layer 2) and Physical (Layer 1) layers of the OSI Model. == See also == * [[Internet Protocol Analysis]] [[Category:Computer Networks]] ga1t894m1taj5x7njvbscyjmotn48ad 0 41213 2693427 1570927 2024-12-26T23:22:31Z Tule-hog 2984180 Tule-hog moved page [[Network+/Standards/Theory and Concepts/Introduction]] to [[Network+/Old guides/Ethernet]]: alter parent 1570927 wikitext text/x-wiki Ethernet is a family of frame-based computer networking technologies for local area networks (LANs). The name comes from the physical concept of the ether. It defines a number of wiring and signaling standards for the physical layer, through means of network access at the Media Access Control (MAC)/Data Link Layer, and a common addressing format. Ethernet is standardized as IEEE 802.3. The combination of the twisted pair versions of Ethernet for connecting end systems to the network, along with the fiber optic versions for site backbones, is the most widespread wired LAN technology. It has been in use from the 1990s to the present, largely replacing competing LAN standards such as token ring, FDDI, and ARCNET. In recent years, Wi-Fi, the wireless LAN standardized by IEEE 802.11, is prevalent in home and small office networks and augmenting Ethernet in larger installations. Ethernet is used in most, but not all, computer networking situations. It is a standard which defines rules that all computers follow to allow successful and efficient communication. [[Category:Computer networks]] 8y7vcdtrtdzwqsh2yp6g33nigra3ndr 2693433 2693427 2024-12-26T23:23:51Z Tule-hog 2984180 add {[[template:bookcat|bookcat]] 2693433 wikitext text/x-wiki Ethernet is a family of frame-based computer networking technologies for local area networks (LANs). The name comes from the physical concept of the ether. It defines a number of wiring and signaling standards for the physical layer, through means of network access at the Media Access Control (MAC)/Data Link Layer, and a common addressing format. Ethernet is standardized as IEEE 802.3. The combination of the twisted pair versions of Ethernet for connecting end systems to the network, along with the fiber optic versions for site backbones, is the most widespread wired LAN technology. It has been in use from the 1990s to the present, largely replacing competing LAN standards such as token ring, FDDI, and ARCNET. In recent years, Wi-Fi, the wireless LAN standardized by IEEE 802.11, is prevalent in home and small office networks and augmenting Ethernet in larger installations. Ethernet is used in most, but not all, computer networking situations. It is a standard which defines rules that all computers follow to allow successful and efficient communication. {{bookcat}} 3n15365xif6r51pfsjcnn708v37b88p 1 41214 2693429 1570929 2024-12-26T23:22:31Z Tule-hog 2984180 Tule-hog moved page [[Talk:Network+/Standards/Theory and Concepts/Introduction]] to [[Talk:Network+/Old guides/Ethernet]]: alter parent 1570929 wikitext text/x-wiki Starting the page with a little info! --[[User:Jezarnold|Jezarnold]] 10:26, 9 September 2007 (UTC) :Thank you, I have added a template. I am hopeful that the content will increase over time. ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] ([[w:de:Benutzer:Erkan_Yilmaz/Bewertung|<small>evaluate me!</small>]], [[User_talk:Erkan_Yilmaz|<small>discussion</small>]]) 10:46, 9 September 2007 (UTC) :Also a category was added [http://en.wikiversity.org/w/index.php?title=Topic%3AEthernet&diff=157554&oldid=157529]. ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] ([[w:de:Benutzer:Erkan_Yilmaz/Bewertung|<small>evaluate me!</small>]], [[User_talk:Erkan_Yilmaz|<small>discussion</small>]]) 11:17, 9 September 2007 (UTC) 8e03vdxiol2bwmge62r6wqb8lqrtqsy 0 41744 2693500 2474929 2024-12-27T00:35:34Z Tule-hog 2984180 Tule-hog moved page [[Network+/Standards/OSI Model/Introduction]] to [[Network+/Old guides/OSI Model]]: alter parent 2474929 wikitext text/x-wiki The Open System Interconnection (OSI) Model is a seven layer model developed by International Organization for Standardization (ISO) in the late 1970s. It is a layered, abstract definition for communications protocol and computer network protocol design. Every layer has certain functionalities which provide services to the layer immediately above it and below it. The seven layers are, from Top to Bottom, Application, Presentation, Session, Transport, Network, Data link and Physical layer. <!--- Not entirely true and slightly biased. Cisco still teaches the OSI model. ~~~ "OSI Model does not reflect the real world scenario and have rather been reclused to academic purpose. It's predecessor, TCP/IP, is now the dominant model in terms of representing real world scenarios and it's associated entities." ---> The OSI Layered Model defines the following seven layers: * 7 - Application Layer * 6 - Presentation layer * 5 - Session layer * 4 - Transport layer * 3 - Network layer * 2 - Data Link layer * 1 - Physical layer The OSI layers are often referred to by their layer number, for example, "IP is a layer 3 protocol". It is important to note that the OSI model has long been retired as a real practical use stack in favor of the TCP/IP stack. However, for educational purposes and for a more complete breakdown of how intercommunication works, the OSI model is used for that purpose. == Application Layer == The Application layer provides network services to the user's applications. In essence, this does not refer to the actual application itself (ie. Email client, Web browser) but the actual protocols that it utilizes such as: HTTP, POP3, IMAP, SMTP, DHCP, DNS and many others in that same category. These higher level protocols are what the application uses to present the information to you. == Presentation Layer == The Presentation Layer exists to make sure that the Application Layer of the sender sends information that can be read by the Application Layer of the receiver. Included in this layer is data conversion, data compression, and data encryption. In typical real world scenarios this layer is not always used so don't be confused and figure out why sometimes it doesn't make a lot of sense that you can't include this in most intercommunication facets. == Session Layer == The Session Layer is responsible for the establishment, management, and termination of communication sessions between two hosts software applications. This may seem confusing as you might believe that this is the job of TCP to establish, maintain and terminate communication between hosts. The session layer deals more with how two software applications establish, maintain and terminate communication among themselves completely separate from the actual practical intercommunication of data packets. The capabilities of the session layer are almost always found in API's (Application Programming Interfaces). Examples of session layer "protocols" are: NetBIOS, TCP/IP Socket, RPC and Unix Sockets. == Transport Layer == In the most basic sense the Transport Layer controls the segmentation and reassembly of the message. This layer is also responsible for error control and flow control. This layer is primarily known for two protocols that you will encounter in real world scenarios: TCP and UDP. Transmission Control Protocol is the famous TCP in TCP/IP. Transmission Control Protocol is a session/connection oriented protocol. UDP (Uniform Datagram Protocol) is a connectionless oriented protocol that provides no error recovery. It does everything else that TCP does minus that one important aspect. This is useful for communications that do not need reliable communication such as: DNS and VOIP. TCP and UDP are responsible for taking many connections coming into a server and forwarding the data correctly based upon port numbers. Thus, TCP and UDP are responsible for port numbering for applications. This is best explained if you have ONE server that is running these services: Web Server (80), DNS (53), SMTP (25), IMAP (143) and POP (110). Many people can connect to this server for several reasons. Some people just want to view a web page, thus, the TCP header will include the destination port as 80 and they will get their web page and if they want to relay mail the TCP header includes 25 and the SMTP service answers forwarding the mail. TCP, unlike UDP, offers error recovery through a process called windowing using SYN and ACK controls. Essentially, a computer send a SYN (Synchronize) to a server and the server responds with an ACK (Acknowledgement) and in that the server itself also includes it's own SYN flag as well for the ACK that will follow from the host and it also includes the size of the window (the amount of unconfirmed packets that can be sent at once before an ACK is sent back). Windowing allows a certain amount of packets to arrive while not receiving an ACK. Once this window is full, in theory, the server responds with an ACK and if that ACK number is in sequence with the next SYN to be sent the sending host knows it received all the packets. If it doesn't get a response it sends all the packets over again and if the ACK from the server is a numeric value LESS than what was sent the host resend the packets that were after that and waits for the correctly numbered ACK. == Network Layer == The Network Layer is responsible for the transport of a packet from one network to another. Logical addressing is used to achieve this. This is not required if two computers are directly connected to one another such as in a LAN setting where MAC addressing can be used to communicate among two computers. What exists at this layer is most famously the IP in TCP/IP (Internet Protcol v4 and v6). This layer deals with IP addresses to provide "logical" addressing of systems both in a LAN and WAN environments. At this layer routing also takes place to route data between two different subnets. The network layer is also responsible for fragmentation of the packets if they are to be sent down a Layer 2 link that has a smaller MTU. Most importantly IP is used for routing and logical addressing of machines. == Data Link Layer == The Data Link Layer is responsible for the transport of ‘frames’ on the same network. Physical addressing is used to achieve this. This layer implements access control to determine which devices are able to transmit on a network with multiple devices using such processes as CSMA/CD (Carrier Sense Multiple Access / Collision Detection) on networks that utilize full duplex communication. This layer refers to the physical addressing (MAC addresses) that are burned into the each NIC (Network Interface Card) and is unique among all cards in the world, although the MAC address of certain network adapters can be changed, such as [http://www.sdadapters.com SpeedDemon network adapters]. == Physical Layer == The Physical Layer controls the transmission and reception of the bit stream over a physical medium. This layer defines mediums such as UTP/STP, Fiber, Coax etc. It defines cabling pin outs, electrical conductivity, light amplification (fiber), cabling distance and the other physical features. == Goals of the OSI Model == The goal of the OSI Model was to produce an open and standardized network model that would allow vendor-independent communication between networked devices. This was to offer an alternative to the many proprietary protocols developed by companies at the time which had effectively tied customers into buying from one main provider. <!--- Think of better way of giving example for this. ~~ "Examples of this are IBM's Systems Network Architecture (SNA) and Digital Equipment Corporation's DECnet." ---> Alongside the TCP/IP Model, the OSI Model also helps to teach students of networking by breaking down network transmissions into easily understandable modules so it is not uncommon to come across either of the models in a learning environment. <!--- ==== The Application Layer - Layer 7 ==== The Application Layer consists of the applications used by everyday users. Telnet, FTP ==== The Presentation Layer - Layer 6 ==== Encoding/encryption JPG, MPG, etc. ==== The Session Layer - Layer 5 ====<blockquote> <blockquote> Block quote </blockquote> </blockquote> Handles communication sessions. ==== The Transport Layer - Layer 4 ==== TCP, UDP, ==== The Network Layer - Layer 3 ==== The Network Layer is the layer at which data from Layer 4 xxxx are encapsulated into units called packets. IP, IPSec, Routing, Path Determination ==== The Data Link Layer - Layer 2 ==== The Data Link Layer is the layer at which data from Layer 3 Packets are encapsulated into units called frames. Link Control, Switching, Bridging ==== The Physical Layer - Layer 1 ==== The Physical Layer is the layer responsible for the transmission of the data frames from Layer 2. This layer breaks the data down into bits (1s and 0s) and transmits them 1 at a time across the physical transmission medium. This sounds very slow at first, but the transmission of a bit can occur hundreds if not thousands of times in just a small fraction of a second. Repeaters and hubs are associated with this layer, since the only operations they perform is repeating the signal out one or many ports. ---> [[Category:Computer Networks]] m825t6mfs2gdj8y0fpdsd6m8xml5uzr 2693509 2693500 2024-12-27T00:42:48Z Tule-hog 2984180 /* Table and acrostics */ transclude 2693509 wikitext text/x-wiki The Open System Interconnection (OSI) Model is a seven layer model developed by International Organization for Standardization (ISO) in the late 1970s. It is a layered, abstract definition for communications protocol and computer network protocol design. Every layer has certain functionalities which provide services to the layer immediately above it and below it. The seven layers are, from Top to Bottom, Application, Presentation, Session, Transport, Network, Data link and Physical layer. <!--- Not entirely true and slightly biased. Cisco still teaches the OSI model. ~~~ "OSI Model does not reflect the real world scenario and have rather been reclused to academic purpose. It's predecessor, TCP/IP, is now the dominant model in terms of representing real world scenarios and it's associated entities." ---> The OSI Layered Model defines the following seven layers: * 7 - Application Layer * 6 - Presentation layer * 5 - Session layer * 4 - Transport layer * 3 - Network layer * 2 - Data Link layer * 1 - Physical layer The OSI layers are often referred to by their layer number, for example, "IP is a layer 3 protocol". It is important to note that the OSI model has long been retired as a real practical use stack in favor of the TCP/IP stack. However, for educational purposes and for a more complete breakdown of how intercommunication works, the OSI model is used for that purpose. == Application Layer == The Application layer provides network services to the user's applications. In essence, this does not refer to the actual application itself (ie. Email client, Web browser) but the actual protocols that it utilizes such as: HTTP, POP3, IMAP, SMTP, DHCP, DNS and many others in that same category. These higher level protocols are what the application uses to present the information to you. == Presentation Layer == The Presentation Layer exists to make sure that the Application Layer of the sender sends information that can be read by the Application Layer of the receiver. Included in this layer is data conversion, data compression, and data encryption. In typical real world scenarios this layer is not always used so don't be confused and figure out why sometimes it doesn't make a lot of sense that you can't include this in most intercommunication facets. == Session Layer == The Session Layer is responsible for the establishment, management, and termination of communication sessions between two hosts software applications. This may seem confusing as you might believe that this is the job of TCP to establish, maintain and terminate communication between hosts. The session layer deals more with how two software applications establish, maintain and terminate communication among themselves completely separate from the actual practical intercommunication of data packets. The capabilities of the session layer are almost always found in API's (Application Programming Interfaces). Examples of session layer "protocols" are: NetBIOS, TCP/IP Socket, RPC and Unix Sockets. == Transport Layer == In the most basic sense the Transport Layer controls the segmentation and reassembly of the message. This layer is also responsible for error control and flow control. This layer is primarily known for two protocols that you will encounter in real world scenarios: TCP and UDP. Transmission Control Protocol is the famous TCP in TCP/IP. Transmission Control Protocol is a session/connection oriented protocol. UDP (Uniform Datagram Protocol) is a connectionless oriented protocol that provides no error recovery. It does everything else that TCP does minus that one important aspect. This is useful for communications that do not need reliable communication such as: DNS and VOIP. TCP and UDP are responsible for taking many connections coming into a server and forwarding the data correctly based upon port numbers. Thus, TCP and UDP are responsible for port numbering for applications. This is best explained if you have ONE server that is running these services: Web Server (80), DNS (53), SMTP (25), IMAP (143) and POP (110). Many people can connect to this server for several reasons. Some people just want to view a web page, thus, the TCP header will include the destination port as 80 and they will get their web page and if they want to relay mail the TCP header includes 25 and the SMTP service answers forwarding the mail. TCP, unlike UDP, offers error recovery through a process called windowing using SYN and ACK controls. Essentially, a computer send a SYN (Synchronize) to a server and the server responds with an ACK (Acknowledgement) and in that the server itself also includes it's own SYN flag as well for the ACK that will follow from the host and it also includes the size of the window (the amount of unconfirmed packets that can be sent at once before an ACK is sent back). Windowing allows a certain amount of packets to arrive while not receiving an ACK. Once this window is full, in theory, the server responds with an ACK and if that ACK number is in sequence with the next SYN to be sent the sending host knows it received all the packets. If it doesn't get a response it sends all the packets over again and if the ACK from the server is a numeric value LESS than what was sent the host resend the packets that were after that and waits for the correctly numbered ACK. == Network Layer == The Network Layer is responsible for the transport of a packet from one network to another. Logical addressing is used to achieve this. This is not required if two computers are directly connected to one another such as in a LAN setting where MAC addressing can be used to communicate among two computers. What exists at this layer is most famously the IP in TCP/IP (Internet Protcol v4 and v6). This layer deals with IP addresses to provide "logical" addressing of systems both in a LAN and WAN environments. At this layer routing also takes place to route data between two different subnets. The network layer is also responsible for fragmentation of the packets if they are to be sent down a Layer 2 link that has a smaller MTU. Most importantly IP is used for routing and logical addressing of machines. == Data Link Layer == The Data Link Layer is responsible for the transport of ‘frames’ on the same network. Physical addressing is used to achieve this. This layer implements access control to determine which devices are able to transmit on a network with multiple devices using such processes as CSMA/CD (Carrier Sense Multiple Access / Collision Detection) on networks that utilize full duplex communication. This layer refers to the physical addressing (MAC addresses) that are burned into the each NIC (Network Interface Card) and is unique among all cards in the world, although the MAC address of certain network adapters can be changed, such as [http://www.sdadapters.com SpeedDemon network adapters]. == Physical Layer == The Physical Layer controls the transmission and reception of the bit stream over a physical medium. This layer defines mediums such as UTP/STP, Fiber, Coax etc. It defines cabling pin outs, electrical conductivity, light amplification (fiber), cabling distance and the other physical features. == Goals of the OSI Model == The goal of the OSI Model was to produce an open and standardized network model that would allow vendor-independent communication between networked devices. This was to offer an alternative to the many proprietary protocols developed by companies at the time which had effectively tied customers into buying from one main provider. <!--- Think of better way of giving example for this. ~~ "Examples of this are IBM's Systems Network Architecture (SNA) and Digital Equipment Corporation's DECnet." ---> Alongside the TCP/IP Model, the OSI Model also helps to teach students of networking by breaking down network transmissions into easily understandable modules so it is not uncommon to come across either of the models in a learning environment. <!--- ==== The Application Layer - Layer 7 ==== The Application Layer consists of the applications used by everyday users. Telnet, FTP ==== The Presentation Layer - Layer 6 ==== Encoding/encryption JPG, MPG, etc. ==== The Session Layer - Layer 5 ====<blockquote> <blockquote> Block quote </blockquote> </blockquote> Handles communication sessions. ==== The Transport Layer - Layer 4 ==== TCP, UDP, ==== The Network Layer - Layer 3 ==== The Network Layer is the layer at which data from Layer 4 xxxx are encapsulated into units called packets. IP, IPSec, Routing, Path Determination ==== The Data Link Layer - Layer 2 ==== The Data Link Layer is the layer at which data from Layer 3 Packets are encapsulated into units called frames. Link Control, Switching, Bridging ==== The Physical Layer - Layer 1 ==== The Physical Layer is the layer responsible for the transmission of the data frames from Layer 2. This layer breaks the data down into bits (1s and 0s) and transmits them 1 at a time across the physical transmission medium. This sounds very slow at first, but the transmission of a bit can occur hundreds if not thousands of times in just a small fraction of a second. Repeaters and hubs are associated with this layer, since the only operations they perform is repeating the signal out one or many ports. ---> ==Table and acrostics== {{/OSI Components}} [[Category:Computer Networks]] lzs03a28yf5ka6t5e5n91ihgcxpitu4 0 41745 2693530 492343 2024-12-27T00:50:42Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693530 wikitext text/x-wiki #REDIRECT [[Voice over Internet Protocol]] == '''Voice Over IP Introduction''' == Stands for: VOICE OVER INTERNET PROTOCOL Voice Over IP is the technology that uses communications protocol to carry voice packets over an IP link. The end-point are typically IP phones or can be computers that uses headset and speaker. Voice before being sent to destination via IP link is converted from analog to IP by means of '''''[[DSP]]''''' resources that reside in the IP phones. VoIP technology has brought a revolution in the networking industry by enabling voice packets to traverse through the same link as the data link, thus helping organization to reduce call cost. Companies are able to remove the need to use a separate voice path, remove recurring cost of PBX installation, since VoIP and more preferably it's subset IP Telephony provides centralised call controlling system. The voice packets traverse over the internet/IP link using UDP protocol riding on RTP packets. The stages that are involved from converting the analog to IP voice and then transmitting it through the IP link are mentioned below: 1. Analog signal (Voice that user generates to speak) is sampled as per Nyquist theorem 2. The sample voice is Quantised (Called Quantization) 3. Quantised sample is encoded using 8 bit code 4. The encoded voice is compressed (optional) 5. The compressed voice is sent across the IP link to the remote site ---- == Sampling Theorem == Sampling is a process of converting Analog signals into a set of numeric value that can be further transformed into bits for the communication systems to understand. It is basically the conversion or representation of voltages into a digitised format. Sampling theory is based on '''Nyquist-Shannon's''' theorem which states that the sampling rate should be twice the rate of highest frequency of Audible voice so that a perfect reconstruction is possible. The maximum frequency of Human voice is approximately 9000 Hz, although the majority of normal human speech falls in the 3500 hz range. Industry standard value for sampling is considered as 4000 Hz (4KHz). So the sampling rate would be 2 times of 4000 Hz = 8000 samples per second. [[Category:Networking]] [[Category:Computer science]] ht9ygbcoeehrzq5kuk5qdq6dev3pcef 0 51982 2693523 1823393 2024-12-27T00:49:32Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693523 wikitext text/x-wiki #REDIRECT [[Network+/Architecture/Media]] ==Network Communication Mediums== === Twisted-Pair Cable === Twisted-pair cabling is a copper wire that comes in two forms, shielded and unshielded. It is the most common form of wiring used in a network. It uses 8 wires twisted into pairs to cancel the effect of crosstalk (Noise from the adjacent wires). === Unshielded Twisted-Pair Cable === UTP relies on the cancellation effect of twisting the wires to reduce ''Electromagnetic Interference'' (EMI). It is required to have a certain amount of twists per meter and it is connected using a ''Registered Jack 45 Connecter'' (RJ-45). UTP can run for 100 meters before the signal needs to be refreshed. UTP has advantages that make it ideal in some networks. * '''Easy to Install''' * '''Small, it does not take up much space in wiring ducts.''' * '''Cheapest type of cable.''' <br> UTP has 6 Categories. *'''Category 1:''' **Only reliable for transmitting telephone communications, not regular data transmissions. *'''Category 2:''' **Previously used in token rings. Speeds only up to 4kbit *'''Category 3:''' **Works in 10BASE-T networks.. Transfer rate of 10mbit *'''Category 4:''' **Used on 16mbit token ring networks. *'''Category 5:''' **Transfer rate of 100mbit. Unreliable for 1000BASE-T networks. *'''Category 5e:''' **Transfer rate of 1000mbit. Used in Gigabit Ethernet networks. *'''Category 6:''' **Same as Cat 5e but made to a higher standard === Shielded Twisted-Pair Cable === Shielded Twisted-Pair Cable uses twisted pairs along with a metallic foil shielding to reduce the crosstalk and EMI. It is usually connected using an STP connecter but can also be connected with an RJ-45. Although it reduces the interference better than UTP, STP has many drawbacks that keep it from having a mainstream use. * '''More Expensive''' * '''Must be grounded at both ends''' * '''Harder to install''' <br> Because of these drawbacks it is rarely implemented in Ethernet networks. It is more common in Europe. === Coaxial Cable === Coaxial Cable uses a copper wire for the conductor, on top of this is insulation for the wire. The third layer consists of a metallic foil or woven copper braid as a shielding, followed by a rubber jacket on the top. It is often referred to as Thicknet or Thinnet, depending on the specification. <br> Benefits: * '''It costs less to buy than Fiber Optic''' * '''It has speeds of 10mb/s to 100mb/s.''' * '''It costs more to install Coaxial cable.''' === Fiber Optic === Transfer rates of up to 10GB/s and a distance of up to 1000 meters. This is an expensive type of medium and takes a special connector to terminate the signal. It has higher bandwidth possibilities and is best suited for backbone installations. It has 3 parts to the cable: Core, Cladding and Buffer. * '''Core:''' ** This is where the light is transmitted * '''Cladding:''' ** Just outside the core it traps the light inside the core and helps guide it around corners. * '''Buffer:''' ** The hard plastic coating on the outside of the cable that protects the core from moisture and physical damage. == 802.11 == Also known as ''wireless''. To use this medium you have to have a wireless router to put out a wireless signal. To receive this signal your PCs have to be equipped with Wireless NICs (Network Interface Cards). There are four types of wireless standards, 802.11b, 802.11a, 802.11g and 802.11n. ==== 802.11 ==== This was the first type of 802.11 signal. It has a maximum transfer rate of 2mb/s, much too slow for applications so it is no longer manufactured. ==== 802.11a ==== Transfer rate of 54mb/s and uses a higher and regulated frequency. Often thought that 802.11a was made after 802.11b. It was actually made at the same time but was slower to gain popularity because of its higher cost. 802.11a is incompatible with 802.11b. 802.11a/b devices actually just run the standards side by side. <br><br> Benefits: * '''Faster Maximum Speed''' * '''No interference from outside devices''' <br> Drawbacks: * '''Higher Cost''' * '''Shorter Range''' * '''Walls and other solids interfere with signal''' ==== 802.11b ==== Transfer rates of up to 11mb/s, it uses an unregulated frequency of 2.4GHz. Since it is unregulated, it gets interference from outside sources such as microwaves and cordless phones. Manufacturers often lower the frequency to reduce the production cost. This is best installed away from other devices and is more useful for homes than it is for business. <br><br> Benefits: * '''Low Cost''' * '''Good Signal Range''' * '''Signal unobstructed by most walls and other solids''' <br> Drawbacks: * '''Slow Maximum Speed''' * '''Gets interference from other devices''' ==== 802.11g ==== Transfer rate of 54mb/s and uses 2.4GHz frequency for greater range. It was designed to get the benefits of both 802.11b and 802.11a and is compatible with 802.11b. <br><br> Benefits: * '''Faster Maximum Speed''' * '''Good Signal Range''' * '''Signal unobstructed by walls and other solids''' <br> Drawbacks: * '''Higher Cost''' * '''Interference from other devices''' ==== 802.11n ==== Transfer rate of 100mb/s and uses an increased signal frequency. 802.11n uses multiple signals and antennas instead of just one, to increase the amount of bandwidth. 802.11n is compatible with 802.11g. <br><br> Benefits: * '''Fastest Maximum Speed for Wireless.''' * '''Best Signal Range''' * '''Less interference from outside devices''' <br> Drawbacks: * '''Most Expensive''' * '''The use of many signals can interfere with 802.11b/g networks''' [[Category:Networking]] baz4ga364fg8zmxf4w4i4o76lvmmh3p 0 54343 2693525 2083992 2024-12-27T00:49:52Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693525 wikitext text/x-wiki #REDIRECT[[Network Administration]] == Introduction == The term Network Topology refers to the logical representation of the layout of your network. While the topology of your network may closely resemble the physical layout of your network, it is entirely possible for the two to be completely different. There are several common network topologies, and they are explained below. == Network Topologies == '''==== Bus ====''' In a bus topology, computers in a data network are connected to each other in a linear fashion, or from network card to network card. This topology is the most prone to failure, as a severed link between any of the computers near the middle of the network would break the network into two segments. ==== Ring ==== In a ring topology, computers are connected in a linear fashion, but either end of the network is connected to the other. This topology is provides more protection against failure than a bus topology, as a severed link would result in traffic traveling in the opposite direction around the ring. ==== Star ==== In a star topology, a computer or device with multiple network cards/ports acts as a central connection point for all other devices on the network. ==== Extended Star ==== An extended star topology functions much like a star topology, but, as the name implies, it offers a hierarchical approach to the network. The best example of an extended star topology is to visualize two or more star networks connected together. ==== Partial Mesh ==== In a partial mesh topology, almost every computer or device has at least one connection to every other device on the network. This is the next best failure resistant topology as it is not as expensive as a full mesh, but more expensive than any of the other topologies. ==== Full Mesh ==== In a full mesh topology, each computer or device has at least one connection to every other device on the network. This is the most failure resistant topology, but also the most expensive as extra network cards and cable is required as the network grows. [[Category:Networking]] qsxe7yb0ikp99l7xiq690m6oyvcpnbc 0 54376 2693524 1046694 2024-12-27T00:49:42Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693524 wikitext text/x-wiki #REDIRECT[[Network Administration]] __NOTOC__ == [[w:Network_card|Network Interface Controller (NIC) Card]] == *Typically when you think of a NIC Card, you should think of an NIC embedded onto the motherboard of your computer or an expansion card that is installed into an ISA,PCI, or PCI-Express slot inside your computer. == [[Repeater]] == *One of the limitations of computer networking is found in the transmission medium. Certain cables are only capable of transmitting a certain distance before a concept called [[w:Attenuation|Attenuation]] comes into play. If the distances of a cable run exceed the physical limitations of the medium, a repeater may be placed before the limitation distance to recondition and repeat the signal so that it may run the rest of the length of the cable. == [[w:Network_hub|Hubs]] == *Hubs are very basic devices that are made up of many NIC ports. They take the electrical signals that a computer transmits into them and repeats them out every port on the device except for the one the signals arrived in. Since hubs offer no services other than repeating signals to multiple ports, they are often called multiport repeaters. == [[w:Network_bridge|Bridges]] == *A '''network bridge''' connects multiple network segments at the data link layer (layer 2) of the OSI model, and the term '''layer 2 switch''' is often used interchangeably with bridge. Bridges are similar to repeaters or network hubs, devices that connect network segments at the physical layer, however a bridge works by using bridging where traffic from one network is managed rather than simply rebroadcast to adjacent network segments. In Ethernet networks, the term "bridge" formally means a device that behaves according to the [[w:802.1d|IEEE 802.1D]] standard—this is most often referred to as a network switch in marketing literature. == [[w:Network_switch|Switches]] == *Low-end network switches appear nearly identical to [[w:Network_hub|network hubs]], but a switch contains more "intelligence" (and comes with a correspondingly slightly higher price tag) than a network hub. Network switches are capable of inspecting data packets as they are received, determining the source and destination device of that packet, and forwarding it appropriately. By delivering each message only to the connected device it was intended for, a network switch conserves network bandwidth and offers generally better performance than a hub. ==[[w:Router|Routers]]== [[File:Router.svg|thumb|75px|right|the schematic Symbol for a [[w:Router|Router ]] ]] *A router allows connectivity to one or more computers, helping create a network. For home users, these are particularly useful for taking a single broadband internet account, and spreading it to at least two or more computers. Standard routers require the internet connection from a standalone modem but modem-routers are increasing in popularity, which can be plugged into any broadband-enabled phone line, reducing cable clutter, and only taking up one power socket. *In the telecoms industry, industrial routers form the backbone of the internet. They work rather like telephone exchanges, passing data between network segments to form a connection. Each router has a configuration table, or routing table, containing information on which connections lead to certain groups of addresses, which connections have priority for usage, and rules for handling different kinds of traffic. A typical home/office router has a very small routing table, but the big routers that handle the main internet traffic can have huge complicated routing tables. Each time a router receives a packer of data it will attempt to send it along the best possible route to its destination, based on its routing table. If that connection is not currently available, it will send it along the next best route. In this way, the routers that form the internet can reconfigure the paths packages take to work around any problems with the network. *The rules for handling traffic are an important part of internet security. A home/office router may have rules limiting how computers outside the network can connect to computers inside the network, as well as preventing private network traffic from spilling into the outside world. Many home routers include additional security features - they scan and filter all traffic that passes through them, usually through an integrated firewall in the hardware. Some may carry out other useful roles such as acting as a print server. *Wireless routers have become more common. A wireless router does exactly the same job in the home as a regular wired (Ethernet) router, with the difference that a computer can be connected to it without needing to run Ethernet cable between the computer and the router. All you need is a wireless network adapter in each PC you want to connect, usually in the form of a card in your PCI slot (or a laptops PCMCIA card slot) or an adapter for USB. Wireless routers generally have four ports to connect Ethernet cable as well, so computers can be connected by whatever means is most convenient - you might want to use a cable for your desktop PC, which sits right next to the router, but use the wireless adapter in your laptop. == [[w:Firewall|Firewalls]] == *A firewall is a dedicated appliance, or software running on another computer, which inspects network traffic passing through it, and denies or permits passage based on a set of rules. A firewall's basic task is to regulate some of the flow of traffic between [[computer network]]s of different trust levels. Typical examples are the [[Internet]] which is a zone with no trust and an [[intranet|internal network]] which is a zone of higher trust. A zone with an intermediate trust level, situated between the Internet and a trusted internal network, is often referred to as a "perimeter network" or [[Demilitarized zone (computing)|Demilitarized zone]] (DMZ). == Other Devices == *[[w:IP_Phone|IP Phone]] ::An IP phone uses Voice over IP technologies allowing telephone calls to be made over an IP network such as the internet instead of the ordinary PSTN system. Calls can traverse the Internet, or a private IP Network such as that of a company. ==Summary== In summary, according to this page, network devices primarily consist of: # Network Interface Controller (NIC) Card # Repeater # Hubs # Bridges # Switches # Routers # Firewalls # Other Devices [[Category:Networking]] saaquunsp3rfcosfql2l2pxo8eoq38p 0 54379 2693355 1570816 2024-12-26T19:35:19Z Tule-hog 2984180 Tule-hog moved page [[Network+/Architecture/Routing/Introduction]] to [[Network+/Old guides/Routing]]: alter parent 1570816 wikitext text/x-wiki Routing Protocols are used to dynamically propagate network routes to other routers on the network. This saves administrative work as each route required by a network does not have to be manually configured on every router in the network. Routing protocols are defined in three broad classes Distance Vector, Link State, and Hybrid. Distance Vector devices calculate the route by how far away (in hops) the destination is, while Link State protocols can use many other values such as bandwidth, latency, reliability and load. Hybrids, as the name implies, use a mixture of Distance Vector and Link State methods. ==Routing Protocols== ===Distance Vector=== Distance Vector routing is sometimes derisively described as routing by rumor. Referring to the fact that no single router really know more about the topology and state of the network than what his neighbour tells/advertises to him. Each neighbour is presenting a list of destinations and promising that that traffic forwarded via itself will reach these destinations. Some form of numeric values, (metrics/hop counts/cost/etc), are attached to these advertisements between neighbours so that the receiving device can compare competing routing to the same destination, from different neighbors, and tie break between them or even determine eligibility for load balancing of traffic over equal cost multiple paths (ECMP). The name Distance Vector was chosen because the neighbour from which you learn a potential path to a destination, (route), represents a possible direction of travel through the network to reach that destination. Hence vector. Distance refers to the numeric values advertised with the routes. Given multiple paths to the same destination but following different vector, neighbours, we can imagine the numbers indicate a distance to the destinations. More over a router would prefer to forward traffic via the shorted path. Administrators can manipulate these values to prefer one path over another. This ability to manipulate the advertised values/metrics mean that there is no hard link between the value and Actual distance to a destination. Distance Vector routing protocols repeatedly announce their known destinations to its neighbours at specific intervals. Routers are considered possible neighbours for distance vector protocols when they are linked together by some transmission medium. For example ethernet, ATM, Packet over Sonet/SDH, Frame relay or DSL links. There are many more possible types of links / interfaces that can connect routers together, some of more complex logical rather than physical links. *'''RIP''' The Routing Information Protocol (RIP) is a routing protocol that was designed for small networks. It can handle a maxiumum of 16 hops and is a classful routing protocol meaning that it does not support Variable Length Subnet Masks (VLSM) and does not support authentication. *'''RIPv2''' The Routing Information Protocol version 2 (RIPv2) makes improvements to the original RIP protocol by adding classless routing (CIDR)and authentication to its list of features, however the 16 hop count limitation still remains to allow for backwards compatibility. *'''IGRP''' The Interior Gateway Routing Protocol (IGRP) is a routing protocol developed by Cisco Systems to make up for shortcomings of RIP version 1. Instead of the 16 hop count limit imposed by RIP, IGRP supports up to 255 hops. ===Link State=== Link State routing protocols gather information about the state and what other devices are connected to it's links, this information is then flooded through the network, to each router in an [[Autonomous System]]. Routers then use this topology information to build it's routing table. Each device then sends periodic messages, "hellos" to each other, to ensure the topology matches their records. If the device detects a change (e.g. A link going down), this change is propagated to all devices, who then recalculate their routing tables, accounting for the changes. When all devices have identical topology records, and have recalculated their routing tables, the network is said to have converged. *'''IS-IS''' *'''OSPF''' ===Hybrid=== *'''EIGRP''' The Enhanced Interior Gateway Routing Protocol (EIGRP) is a routing protocol developed by Cisco Systems to expand upon IGRP. It utilizes the Diffusing Update Algorithm (DUAL) to determine the best possible loop-free path through a routed network. EIGRP maintains routing data in three tables known as the Neighbor Table, Topology Table, and Routing Table. ===Gateway Protocols=== *'''IS-IS''' *'''BGP''' *'''Border Gateway Protocol (BGP) is currently in version 4. Successor of Version 2/3 and replace Exterior Gateway Protocol (EGP) in the early 1990s.''' *'''Designed to route IP through Autonomous Systems (AS).''' *'''There are two types of BGP as per there use Interior Border Gateway Protocol (IBGP) and Exterior Border Gateway Protocol (EBGP).''' *'''By default BGP finds the best path to a network by using the best Autonomous Systems (AS) path.''' *'''Routing policies are configured using BGP attributes.''' *'''BGP converges slowly, batch updates are sent once every 5 seconds for IBGP peers, once every 30 seconds for EBGP peers.''' *'''BGP does not enable one AS to send traffic to a neighboring AS intending that the traffic take a different route from that taken by traffic originating in the neighboring AS. - RFC1771 ''' ==See also== * [[Internet Protocol Analysis/Routing | Internet Protocol Analysis - Routing]] [[Category:Computer networks]] frvljgduzds3xh7x294xtgji3opj4ys 2693435 2693355 2024-12-26T23:24:38Z Tule-hog 2984180 add {[[template:bookcat|bookcat]] 2693435 wikitext text/x-wiki Routing Protocols are used to dynamically propagate network routes to other routers on the network. This saves administrative work as each route required by a network does not have to be manually configured on every router in the network. Routing protocols are defined in three broad classes Distance Vector, Link State, and Hybrid. Distance Vector devices calculate the route by how far away (in hops) the destination is, while Link State protocols can use many other values such as bandwidth, latency, reliability and load. Hybrids, as the name implies, use a mixture of Distance Vector and Link State methods. ==Routing Protocols== ===Distance Vector=== Distance Vector routing is sometimes derisively described as routing by rumor. Referring to the fact that no single router really know more about the topology and state of the network than what his neighbour tells/advertises to him. Each neighbour is presenting a list of destinations and promising that that traffic forwarded via itself will reach these destinations. Some form of numeric values, (metrics/hop counts/cost/etc), are attached to these advertisements between neighbours so that the receiving device can compare competing routing to the same destination, from different neighbors, and tie break between them or even determine eligibility for load balancing of traffic over equal cost multiple paths (ECMP). The name Distance Vector was chosen because the neighbour from which you learn a potential path to a destination, (route), represents a possible direction of travel through the network to reach that destination. Hence vector. Distance refers to the numeric values advertised with the routes. Given multiple paths to the same destination but following different vector, neighbours, we can imagine the numbers indicate a distance to the destinations. More over a router would prefer to forward traffic via the shorted path. Administrators can manipulate these values to prefer one path over another. This ability to manipulate the advertised values/metrics mean that there is no hard link between the value and Actual distance to a destination. Distance Vector routing protocols repeatedly announce their known destinations to its neighbours at specific intervals. Routers are considered possible neighbours for distance vector protocols when they are linked together by some transmission medium. For example ethernet, ATM, Packet over Sonet/SDH, Frame relay or DSL links. There are many more possible types of links / interfaces that can connect routers together, some of more complex logical rather than physical links. *'''RIP''' The Routing Information Protocol (RIP) is a routing protocol that was designed for small networks. It can handle a maxiumum of 16 hops and is a classful routing protocol meaning that it does not support Variable Length Subnet Masks (VLSM) and does not support authentication. *'''RIPv2''' The Routing Information Protocol version 2 (RIPv2) makes improvements to the original RIP protocol by adding classless routing (CIDR)and authentication to its list of features, however the 16 hop count limitation still remains to allow for backwards compatibility. *'''IGRP''' The Interior Gateway Routing Protocol (IGRP) is a routing protocol developed by Cisco Systems to make up for shortcomings of RIP version 1. Instead of the 16 hop count limit imposed by RIP, IGRP supports up to 255 hops. ===Link State=== Link State routing protocols gather information about the state and what other devices are connected to it's links, this information is then flooded through the network, to each router in an [[Autonomous System]]. Routers then use this topology information to build it's routing table. Each device then sends periodic messages, "hellos" to each other, to ensure the topology matches their records. If the device detects a change (e.g. A link going down), this change is propagated to all devices, who then recalculate their routing tables, accounting for the changes. When all devices have identical topology records, and have recalculated their routing tables, the network is said to have converged. *'''IS-IS''' *'''OSPF''' ===Hybrid=== *'''EIGRP''' The Enhanced Interior Gateway Routing Protocol (EIGRP) is a routing protocol developed by Cisco Systems to expand upon IGRP. It utilizes the Diffusing Update Algorithm (DUAL) to determine the best possible loop-free path through a routed network. EIGRP maintains routing data in three tables known as the Neighbor Table, Topology Table, and Routing Table. ===Gateway Protocols=== *'''IS-IS''' *'''BGP''' *'''Border Gateway Protocol (BGP) is currently in version 4. Successor of Version 2/3 and replace Exterior Gateway Protocol (EGP) in the early 1990s.''' *'''Designed to route IP through Autonomous Systems (AS).''' *'''There are two types of BGP as per there use Interior Border Gateway Protocol (IBGP) and Exterior Border Gateway Protocol (EBGP).''' *'''By default BGP finds the best path to a network by using the best Autonomous Systems (AS) path.''' *'''Routing policies are configured using BGP attributes.''' *'''BGP converges slowly, batch updates are sent once every 5 seconds for IBGP peers, once every 30 seconds for EBGP peers.''' *'''BGP does not enable one AS to send traffic to a neighboring AS intending that the traffic take a different route from that taken by traffic originating in the neighboring AS. - RFC1771 ''' ==See also== * [[Internet Protocol Analysis/Routing | Internet Protocol Analysis - Routing]] {{bookcat}} d4udp113pnkqwjfi1t61wai3o8u2jye 0 54381 2693522 2018526 2024-12-27T00:49:22Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693522 wikitext text/x-wiki [[Topic:Computer networks]] : [[Network Classifications]] Computer networks are typically classified by scale, ranging from small, personal networks to global wide-area networks and the Internet itself. The following readings provide detailed information on the common network classifications: * [[w:Computer_network#Scale | Wikipedia: Computer Network - Scale]] * [[w:Personal_area_network | Wikipedia: Personal Area Network (PAN)]] * [[w:Storage_Area_Network | Wikipedia: Storage Area Network (SAN)]] * [[w:LAN | Wikipedia: Local Area Network (LAN)]] * [[w:Campus_area_network | Wikipedia: Campus Area Network (CAN)]] * [[w:Metropolitan_area_network | Wikipedia: Metropolitan Area Network (MAN)]] * [[w:Wide_area_network | Wikipedia: Wide Area Network (WAN)]] [[Category:Networking]] nh3k25ntwrzg2j3mysmhr5va84sy56i 0 54383 2693531 1518461 2024-12-27T00:50:52Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693531 wikitext text/x-wiki #REDIRECT[[Mobile Networks]] ==Introduction== Wireless networking can entail a number of technologies. Some are listed below. ==Infrafred== ==Bluetooth== Typically used to create Personal Area Networks (PANs), Bluetooth has a distance limitation of up to 100 meters. The Bluetooth standard defines 3 classes of transmitters and recievers. *Class 1 defines a maximum permitted power of 100mW (20 dBm) and can transmit up to approximately 100 meters. *Class 2 defines a maximum permitted power of 2.5mW (4 dBm) and can transmit up to approximately 10 meters. *Class 3 defines a maximum permitted power of 1mW (0 dBm) and can transmit up to approximately 1 meter. ==IEEE 802.11== As described in [[Network communication medium]], IEEE 802.11 refers to the standards of wireless technology most commonly used today in network connections. ==Security== *Wired Equivalent Privacy (WEP) *Wi-Fi Protected Access (WPA) *Wi-Fi Protected Access Version 2 (WPA2) *802.1x [[Category:Networking]] e4y5c0a0pwg9q7vaivkou8ms9p0urrr 0 57461 2693518 1510564 2024-12-27T00:48:42Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693518 wikitext text/x-wiki #REDIRECT[[Internet Layer]] IPv4 Addressing refers to an address that conforms to the Version 4 standard of the [[w:Internet_Protocol|Internet Protocol]] (IP). These addresses are assigned to computers to identify them on an IP Network. IPv4 addresses are 32-bits long, with 4 [[w:Octet_%28computing%29|octets]] of 8 [[w:Binary_digit|bits]] each. While IP Addresses are [[w:Binary_numeral_system|binary numbers]], IP Addresses are usually written in what is called a [[w:Dotted_decimal|dotted decimal]] notation so that they are human readable. [[w:Dotted_decimal|Dotted Decimal]] notation is accomplished by [[w:Binary_numeral_system#Conversion_to_and_from_other_numeral_systems|converting]] the binary values of each octet to its decimal number equivilent and placing a decimal point between each octet (i.e. 159.203.155.100). ==Subnet Masks== <!--- Explain Subnet Masks ---> ===Dotted Decimal Notation=== Like the IPv4 Address itself, Subnet Masks are sometimes represented in [[w:Dotted_decimal|dotted decimal]] notation. <!--- Expand and reword ---> ===CIDR Notation=== Classless Interdomain Routing (CIDR) Notation refers to using a two to three character representation of a subnet mask. CIDR is written as a slash (\) and then a number that represents the number of bits used in the subnet mask. For example, if a subnet mask in binary format is 11111111111111110000000000000000, there are 16 total on bits, therefore the CIDR Notation for this mask would be \16. ==Classes== *Class A **Contains the IPv4 Addresses 0.0.0.0-127.255.255.255. Default subnet mask is 255.0.0.0. *Class B **Contains the IPv4 Addresses 128.0.0.0-191.255.255.255. Default subnet mask is 255.255.0.0. *Class C **Contains the IPv4 Addresses 192.0.0.0-223.255.255.255. Default subnet mask is 255.255.255.0. *Class D **Contains the IPv4 Addresses 224.0.0.0-239.255.255.255. This address is for [[w:Multicasting|multicasting]]. *Class E **Contains the IPv4 Addresses 240.0.0.0-255.255.255.255. These addresses are reserved by the [[w:Internet_Assigned_Numbers_Authority|Internet Assigned Numbers Authority]] (IANA) and should not be used on IP Networks. ==Loopback Addresses== The 127.0.0.0/8 network is a part of the Class A classification, but it is designated for [[w:Loopback#Virtual_Internet_Protocol_Network_Interface|loopback addressing]] and cannot be assigned to a network. The loopback subnet is not routable on the internet. ==Private Addresses== There are 3 IPv4 address ranges that are considered private addresses, and these ranges are: *Class A: 10.0.0.0/8 *Class B: 172.16.0.0-172.32.0.0/16 *Class C: 192.168.0.0/24 These addresses are not routable on the internet, meaning that in order for computers that are assigned a private address to communicate with the internet, a Network Address Translation (NAT) service must be put in place between the privately addressed computer and the internet. ==Subnetting== Subnetting describes the process of dividing IP Addresses into logical subnetworks. <!--- Finish Example and Review before posting ===Variable Length Subnet Masking=== Variable Length Subnet Masking, or VLSM, allows a network administrator to divide his network address range into subnetworks that vary in size. For example, if Jim Bob were to have a Class C network address range such as 192.168.1.0/24 with standard subnetting he could split his network into 2, 4, 8, 16, 32, 64, 128, or 256 equally sized networks. This is not always the most efficient way of subnetting, as sometimes you may have a network that does not require all of the available host addresses in a subnet. That is where VLSM comes into play. VLSM allows you to vary the length of a subnet mask used to subnet your network so that you may custom tune the size of that subnetwork to your needs. ---> [[Category:Networking]] [[Category:Engineering and Technology|{{PAGENAME}}]] aaiyhjh3fsy2lsdamzxb8idyee33x9h 14 60408 2693465 275879 2024-12-26T23:59:49Z Tule-hog 2984180 [[wv:bold|boldly]] tag {[[template:category redirect|category redirect]]} 2693465 wikitext text/x-wiki {{category redirect|Networking}} dfru00ja0ptjjbsfd7txd4m4jcvsx1f 0 60487 2693520 2485710 2024-12-27T00:49:02Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693520 wikitext text/x-wiki Cisco networking includes a variety of computer networking concepts, including network operations, switching, routing, security, and wide area networks based on [[Wikipedia:Cisco Systems|Cisco Systems]] devices and [[Cisco IOS]]. == Courses == * [[/CCENT/]] * [[/CCNA/]] * [[/CCNA Voice/]] * [[/CCNP/]] * [[/CCVP/]] == See Also == * [[Wikipedia: Cisco Systems]] * [[Cisco IOS]] == References == {{CourseCat}} [[Category:Networking]] [[Category:Cisco]] 7qvg4v7nkspksbucvvig5j8wmy0u3ia 0 79365 2693519 1046644 2024-12-27T00:48:52Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693519 wikitext text/x-wiki #REDIRECT[[Link Layer]] ==Introduction== The Spanning-Tree Protocol (STP) is an algorithm employed by bridges and switches to try and ensure that switching loops do not occur in the network topology. Capable of shutting down redundant links until they are needed, STP is a staple for enterprise networks. There have been many different types of the spanning-tree protocol to appear over the years such as the Multiple Spanning-Tree Protocol (MSTP), Per-VLAN Spanning-Tree Protocol (PVST), Rapid Spanning-Tree Protocol (RSTP), and Rapid Per-VLAN Spanning-Tree Protocol (RPVST). == Spanning-Tree Protocol (IEEE 802.1D) == === Multiple Spanning-Tree Protocol (IEEE 802.1Q-2003) === === Per-VLAN Spanning-Tree Protocol (Cisco Proprietary) === === Rapid Spanning-Tree Protocol (IEEE 802.1w) === === Rapid Per-VLAN Spanning-Tree Protocol (Cisco Proprietary) === [[Category:Networking]] [[Category:Engineering and Technology|{{PAGENAME}}]] ijqnc1aergvjyxiax5h3f1ned15uagy 0 85319 2693529 1824185 2024-12-27T00:50:32Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693529 wikitext text/x-wiki {|class="wikitable" border="1" style="float:right;margin:0 0 1em 1em" |- ! colspan="5" | OSI Model |- ! !Data unit !Layer !style="width:9em;"|Function |- !rowspan="4"|Host<br>layers | bgcolor="#D8EC9B" rowspan="3"|Data | bgcolor="#D8EC9B" | 7. [[UML/OSI#Application|Application]] | bgcolor="#D8EC9B" |<small>Network process to application</small> |- | bgcolor="#D8EC9B" |6. [[UML/OSI#Presentation|Presentation]] | bgcolor="#D8EC9B" |<small>Data representation and encryption</small> |- | bgcolor="#D8EC9B" |5. [[UML/OSI#Session|Session]] | bgcolor="#D8EC9B" |<small>Interhost communication</small> |- | bgcolor="#E7ED9C" |Segment | bgcolor="#E7ED9C" |4. [[UML/OSI#Transport|Transport]] | bgcolor="#E7ED9C" |<small>End-to-end connections and reliability</small> |- !rowspan="3"|Media<br>layers | bgcolor="#EDDC9C" |Packet | bgcolor="#EDDC9C" |3. [[UML/OSI#Network|Network]] | bgcolor="#EDDC9C" |<small>Path determination and [[w:logical address|logical address]]ing</small> |- | bgcolor="#E9C189" |Frame | bgcolor="#E9C189" |2. [[UML/OSI#Data Link|Data Link]] | bgcolor="#E9C189" |<small>Physical addressing</small> |- | bgcolor="#E9988A" |Bit | bgcolor="#E9988A" |1. [[UML/OSI#Physical|Physical]] | bgcolor="#E9988A" |<small>Media, signal and binary transmission</small> |} <div style="background:LightYellow;Padding:10px"> The '''[[w:OSI Model|OSI Model]]''' is the Open System Interconnection Reference Model. In its most basic form, it divides network architecture into seven layers which, from top to bottom, are the Application, Presentation, Session, Transport, Network, Data-Link, and Physical Layers. It is therefore often referred to as the OSI Seven Layer Model.</div> The OSI model can be used to conceptualize complex networks of people, places, and machines. UML diagram elements can be structured in layers that correspond to the 7-layer OSI model. <ref>[http://infolab.stanford.edu/~melnik/pub/sw00/ A Layered Approach to Information Modeling and Interoperability on the Web] 2000 - Melnik, Decker, Database Group, Stanford University </ref> Currently there is growing interest in the use of [[UML]] for system modeling. Structuring specifications is difficult, but The ITU (International Telecommunications Union), the IEC (International Electrotechnical Commission) and the ISO have jointly defined some UML profiles for a Reference Model of Open Distributed Processing ([[w:RM-ODP|RM-ODP]]). <ref>[http://www.lcc.uma.es/~av/download/MOS-036%20UML4ODP_FDIS_v00-78.pdf X.906 : UML4ODP - Use of UML for ODP system specifications] ITU Recommendation X.906 (11/07) [http://www.itu.int/rec/T-REC-X.906-200711-I/en itu subscription version]</ref> ==Physical== Media, signal and binary transmission Protocols: [[w:RS-232|RS-232]], [[w:V.35|V.35]], [[w:V.34|V.34]], [[w:I.430|I.430]], [[w:I.431|I.431]], [[w:T1(Networking)|T1]], [[w:E-carrier#E1|E1]], [[w:IEEE_802.3|802.3 Ethernet]], [[w:10BASE-T|10BASE-T]], [[w:100BASE-TX|100BASE-TX]], [[w:POTS|POTS]], [[w:SONET|SONET]], [[w:DSL|DSL]], [[w:IEEE_802.11|802.11a/b/g/n PHY]], [[w:G.hn|ITU-T G.hn PHY]] ==Data Link== Physical addressing Protocols: [[w:Address Resolution Protocol|ARP]], [[w:CSLIP|CSLIP]], [[w:SLIP|SLIP]], [[w:Frame Relay|Frame Relay]], [[w:G.hn|ITU-T G.hn DLL]] ==Network== Path determination and [[w:logical address|logical address]]ing Protocols: [[w:Internet Protocol|IP]], [[w:Internet Control Message Protocol|ICMP]], [[w:IPsec|IPsec]], [[w:IGMP|IGMP]] ==Transport== End-to-end connections and reliability Protocols: [[w:Transmission Control Protocol|TCP]], [[w:User Datagram Protocol|UDP]], [[w:PPTP|PPTP]], [[w:Layer 2 tunneling Protocol|L2TP]], [[w:Stream Control Transport protocol|SCTP]] ==Session== Interhost communication Protocols: [[w:Named Pipes|Named Pipes]], [[w:NetBIOS|NetBIOS]], [[w:Session Announcement Protocol|SAP]] ==Presentation== Data representation and encryption Protocols: [[w:MIME|MIME]], [[w:External Data Representation|XDR]], [[w:Secure Sockets Layer|SSL]], [[w:Transport Layer Security|TLS]] ==Application== Network process to application Protocols: [[w:NNTP|NNTP]], [[w:Session Initiation Protocol|SIP]], [[w:Simple Sensor Interface protocol|SSI]], [[w:DNS|DNS]], [[w:FTP|FTP]], [[w:Gopher (protocol)|Gopher]], [[w:HTTP|HTTP]], [[w:Network File System (protocol)|NFS]], [[w:Network Time Protocol|NTP]], [[w:SMPP|SMPP]], [[w:SMTP|SMTP]], [[w:Simple Network Management Protocol|SNMP]], [[w:Telnet|Telnet]], [[w:Category:Application layer protocols|(more)]] ==notes== <references /> [[Category:Networking]] 2evdogtvpkafg2lhplpjub59re3vpxr 1 87789 2693541 1570923 2024-12-27T00:58:23Z Tule-hog 2984180 Tule-hog moved page [[Talk:Network+/Standards/TCP/IP Model/Introduction]] to [[Talk:Network+/Old guides/IP Model]]: alter parent 1570923 wikitext text/x-wiki ==Purpose of the page== I see this page as a gateway to introducing the overall structure of the TCP/IP model, without going too far in depth, since topics for examining each specific layer exist. Basic explanations and applications should be described here, more detailed theory should be left to the other pages. <!-- retained just in case. Why this is here, I have no idea. This ISN'T a general WikiVersity Q&A page, it is about computer networking. == More Information on investing in real estate == Hey en.wikiversity.org, does anyone know where to get more information on investing in real estate? Thanks. --> ==Nomination for renaming== Why people seem to insist on calling this "TCP/IP" I have no clue. IP is on its own, TCP happens to be a protocol one layer deeper. Can it even be honestly said it's the predominant use of IP anymore, what with the increased use of things like RTP (which uses UDP)? Networking by its nature is layered, and calling it "TCP/IP" suggests TCP and IP are on equal levels when they are very clearly not. How about something more like "IP Fundamentals" or "IP Networking Fundamentals?" [[User:Rchandra|Rchandra]] ([[User talk:Rchandra|discuss]] • [[Special:Contributions/Rchandra|contribs]]) 17:06, 18 June 2013 (UTC) :Wikipedia calls it [[Wikipedia:Internet_protocol_suite | Internet protocol suite]]. Something like that makes sense to me. Internet Protocol Fundamentals or Internet Protocol Suite Fundamentals or Internet Protocol Networking Fundamentals. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:24, 19 June 2013 (UTC) emkc8cjqk3x90qlexmkngok7gf1bups 0 87792 2693467 1911347 2024-12-27T00:04:43Z Tule-hog 2984180 mv from [[:Category:Mobile Networks]] 2693467 wikitext text/x-wiki This topic page is for organizing the development of content related to '''wireless systems'''. If you are knowledgeable in any area Wireless systems, [[Help:Be bold|feel free to improve]] upon what you see, we would greatly appreciate your contributions. ==Introduction== Wireless networks have significantly impacted the world as far back as World War II. With the use of wireless networks, information could be sent overseas or behind enemy lines easily and quickly and was more reliable. Since then wireless networks have continued to develop and its uses have significantly grown. Cellular phones are part of huge wireless network systems. People use these phones daily to communicate with one another. Sending information over seas is only possible through wireless network systems using satellites and other signals to communicate across the world otherwise getting information Emergency services such as the police department utilize wireless networks to communicate important information quickly. People and businesses use wireless networks to send and share data quickly whether it be in a small office building or across the world. Another important use for wireless networks is as an inexpensive and rapid way to be connected to the Internet in countries and regions where the telecom infrastructure is poor or there is a lack of resources, like most Developing Countries. Wireless networks allow you to eliminate messy cables. Wireless connections offer more mobility, the downside is there can sometimes be interference that might block the radio signals from passing through. One way to avoid this is by putting the source of your wireless connection in a place where the signal will have as little interference as possible. Sometimes nearby networks are using the same frequencies, this can also cause interference within the network and can reduce its performance. Compatibility issues also arise when dealing with wireless networks. Different components not made by the same company may not work together, or might require extra work to fix compatibility issues. To avoid this, purchase products made by the same company so that there are fewer compatibility issues. Wireless networks, in terms of internet connections, are typically slower than those that are directly connected through an Ethernet cable. Though the speed is slower, most things will still move at the same speed except for things like video downloads. Though wireless technology continues to develop, it is now easier to get networks up and running cheaper and faster than ever before. A wireless network is more vulnerable because anyone can try to break into a network broadcasting a signal. Many networks offer WEP - Wired Equivalent Privacy - security systems which have been found to be vulnerable to intrusion. Though WEP does block some intruders, the security problems have caused some businesses to stick with wired networks until security can be improved. Another type of security for wireless networks is WPA - Wi-Fi Protected Access. WPA provides more security to wireless networks than a WEP security set up. The use of firewalls will help with security breaches which can help to fix security problems in some wireless networks that are more vulnerable. Wireless networking can entail a number of technologies. Some are listed below. ==Infrared== ==Bluetooth== Typically used to create Personal Area Networks (PANs), Bluetooth has a distance limitation of up to 100 meters. The Bluetooth standard defines 3 classes of transmitters and receivers. *Class 1 defines a maximum permitted power of 100mW (20 dBm) and can transmit up to approximately 100 meters. *Class 2 defines a maximum permitted power of 2.5mW (4 dBm) and can transmit up to approximately 10 meters. *Class 3 defines a maximum permitted power of 1mW (0 dBm) and can transmit up to approximately 1 meter. ==IEEE 802.11== As described in [[Network communication medium]], IEEE 802.11 refers to the standards of wireless technology most commonly used today in network connections. The current standard for this the IEEE 802.11x standards also called Wi-Fi for Wireless Fidelity. Wi-Fi is being used by both business and consumer as a way to share a internet connection as well as P2P connections. Currently there are three Wi-Fi Standards a, b and g, the n and i standards are currently being developed. 802.11b - 802.11b was the first wi-fi standard to be introduced and has a maximum speed of 11Mbps. 802.11a - 802.11a was the second wi-fi standard and had a maximum speed of 54Mbps with the same range transmission and is not compatible with 802.11b because it used the 5GHz band and which has less interference. 802.11g - 802.11g was the third standard and had a maximum speed of 54Mbps and much farther range than both a and b but is backward compatible with 802.11b. ==Security== *Wired Equivalent Privacy (WEP) *Wi-Fi Protected Access (WPA) *Wi-Fi Protected Access Version 2 (WPA2) *802.1x ==Further introduction== Today being able to connect with the world while you are on the go is a natural habit by virtue of mobile networks and services you enjoy from service providers of GSM 2G, CDMA, WCDMA or 3G/UMTS and or HSPA. A few of us may be already using LTE or 4G devices for data connectivity already.What makes your user equipment work?, how are you able to talk to others as easily as you could on your plain old telephone system of PSTN? In this introductory discussion on mobile networks you can obtain answers to such questions. Way back during 1980s, world has seen Advanced Mobile Phone Systems in the USA and primitive GSM networks emerged in Europe. The notion of cellular mobile communications emerged with studies performed by Bell Labs and conserted efforts of EU like strong bodies to make it a sustainable globally roaming friendly wireless communication system that is in practice still after more than three decades since its inception. While North American region has welcomed all emerging wireless technologies and standards to be incorporated in their nations, Europe predominantly retained strong affiliations with Groupe Spaciale mobile or GSM standard and its data versions such as GPRS, EDGE. 3GPP and 3GPP2 partnership projects have emerged giving a sense of project managerial support to promote standards and currently we have Release 15 being updated during 2014 June time frame as we read this article From a 200 khz base band as used in GSM full duplex at 900 mhz range of spectrum, CDMA evolved as 1.23 mhz spectrum with a distictly different wireless paradigm of unique codes being used for users,Base Station, Mobile device networking elements and traffic channels. UMTS emerged as a wideband version of CDMA with a typical 5 Mhz bandwidth to cater for both voice and data requirements as per IMT2000 guidelines of ITU-R. With the advent of Orthogonal Frequency Division Multiplexing concepts from Wavelan of AT&T, primitive implementations of 1.4 mhz LTE have emerged during 2008 especially after WiMAX standard of IEEE 802.16 has started not being able to overcome Peak to Average Power Ratio issue in the uplink. Thus the wirless LTE uplink now uses SC-FDMA standard with 15 khz sub carriers. A Resource Block in LTE is a spectrum entity of 12 times 15000 hz sinusoidal waves that makes 180 khz just almost similar in size of a radio transeiver or RT with big boost in terms of being able to generate seven symbols per a slot on each and every sub carrier yielding two chunks of 84 resource elements per a milli second Thus with 20 Mhz spectrum allocation of LTE one gets a hundred resource blocks and 100.8 megabits per second throughput in the downlink.This can be accumulated upto 300 megabits per second using 4T4R MIMO. An advanced implementation of the same technology with Coordinated Multipoint and Carrier Aggregation can take the datarate to as high as 1 gbps and beyond as 3 gbps with 100 mhz spectrum allocation and coordinated beam forming of nearby eNodeBs towards the target UE to make such a huge data rate possible ==Learning resources== A good beginning is ftp://ftp.3gpp.org Specs Year 2010 Dec for instance Chose Release 8 or above Pick up a maiden document such as 36.300 Subsequently as you delve deeper on a specific subject such as PHY layer, you might want to download specs 36.211,36.212.36.213 or 36.214 ==Wikipedia== *[[w:Ralink|Ralink]] ==External links== * [http://sourceforge.net/projects/sinalgo/ Sinalgo] - Wireless network simulator * [http://sourceforge.net/project/showfiles.php?group_id=134759 Generic 802.11 Networking Subsystem] * [http://sourceforge.net/projects/straw/ SWANS++] * [[w:NetSim | NetSim]] - Useful network simulator for Research and Network lab experimentation * [http://www.sensorsmag.com/articles/0203/38/main.shtml Wireless Mesh Networks] ==Related news== * April 2007 [http://www.physorg.com/news95839153.html Spectrum formally devoted to television signals to be auctioned by Feds...] * March 2007 [http://www.epicos.com/epicos/portal/media-type/html/user/anon/page/default.psml/js_panename/News+Information+Article+View;jsessionid=FCC6403311A50CF65F58FA9D048E0942.tomcat2?articleid=74689&showfull=false Mesh networking patents to be sold at auction.] [[Category:Mobile Networks]] k0c5rce9kglldh40bkhd1d1zjgt3vrt 1 91097 2693488 1570810 2024-12-27T00:25:51Z Tule-hog 2984180 Tule-hog moved page [[Talk:Network+/Architecture/Media/Introduction]] to [[Talk:Network+/Old guides/Network media]]: alter parent 1570810 wikitext text/x-wiki I am not that good at computer stuff, but this passage sounds a bit off to me "Coaxial Cable [edit]Coaxial Cable uses a copper wire for the conductor, on top of this is insulation for the wire. The third layer consists of a metallic foil or woven copper braid as a shielding, followed by a rubber jacket on the top. It is often referred to as Thicknet or Thinnet, depending on the specification. Benefits: It costs less to buy than Fiber Optic It has speeds of 10mb/s to 100mb/s. It costs more to install Coaxial cable. " does the fact that "It costs more to install Coaxial cable." qualify as "Benefits:"? [[Special:Contributions/60.37.240.76|60.37.240.76]] 08:50, 27 January 2010 (UTC) cf44vldqg2vvlbwkzijld7qzormzqwo 0 113381 2693305 2693115 2024-12-26T13:15:29Z Eshaa2024 2993595 /* Complex Analysis Part 2 */ 2693305 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysis Maximum Principle|Complex Analysis Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis%20Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Analysis%20Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> 4h089cbu9jjevgusk4qiy305s8z01ni 2693306 2693305 2024-12-26T13:47:31Z Eshaa2024 2993595 /* Chapter 3 - Complex Derivative */ 2693306 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysis Maximum Principle|Complex Analysis Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis%20Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Analysis%20Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> gspb2rpbapbpck7o079eb2ib4fskw07 2693310 2693306 2024-12-26T15:23:19Z Eshaa2024 2993595 /* Complex Analysis Part 2 */ 2693310 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> 25qhynqkcc2jhvmt2odc0b3rhtggl0u 2693385 2693310 2024-12-26T21:14:46Z Eshaa2024 2993595 /* Complex Analysis Part 2 */ 2693385 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Openness theorem/theorem of territorial loyalty|Openness theorem/theorem of territorial loyalty]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Openness%20theorem/theorem%20of%20territorial%20loyalty&author=Openness%20theorem&language=en&audioslide=yes&shorttitle=theorem%20of%20territorial%20loyalty&coursetitle=Openness%20theorem Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> dawc7sml6g1ix052fyi1rxhfc6zbzi2 2693388 2693385 2024-12-26T21:45:52Z Eshaa2024 2993595 /* Complex Analysis Part 2 */ 2693388 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Openness theorem theorem of territorial loyalty|Openness theorem theorem of territorial loyalty]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Openness%20theorem%20theorem%20of%20territorial%20loyalty&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> ax27zqyuq5700uvup8bfdue9reveyjf 2693396 2693388 2024-12-26T22:20:47Z Eshaa2024 2993595 /* Complex Analysis Part 2 */ 2693396 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Openness theorem theorem of territorial loyalty|Openness theorem theorem of territorial loyalty]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Openness%20theorem%20theorem%20of%20territorial%20loyalty&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> n74efnznkdwscaf8n0rrouu3wjq95ka 2693401 2693396 2024-12-26T22:33:21Z Eshaa2024 2993595 /* Complex Analysis Part 2 */ 2693401 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Openness theorem theorem of territorial loyalty|Openness theorem theorem of territorial loyalty]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Openness%20theorem%20theorem%20of%20territorial%20loyalty&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> o81tnjse5solq31yifcnt1mizrygk29 2693413 2693401 2024-12-26T22:56:44Z Eshaa2024 2993595 2693413 wikitext text/x-wiki [[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]] [[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]] [[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]] '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] ** [[w:en:Integral|Wikipedia:Integral ]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Curve integral |Wikipedia: Curve integral]] * [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]] * '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 5 - Holomorphic Functions === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]] ** [[/Differences from real differentiability/]] ** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>, ** [[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Complex Analysis Part 2 === * '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/cycle|cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] *[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]] * '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Openness theorem theorem of territorial loyalty|Openness theorem theorem of territorial loyalty]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Openness%20theorem%20theorem%20of%20territorial%20loyalty&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ===Singularity and Residues - Part 3=== * '''[[Complex Analysis/Singularities|Singularities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> 3z7pu7qnymeqycyibzubp22ep6e2q3t 0 115211 2693528 2461715 2024-12-27T00:50:22Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693528 wikitext text/x-wiki '''Wired technologies''' * ''Twisted pair'' wire is the most widely used medium for telecommunication. Twisted-pair cabling consist of copper wires that are twisted into pairs. Ordinary telephone wires consist of two insulated copper wires twisted into pairs. Computer networking cabling consist of 4 pairs of copper cabling that can be utilized for both voice and data transmission. The use of two wires twisted together helps to reduce crosstalk and electromagnetic induction. The transmission speed ranges from 2 million bits per second to 100 million bits per second. Twisted pair cabling comes in two forms which are Unshielded Twisted Pair (UTP) and Shielded twisted-pair (STP) which are rated in categories which are manufactured in different increments for various scenarios. * ''Coaxial cable'' is widely used for cable television systems, office buildings, and other work-sites for local area networks. The cables consist of copper or aluminum wire wrapped with insulating layer typically of a flexible material with a high dielectric constant, all of which are surrounded by a conductive layer. The layers of insulation help minimize interference and distortion. Transmission speed range from 200 million to more than 500 million bits per second. *'' Optical fiber cable'' consists of one or more filaments of glass fiber wrapped in protective layers that carries a data by means of pulses of light. It transmits light which can travel over extended distances. Fiber-optic cables are not affected by electromagnetic radiation. Transmission speed may reach trillions of bits per second. The transmission speed of fiber optics is hundreds of times faster than for coaxial cables and thousands of times faster than a twisted-pair wire. A recent innovation in fiber-optic cable is the use of colored light. Instead of carrying one message in a stream of white light impulses, this technology can carry multiple signals in a single strand. '''Wireless technologies''' * ''Terrestrial microwave'' – Terrestrial microwaves use Earth-based transmitter and receiver. The equipment looks similar to satellite dishes. Terrestrial microwaves use low-gigahertz range, which limits all communications to line-of-sight. Path between relay stations spaced approx, 48 km (30 miles) apart. Microwave antennas are usually placed on top of buildings, towers, hills, and mountain peaks. * ''Communications satellites'' – The satellites use microwave radio as their telecommunications medium which are not deflected by the Earth's atmosphere. The satellites are stationed in space, typically 35,400 km (22,200 miles) (for geosynchronous satellites) above the equator. These Earth-orbiting systems are capable of receiving and relaying voice, data, and TV signals. * ''Cellular and PCS systems'' – Use several radio communications technologies. The systems are divided to different geographic areas. Each area has a low-power transmitter or radio relay antenna device to relay calls from one area to the next area. * ''Wireless LANs'' – Wireless local area network use a high-frequency radio technology similar to digital cellular and a low-frequency radio technology. Wireless LANs use spread spectrum technology to enable communication between multiple devices in a limited area. An example of open-standards wireless radio-wave technology is IEEE. * ''Infrared communication'' , which can transmit signals between devices within small distances not more than 10 meters peer to peer or ( face to face ) without any body in the line of transmitting. [[Category:Networking]] tp0t7etqmkyg5v98rrsqfp1zby2m18w 2 125235 2693470 1139707 2024-12-27T00:08:36Z Tule-hog 2984180 Bot: Replacing category Networks with [[:Category:Networking|Networking]] 2693470 wikitext text/x-wiki {{Research project}} (fix the formating latter) P2P internet can be achieved by small increments 1. Wi-Fi stage - software. You load special software in your wifi capable router, the packets will travel through a chain of wifi routers in order to attain its final destination. It should be feasible to reach ... Portugal from Finland, by bouncing from router to router. At the very beginning, the first chains will connect to the internet through free wifi hot spots, and users donating some of there spare bandwidth. Initial quality will be terrible, but it will gradually build it self up. People can have alredy some dissent access to the internet through donated bandwidth. But they can enjoy various services by the local network alone. We can have local Radio, mobile telephony, local TV, mirrored websites( especially the very big ones) and other stuff. From the normal internet we can get some relatively low bandwidth services, multicasting stuff (internet radio, TV), e-mail, etc.... Wifi radio on cars could transform highways to literal highways of information, by having packets bouncing from one moving car to the other. Ionosphere bouncing as used in amateur radio could also help a bit in long distance transmitions. It will still be crap for long distances. 2. Wi-Fi stage - antenna upgrade. The little tiny winy and cheap default antenna wifi stuff come with are crap. Even a very small investment in a beater antenna increases the radius by an important factor. 3. Wi-Fi stage - Router upgrade. Today routers weren’t meant to be used this way 4. Optics. Optical communication( simple LED, then laser) can help in increasing bandwidth if there’s no more room on the frequencies. Its more expensive then the previous stages, but it now worth it, with this trick we avoid installing cables, it would be even more expensive doing that. 5. Fibre optics . Final stage. People connect to each other with fibre optics (when applicable). Bandwidth between routers increase explosively. Any Overhead delay by bouncing through routers can be cancelled, by establishing enough fibre optic connections between the routers. The number of fibber optics between routers can be arbitrary. Anything that is real time can have a special protocol with high priority of transition. On long distances: If i'm correct about the economics of the internet. The most expensive part of the transition is the near the internautes, not the long distances. One possibility is that the public authorities nationalise or become unique clients and offer as a public good that part of the network. An other possibility, is that cities and countries start doing the same thing as in the local level, they heavily interconnect with there immediate neighbours. Summary: First just a software upgrade, then an antenna upgrade, then router upgrade, then optic communication upgrade(led, then laser), then direct fibber optics connections. Quality increases from very bad to adequate, working in parallel with normal internet, and gradually taking on at least some of the load. Its problem: Too few people are on it right now rendering it useless, and people don't join because its useless, the uselessness cycle need to be broken. Even with low % of people it could still work in a minimalist way. Technical note: ipv6 will be handy here. [[Category:Original research]] [[Category:Research projects]] [[Category:Engineering]] [[Category:Internet]] [[Category:Networking]] p7matbxgq72xzmd32vk0454q2pghk2m 0 139648 2693532 2580230 2024-12-27T00:51:02Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693532 wikitext text/x-wiki {{TOCright}} Wireshark is a free and open source packet analyzer used for network troubleshooting and analysis. These activities will show you how to use Wireshark to capture and analyze Dynamic Host Configuration Protocol (DHCP) traffic. == Readings == * [[Wikipedia: Dynamic Host Configuration Protocol]] == Preparation == To prepare for this activity: # Start Windows. # Log in if necessary. # [[Wireshark/Install | Install Wireshark]]. == Activity 1 - Capture DHCP Traffic == To capture DHCP traffic: # [[Wireshark/Start | Start a Wireshark capture]]. # [[Command_Prompt/Open | Open a command prompt]]. # Type '''ipconfig /renew''' and press '''Enter'''. # Type '''ipconfig /release''' and press '''Enter'''. # Type '''ipconfig /renew''' and press '''Enter'''. # Close the command prompt. # [[Wireshark/Stop | Stop the Wireshark capture]]. == Activity 2 - Analyze DHCP Request Traffic == To analyze DHCP Request (lease renewal) traffic: # Observe the traffic captured in the top Wireshark packet list pane. To view only DHCP traffic, type '''udp.port == 68''' (lower case) in the Filter box and press '''Enter.''' # In the top Wireshark packet list pane, select the first DHCP packet, labeled '''DHCP Request'''. # Observe the packet details in the middle Wireshark packet details pane. Notice that it is an Ethernet II / Internet Protocol Version 4 / User Datagram Protocol / Bootstrap Protocol frame. # Expand Ethernet II to view Ethernet details. # Observe the Destination and Source fields. The destination should be your DHCP server's MAC address and the source should be your MAC address. You can use [[Ipconfig/All | ipconfig /all]] and [[Computer Networks/Management/Utilities/Arp/View | arp -a]] to confirm. # Expand Internet Protocol Version 4 to view IP details. # Observe the Source address. Notice that the source address is your IP address. # Observe the Destination address. Notice that the destination address is the IP address of the DHCP server. # Expand User Datagram Protocol to view UDP details. # Observe the Source port. Notice that it is bootpc (68), the BOOTP client port. # Observe the Destination port. Notice that it is bootps (67), the BOOTP server port. # Expand Bootstrap Protocol to view BOOTP details. # Observe the DHCP Message Type. Notice that it is a Request (3). # Observe the Client IP address, Client MAC address, and DHCP option fields. This is the request to the DHCP server. == Activity 3 - Analyze DHCP ACK Traffic == To analyze DHCP ACK (server acknowledgement) traffic: # In the top Wireshark packet list pane, select the second DHCP packet, labeled '''DHCP ACK'''. # Observe the packet details in the middle Wireshark packet details pane. Notice that it is an Ethernet II / Internet Protocol Version 4 / User Datagram Protocol / Bootstrap Protocol frame. # Expand Ethernet II to view Ethernet details. # Observe the Destination and Source fields. The destination should be your MAC address and the source should be your DHCP server's MAC address. # Expand Internet Protocol Version 4 to view IP details. # Observe the Source address. Notice that the source address is the DHCP server IP address. # Observe the Destination address. Notice that the destination address is your IP address. # Expand User Datagram Protocol to view UDP details. # Observe the Source port. Notice that it is bootps (67), the BOOTP server port. # Observe the Destination port. Notice that it is bootpc (68), the BOOTP client port. # Expand Bootstrap Protocol to view BOOTP details. # Observe the DHCP Message Type. Notice that it is an ACK (5). # Observe the Client IP address and Client MAC address fields. This is the acknowledgement from the DHCP server. # Observe the DHCP options and expand to view the details for IP Address Lease Time, Subnet Mask, Router (Default Gateway), Domain Name Server, and Domain Name, as well as any other options if included. == Activity 4 - Analyze DHCP Release Traffic == To analyze DHCP Release traffic: # In the top Wireshark packet list pane, select the third DHCP packet, labeled '''DHCP Release'''. # Observe the packet details in the middle Wireshark packet details pane. Notice that it is an Ethernet II / Internet Protocol Version 4 / User Datagram Protocol / Bootstrap Protocol frame. # Expand Ethernet II to view Ethernet details. # Observe the Destination and Source fields. The destination should be your DHCP server's MAC address and the source should be your MAC address. You can use [[Ipconfig/All | ipconfig /all]] and [[Computer Networks/Management/Utilities/Arp/View | arp -a]] to confirm. # Expand Internet Protocol Version 4 to view IP details. # Observe the Source address. Notice that the source address is your IP address. # Observe the Destination address. Notice that the destination address is the IP address of the DHCP server. # Expand User Datagram Protocol to view UDP details. # Observe the Source port. Notice that it is bootpc (68), the BOOTP client port. # Observe the Destination port. Notice that it is bootps (67), the BOOTP server port. # Expand Bootstrap Protocol to view BOOTP details. # Observe the DHCP Message Type. Notice that it is a Release (7). # Observe the Client IP address and Client MAC address fields. This is the address that will be released on the DHCP server. == Activity 5 - Analyze DHCP Discover Traffic == To analyze DHCP Discover (lease request) traffic: # In the top Wireshark packet list pane, select the fourth DHCP packet, labeled '''DHCP Discover'''. # Observe the packet details in the middle Wireshark packet details pane. Notice that it is an Ethernet II / Internet Protocol Version 4 / User Datagram Protocol / Bootstrap Protocol frame. # Expand Ethernet II to view Ethernet details. # Observe the Destination and Source fields. The destination should be the broadcast address ff:ff:ff:ff:ff:ff and the source should be your MAC address. When the client doesn't have an IP address or server information, it has to broadcast to discover a DHCP server. # Expand Internet Protocol Version 4 to view IP details. # Observe the Source address. Notice that the source address is 0.0.0.0, indicating no current IP address. # Observe the Destination address. Notice that the destination address 255.255.255.255, the broadcast IP address. # Expand User Datagram Protocol to view UDP details. # Observe the Source port. Notice that it is bootpc (68), the BOOTP client port. # Observe the Destination port. Notice that it is bootps (67), the BOOTP server port. # Expand Bootstrap Protocol to view BOOTP details. # Observe the DHCP Message Type. Notice that it is a Discover (1). # Observe the Client IP address, Client MAC address, and DHCP option fields. This is the request to the DHCP server. == Activity 6 - Analyze DHCP Offer Traffic == To analyze DHCP Offer (server offer) traffic: # In the top Wireshark packet list pane, select the fifth DHCP packet, labeled '''DHCP Offer'''. # Observe the packet details in the middle Wireshark packet details pane. Notice that it is an Ethernet II / Internet Protocol Version 4 / User Datagram Protocol / Bootstrap Protocol frame. # Expand Ethernet II to view Ethernet details. # Observe the Destination and Source fields. The destination should be your MAC address and the source should be your DHCP server's MAC address. # Expand Internet Protocol Version 4 to view IP details. # Observe the Source address. Notice that the source address is the DHCP server's IP address. # Observe the Destination address. Notice that the destination address is 255.255.255.255 (broadcast) address. # Expand User Datagram Protocol to view UDP details. # Observe the Source port. Notice that it is bootps (67), the BOOTP server port. # Observe the Destination port. Notice that it is bootpc (68), the BOOTP client port. # Expand Bootstrap Protocol to view BOOTP details. # Observe the DHCP Message Type. Notice that it is an Offer (2). # Observe the Client IP address and Client MAC address fields. This is the offer from the DHCP server. # Observe the DHCP options and expand to view the details for IP Address Lease Time, Subnet Mask, Router (Default Gateway), Domain Name Server, and Domain Name, as well as any other options if included. == Activity 7 - Analyze DHCP Request Traffic == To analyze DHCP Request (lease request) traffic: # In the top Wireshark packet list pane, select the sixth DHCP packet, labeled '''DHCP Request'''. # Observe the packet details in the middle Wireshark packet details pane. Notice that it is an Ethernet II / Internet Protocol Version 4 / User Datagram Protocol / Bootstrap Protocol frame. # Expand Ethernet II to view Ethernet details. # Observe the Destination and Source fields. The destination should be the broadcast address ff:ff:ff:ff:ff:ff and the source should be your MAC address. When the client doesn't have an IP address or server information, it has to broadcast to request an address lease. # Expand Internet Protocol Version 4 to view IP details. # Observe the Source address. Notice that the source address is 0.0.0.0, indicating no current IP address. # Observe the Destination address. Notice that the destination address 255.255.255.255, the broadcast IP address. # Expand User Datagram Protocol to view UDP details. # Observe the Source port. Notice that it is bootpc (68), the BOOTP client port. # Observe the Destination port. Notice that it is bootps (67), the BOOTP server port. # Expand Bootstrap Protocol to view BOOTP details. # Observe the DHCP Message Type. Notice that it is a Request (3). # Observe the Client IP address, Client MAC address, and DHCP option fields. This is the request to the DHCP server. == Activity 8 - Analyze DHCP ACK Traffic == To analyze DHCP ACK (server acknowledgement) traffic: # In the top Wireshark packet list pane, select the seventh DHCP packet, labeled '''DHCP ACK'''. # Observe the packet details in the middle Wireshark packet details pane. Notice that it is an Ethernet II / Internet Protocol Version 4 / User Datagram Protocol / Bootstrap Protocol frame. # Expand Ethernet II to view Ethernet details. # Observe the Destination and Source fields. The destination should be your MAC address and the source should be your DHCP server's MAC address. # Expand Internet Protocol Version 4 to view IP details. # Observe the Source address. Notice that the source address is the DHCP server IP address. # Observe the Destination address. Notice that the destination address is the broadcast address 255.255.255.255. # Expand User Datagram Protocol to view UDP details. # Observe the Source port. Notice that it is bootps (67), the BOOTP server port. # Observe the Destination port. Notice that it is bootpc (68), the BOOTP client port. # Expand Bootstrap Protocol to view BOOTP details. # Observe the DHCP Message Type. Notice that it is an ACK (5). # Observe the Client IP address and Client MAC address fields. This is the acknowledgement from the DHCP server. # Observe the DHCP options and expand to view the details for IP Address Lease Time, Subnet Mask, Router (Default Gateway), Domain Name Server, and Domain Name, as well as any other options if included. # Close Wireshark to complete this activity. '''Quit without Saving''' to discard the captured traffic. == References == * [http://www.wireshark.org/docs/wsug_html_chunked/ Wireshark: User's Guide] * [http://wiki.wireshark.org/DHCP Wireshark: DHCP] [[Category:Wireshark]] [[Category:Networking]] [[Category:Activities]] [[Category:Dynamic Host Configuration Protocol]] h21d3ai168aamswhxtgfn296lqavfhf 10 145619 2693582 2392758 2024-12-27T04:27:42Z Tule-hog 2984180 mv cat 2693582 wikitext text/x-wiki {{Documentation subpage}} <!-- Categories and interwikis go at the bottom of this page. --> This template indicates that a page may qualify for speedy deletion and adds it to [[:Category:Candidates for speedy deletion]] for review by custodians. You should be familiar with the [[Wikiversity:Deletions|deletion guideline]] before using this template on any pages. == Usage == {{tlx|delete}} Parameters: * {{para|soon|<anything>}} to add a "soon" qualifier to the message. * {{para|reason|<why>}} to explain why this page fits the criteria for speedy deletion. == See also == * {{tlx|deletion request}} * {{tlx|proposed deletion}} <includeonly> <!-- Categories and interwikis go here: --> [[Category:Deletion templates|{{PAGENAME}}]] </includeonly> bpomcicxm66wbc0hs5mpv2inrom5xje 0 148182 2693472 1581928 2024-12-27T00:08:56Z Tule-hog 2984180 Bot: Replacing category Networks with [[:Category:Networking|Networking]] 2693472 wikitext text/x-wiki <noinclude> {{:Web_Science/Templates:TopNavigation}} </noinclude> {| class="wikitable sortable" style="text-align: center" |- ! Video<br>and script !! Associated Lesson |- | [[File:General_Problems_of_Communication_over_a_Shared_Medium.ogv|thumb|120px]] || [[Web_Science/Part1:_Foundations_of_the_web/Internet_Architecture/Ethernet/Communication over a shared Medium|Communication over a shared Medium]] |- | [[File:How_to_build_an_Ethernet_Frame.webm|thumb|120px]] || [[Web_Science/Part1:_Foundations_of_the_web/Internet_Architecture/Ethernet/Ethernet Header|Ethernet Header]] |- | [[File:Minimum_Frame_Length_in_Ethernet_explained.webm|thumb|120px]] || [[Web_Science/Part1:_Foundations_of_the_web/Internet_Architecture/Ethernet/Minimum Package length vs Maximum cable length|Minimum Package length vs Maximum cable length]] |- | [[File:Ethernet_Carrier_sense_multiple_access_with_collision_detection.ogv|thumb|120px]] || [[Web_Science/Part1:_Foundations_of_the_web/Internet_Architecture/Ethernet/Collision Detection|Collision Detection]] |- | no video || [[Web_Science/Part1:_Foundations_of_the_web/Internet_Architecture/Ethernet/Summary, Further readings, Homework|Summary, Further readings & Homework]] |} The following video of the flipped classroom associated with this topic are available: You can find more information on [[:File:Web_Science_MOOC_Ethernet_flipped_classroom_session_-_edited.webm|wiki commons]] and also [http://upload.wikimedia.org/wikipedia/commons/6/67/Web_Science_MOOC_Ethernet_flipped_classroom_session_-_edited.webm directly download this file] [[Category:Topic:Web Science]] [[Category:MOOC]] [[Category:Networking]] 9qkfr6057cepu4u09bwmw0efnmtqc67 0 149997 2693471 2581883 2024-12-27T00:08:46Z Tule-hog 2984180 Bot: Replacing category Networks with [[:Category:Networking|Networking]] 2693471 wikitext text/x-wiki <noinclude> {{#invoke:Mooc|render|base=Web Science/Part1: Foundations of the web}} </noinclude> [[Category:Topic:Web Science]] [[Category:MOOC]] [[Category:Networking]] [[Category:Domain Name System]] m11oeelbhz36kp3f4h2ympusnrfx72b 0 157678 2693322 2681830 2024-12-26T18:39:35Z Tule-hog 2984180 /* Objectives */ bump stages 2693322 wikitext text/x-wiki {{status|0%}}<noinclude> {{:{{FULLPAGENAME}}/Sidebar}} </noinclude> '''Computer Support''' is a computer hardware and software topic that includes computer hardware, networking, laptops, printers, operational procedures, operating systems, security, mobile devices, and troubleshooting. This course comprises 2 parts, 9 sections, and 58 lessons covering computer support. Each lesson includes a combination of Wikipedia readings, YouTube videos, and hands-on learning activities. The course also assists learners in preparing for [[Wikipedia:CompTIA|CompTIA]] A+ Certification. {{noprint | This entire Wikiversity course can be downloaded in book form by selecting Download Learning Guide in the sidebar.}} == Preparation == This is a second-semester, college-level course. Learners should already be familiar with [[IC3|introductory computer concepts]] and [[IT Fundamentals]]. ==Objectives== See the [[Computer Support/Objectives|list of all objectives]]. For specific sections: {{stages}} ===Core 1=== # [[Computer Support/Objectives/Mobile Devices|Mobile Devices]] {{stage|25%}} # [[Computer Support/Objectives/Networking|Networking]] {{stage|25%}} # [[Computer Support/Objectives/Hardware|Hardware]] {{stage|25%}} # [[Computer Support/Objectives/Virtualization and Cloud Computing|Virtualization and Cloud Computing]] {{stage|25%}} # [[Computer Support/Objectives/Hardware and Network Troubleshooting|Hardware and Network Troubleshooting]] {{stage|25%}} ===Core 2=== # [[Computer Support/Objectives/Operating Systems|Operating Systems]] {{stage|25%}} # [[Computer Support/Objectives/Security|Security]] {{stage|25%}} # [[Computer Support/Objectives/Software Troubleshooting|Software Troubleshooting]] {{stage|25%}} # [[Computer Support/Objectives/Operational Procedures|Operational Procedures]] {{stage|25%}} Also see the [[Computer Support/Acronyms|list of acronyms]] {{stage|25%}} and the [[Computer Support/Technologies|list of technologies]] {{stage|25%}}. ==Exam Details== Exam description: <blockquote>220-1101 covers mobile devices, networking technology, hardware, virtualization and cloud computing. <br/> 220-1102 covers operating systems, security, software and operational procedures.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: * 220-1101: 675 (on scale of 900) * 220-1101: 700 (on a scale of 900) Recommended experience: <blockquote>9 to 12 months hands-on experience in the lab or field</blockquote> Exam codes: 220-1101, 220-1102 Languages: English, German, Japanese, Portuguese, Thai, Spanish, French == Lessons == ==== Part 1 ==== * [[/Hardware/]] **'''Hardware Module 1''' *** [[/Hardware/Motherboards|Motherboards]] *** [[/Hardware/CPUs|CPUs]] *** [[/Hardware/BIOS and UEFI|BIOS and UEFI]] *** [[/Hardware/RAM|RAM]] *** [[/Hardware/Expansion Cards|Expansion Cards]] *** [[/Hardware/Storage|Storage]] *** [[/Hardware/Power Supplies|Power Supplies]] ** '''Hardware Module 2''' *** [[/Hardware/Interfaces|Interfaces]] *** [[/Hardware/Displays|Displays]] *** [[/Hardware/Connectors|Connectors]] *** [[/Hardware/Peripheral Devices|Peripheral Devices]] *** [[/Networking/Media|Media]] **'''Hardware Module 3''' *** [[/Hardware/Multifunction Devices|Multifunction Devices]] *** [[/Hardware/Print Technologies|Print Technologies]] *** [[/Hardware/Printer Maintenance|Printer Maintenance]] *** [[/Hardware/Components|Components]] * [[/Mobile Devices/]] **'''Mobile Devices Module 1''' *** [[/Mobile Devices/Laptop Components|Laptop Components]] *** [[/Mobile Devices/Laptop Features|Laptop Features]] **'''Mobile Devices Module 2''' *** [[/Mobile Devices/Other Devices|Other Devices]] *** [[/Mobile Devices/Ports and Accessories|Ports and Accessories]] *** [[/Mobile Devices/Connectivity|Mobile Connectivity]] *** [[/Mobile Devices/Synchronization|Mobile Synchronization]] **'''Virtualization and Cloud Computing''' *** [[/Other Technologies/Virtualization|Virtualization]] *** [[/Other Technologies/Cloud Concepts|Cloud Concepts]] * [[/Networking/]] **'''Networking Module 1''' *** [[/Networking/Devices|Devices]] *** [[/Networking/Tools|Tools]] **'''Networking Module 2''' *** [[/Networking/Addressing|Addressing]] *** [[/Networking/Protocols|Protocols]] *** [[/Networking/Wireless|Wireless]] *** [[/Other Technologies/Network Services|Network Services]] **'''Networking Module 3''' *** [[/Networking/Routers|Routers]] *** [[/Networking/Connectivity|Connectivity]] * [[/Hardware Troubleshooting/]] **'''Hardware & Networking Troubleshooting Module 1''' *** [[/Procedures/Troubleshooting Theory|Troubleshooting Theory]] *** [[/Hardware Troubleshooting/Systems|Systems]] *** [[/Hardware Troubleshooting/Storage|Storage]] **'''Hardware & Networking Troubleshooting Module 2''' *** [[/Hardware Troubleshooting/Displays|Displays]] *** [[/Hardware Troubleshooting/Printers|Printers]] **'''Hardware & Networking Troubleshooting Module 3''' *** [[/Hardware Troubleshooting/Networks|Networks]] *** [[/Hardware Troubleshooting/Mobile Devices|Mobile Devices]] ==== Part 2 ==== * [[/Operating Systems/]] **'''Operating Systems Module 1''' *** [[/Windows/Installation|Installation]] *** [[/Other Technologies/Mobile Operating Systems|Mobile Operating Systems]] **'''Operating Systems Module 2''' *** [[/Windows/Features|Features]] **'''Operating Systems Module 3''' *** [[/Windows/Control Panel|Control Panel]] *** [[/Windows/GUI Tools|GUI Tools]] **'''Operating Systems Module 4''' *** [[/Windows/Maintenance|Maintenance]] *** [[/Windows/Command Line Tools|Command Line Tools]] *** [[/Other Technologies/macOS and Linux|macOS and Linux]] *** [[/Windows/Networking|Networking]] * [[/Procedures/]] **'''Operational Procedures''' *** [[/Procedures/Safety|Safety]] *** [[/Procedures/Environment|Environment]] *** [[/Procedures/Policies|Policies]] *** [[/Procedures/Communication|Communication]] * [[/Security/]] **'''Security Module 1''' *** [[/Security/Prevention|Prevention]] *** [[/Security/Mobile|Mobile]] *** [[/Security/Data Destruction|Data Destruction]] **'''Security Module 2''' *** [[/Security/Threats|Threats]] *** [[/Security/Best Practices|Best Practices]] **'''Security Module 3''' *** [[/Security/Windows|Windows]] *** [[/Security/Wired and Wireless|Wired and Wireless]] * [[/Software Troubleshooting/]] **'''Software Troubleshooting Module 1''' *** [[/Software Troubleshooting/PC Operating Systems|PC Operating Systems]] **'''Software Troubleshooting Module 2''' ***[[/Software Troubleshooting/PC Security|PC Security]] **'''Software Troubleshooting Module 3''' *** [[/Software Troubleshooting/Mobile Operating Systems|Mobile Operating Systems]] *** [[/Software Troubleshooting/Mobile Security|Mobile Security]] == See Also == {{Wikibooks|A+ Certification}} * [[IT Fundamentals]] * [[Wikipedia: CompTIA]] * [[Computer Networks]] * [[Network+ Certification]] * [[Security+ Certification]] == External Links == * [https://www.cybrary.it/course/comptia-aplus/ Cybrary: Free CompTIA A+ Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-a-220-1101-exam-objectives-(3-0) CompTIA: A+ Certification Exam Objectives - Exam 220-1101] * [https://partners.comptia.org/docs/default-source/resources/comptia-a-220-1102-exam-objectives-(3-0) CompTIA: A+ Certification Exam Objectives - Exam 220-1102] {{Hide|{{Information technology|theme=14}}}} {{Hide|{{Tertiary|theme=14}}}} [[Category:Technology courses]] [[Category:Certifications]] [[Category:Study guides]] {{CourseCat}} lhzam4hpbrvo4w3m6umak642wszifga 0 160164 2693383 2068277 2024-12-26T21:12:32Z 129.176.64.37 grammar (an) - please check this 2693383 wikitext text/x-wiki '''Acceleration stress-energy tensor''' is a symmetric four-dimensional tensor of the second valence (rank), which describes the density and flux of energy and momentum of an acceleration field in matter. This tensor in the covariant theory of gravitation is included in the equation for determining the metric along with the [[gravitational stress-energy tensor]], the [[pressure stress-energy tensor]], the [[dissipation stress-energy tensor]] and the stress-energy tensor of an electromagnetic field. The covariant derivative of the acceleration stress-energy tensor determines the density of the [[four-force]] acting on the matter. == Covariant theory of gravitation== === Definition === In [[covariant theory of gravitation]] (CTG) the acceleration field is not a scalar field and considered as 4-vector field, 4-potential of which consists of the scalar and 3-vector components. In CTG the acceleration stress-energy tensor was defined by [[User:Fedosin | Fedosin]] through the [[acceleration tensor]] <math> ~ u_{ik} </math> and the metric tensor <math> ~ g^{ik} </math> by the principle of least action: <ref name="f"> [[User:Fedosin | Fedosin S.G.]] [http://journals.yu.edu.jo/jjp/Vol9No1Contents2016.html About the cosmological constant, acceleration field, pressure field and energy.] Jordan Journal of Physics. Vol. 9 (No. 1), pp. 1-30 (2016).</ref> :<math>~ B^{ik} = \frac{c^2} {4 \pi \eta } \left( - g^{im} u_{nm} u^{nk}+ \frac {1} {4} g^{ik}u_{mr}u^{mr}\right) ,</math> where <math> ~ \eta </math> is the acceleration field constant defined in terms of the fundamental constants and physical parameters of the system. [[Acceleration field]] is considered as a component of the [[general field]]. === Components of the acceleration stress-energy tensor === Since acceleration tensor consists of the components of the acceleration field strength <math> ~ \mathbf {S} </math> and the solenoidal acceleration vector <math> ~ \mathbf {N} </math>, then the acceleration stress-energy tensor can be expressed through these components. In the limit of [[special relativity]] the metric tensor ceases to depend on the coordinates and time, and in this case the acceleration stress-energy tensor gains the simplest form: :<math>~ B^{ik} = \begin{vmatrix} \varepsilon_a & \frac {K_x}{c} & \frac {K_y}{c} & \frac {K_z}{c} \\ c P_{ax} & \varepsilon_a - \frac{S^2_x+c^2 N^2_x}{4\pi \eta } & -\frac{S_x S_y+c^2 N_x N_y }{4\pi\eta } & -\frac{S_x S_z+c^2 N_x N_z }{4\pi\eta } \\ c P_{ay} & -\frac{S_x S_y+c^2 N_x N_y }{4\pi\eta } & \varepsilon_a -\frac{S^2_y+c^2 N^2_y }{4\pi\eta } & -\frac{S_y S_z+c^2 N_y N_z }{4\pi\eta } \\ c P_{az} & -\frac{S_x S_z+c^2 N_x N_z }{4\pi\eta } & -\frac{S_y S_z+c^2 N_y N_z }{4\pi\eta } & \varepsilon_a -\frac{S^2_z+c^2 N^2_z }{4\pi\eta } \end{vmatrix}. </math> The time-like components of the tensor denote: 1) The volumetric energy density of acceleration field :<math>~ B^{00} = \varepsilon_a = \frac{1}{8 \pi \eta }\left(S^2+ c^2 N^2 \right).</math> 2) The vector of momentum density of acceleration field <math> ~\mathbf{P_a} =\frac{ 1}{ c^2} \mathbf{K}, </math> where the vector of energy flux density of acceleration field is :<math>~\mathbf{K} = \frac{ c^2 }{4 \pi \eta }[\mathbf{S}\times \mathbf{N}].</math> Due to the symmetry of the tensor indices, <math> P^{01}= P^{10}, P^{02}= P^{20}, P^{03}= P^{30}</math>, so that <math> \frac{ 1}{ c} \mathbf{K}= c \mathbf{P_a} .</math> 3) The space-like components of the tensor form a submatrix 3 x 3, which is the 3-dimensional acceleration stress tensor, taken with a minus sign. The acceleration stress tensor can be written as :<math>~ \sigma^{p q} = \frac {1}{4 \pi \eta } \left( S^p S^q + c^2 N^p N^q - \frac {1}{2} \delta^{pq} (S^2 + c^2 N^2 ) \right) ,</math> where <math>~p,q =1,2,3, </math> the components <math>S^1=S_x, </math> <math>S^2=S_y, </math> <math>S^3=S_z, </math> <math> N^1=N_x, </math> <math>N^2=N_y, </math> <math>N^3=N_z, </math> the [[w:Kronecker delta |Kronecker delta]] <math>~\delta^{pq}</math> equals 1 if <math>~p=q, </math> and equals 0 if <math>~p \not=q. </math> Three-dimensional divergence of the stress tensor of acceleration field connects the force density and rate of change of momentum density of the acceleration field: :<math>~ \partial_q \sigma^{p q} = - f^p +\frac {1}{c^2} \frac{ \partial K^p}{\partial t}, </math> where <math>~ f^p </math> denote the components of the three-dimensional acceleration force density, <math>~ K^p </math> – the components of the energy flux density of the acceleration field. === 4-force density and field equation === The principle of least action implies that the 4-vector of force density <math> ~ f_\alpha </math> can be found through the acceleration stress-energy tensor, either through the product of acceleration tensor and mass 4-current: :<math>~ f_\alpha = \nabla_\beta {B_\alpha}^\beta = - u_{\alpha k} J^k. \qquad (1) </math> The field equations of acceleration field are as follows: :<math>~ \nabla_n u_{ik} + \nabla_i u_{kn} + \nabla_k u_{ni}=0, </math> :<math>~\nabla_k u^{ik} = -\frac {4 \pi \eta }{c^2} J^i .</math> In the special theory of relativity, according to (1) for the components of the [[four-force]] density can be written: :<math>~ f_\alpha = (- \frac {\mathbf{S} \cdot \mathbf{J} }{c}, - \mathbf{f} ),</math> where <math>~ \mathbf{f}= - \rho \mathbf{S} - [\mathbf{J} \times \mathbf{N} ]</math> is the 3-vector of the force density, <math>~\rho</math> is the density of the moving matter, <math>~\mathbf{J} =\rho \mathbf{v} </math> is the 3-vector of the mass current density, <math>~\mathbf{v} </math> is the 3-vector of velocity of the matter unit. In Minkowski space, the field equations are transformed into four equations for the acceleration field strength <math> ~ \mathbf {S} </math> and solenoidal acceleration vector <math> ~ \mathbf {N} </math> :<math>~\nabla \cdot \mathbf{ S} = 4 \pi \eta \rho,</math> :<math>~\nabla \times \mathbf{ N} = \frac {1 }{c^2}\frac{\partial \mathbf{ S}}{\partial t}+\frac {4 \pi \eta \rho \mathbf{ v}}{c^2},</math> :<math>~\nabla \cdot \mathbf{ N} = 0,</math> :<math>~\nabla \times \mathbf{ S} = - \frac{\partial \mathbf{ N}}{\partial t}.</math> === Equation for the metric === In the covariant theory of gravitation the acceleration stress-energy tensor in accordance with the principles of [[metric theory of relativity]] is one of the tensors defining metrics inside the bodies by the equation for the metric: :<math>~ R_{ik} - \frac{1} {4 }g_{ik}R = \frac{8 \pi G \beta }{ c^4} \left( B_{ik}+ P_{ik}+ U_{ik}+ W_{ik} \right), </math> where <math>~ \beta </math> is the coefficient to be determined, <math>~ B_{ik}</math>, <math>~ P_{ik}</math>, <math>~ U_{ik}</math> and <math>~ W_{ik}</math> are the stress-energy tensors of the acceleration field, [[pressure field]], gravitational and electromagnetic fields, respectively, <math>~ G </math> is the [[gravitational constant]]. === Equation of motion === The equation of motion of a point particle inside or outside matter can be represented in tensor form, with acceleration stress-energy tensor <math> B^{ik}</math> or acceleration tensor <math> u_{nk}</math> : :<math>~ - \nabla_k \left( B^{ik}+ U^{ik} +W^{ik}+ P^{ik} \right) = g^{in}\left(u_{nk} J^k + \Phi_{nk} J^k + F_{nk} j^k + f_{nk} J^k \right) =0. \qquad (2)</math> where <math> ~ \Phi_{nk}</math> is the [[gravitational tensor ]], <math> ~F_{nk}</math> is the [[w:electromagnetic tensor |electromagnetic tensor]], <math> ~ f _{nk}</math> is the [[pressure field tensor]], <math>~j^k = \rho_{0q} u^k </math> is the charge 4-current, <math>~\rho_{0q}</math> is the density of electric charge of the matter unit in the reference frame at rest, <math>~ u^k </math> is the 4-velocity. We now recognize that <math> ~ J^k = \rho_{0} u^k </math> is the mass 4-current and the acceleration tensor is defined through the covariant 4-potential as <math>~ u _{nk}= \nabla_n U_k - \nabla_k U_n. </math> This gives the following: <ref> Fedosin S.G. Equations of Motion in the Theory of Relativistic Vector Fields. International Letters of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12. </ref> :<math>~ \nabla_\beta {B_n}^\beta = - u_{n k} J^k = - \rho_{0} u^k (\nabla_n U_k - \nabla_k U_n)= \rho_{0} \frac {DU_n}{D \tau } - \rho_{0} u^k \nabla_n U_k . \qquad (3)</math> Here [[operator of proper-time-derivative]] <math>~ u^k \nabla_k = \frac {D}{D \tau }</math> is used, where <math> ~ D </math> is the symbol of 4-differential in curved spacetime, <math> ~ \tau </math> is the [[proper time]], <math> ~ \rho_0 </math> is the mass density in the comoving frame. Accordingly, the equation of motion (2) becomes: :<math>~ \rho_{0} \frac {DU_n}{D \tau }- \rho_{0} u^k \nabla_n U_k = - \nabla^k \left(U_{nk} +W_{nk}+ P_{nk} \right) = \Phi_{nk} J^k + F_{nk} j^k + f_{nk} J^k. </math> Time-like component of the equation at <math>~ n=0</math> describes the rate of change of the scalar potential of the acceleration field, and spatial component at <math>~ n=1{,}2{,}3</math> connects the rate of change of the vector potential of the acceleration field with the force density. === Conservation laws=== When the index <math> ~ i = 0 </math> in (2), i.e. for the time-like component of the equation, in the limit of special relativity from the vanishing of the left side of (2) follows: :<math>~ \nabla \cdot (\mathbf{ K }+ \mathbf{H}+\mathbf{P}+ \mathbf{F} ) = -\frac{\partial (B^{00}+U^{00}+W^{00}+P^{00} )}{\partial t},</math> where <math>~ \mathbf{ K }</math> is the vector of the acceleration field energy flux density, <math>~ \mathbf{H}</math> is the [[Heaviside vector]], <math>~ \mathbf{ P }</math> is the [[w:Poynting vector |Poynting vector]], <math>~ \mathbf{F}</math> is the vector of the pressure field energy flux density. This equation can be regarded as a local conservation law of energy-momentum of the four fields. <ref name="gen"> Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. International Frontier Science Letters, Vol. 14, pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19. </ref> The integral form of the law of conservation of energy-momentum is obtained by integrating (2) over the 4-volume. By the [[w:Divergence theorem |divergence theorem]] the integral of the 4-divergence of some tensor over the 4-space can be replaced by the integral of time-like tensor components over 3-volume. As a result, in Lorentz coordinates the integral vector equal to zero may be obtained: <ref> Fedosin S.G. [http://vixra.org/abs/1403.0973 The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field.] American Journal of Modern Physics. Vol. 3, No. 4, 2014, pp. 152-167. http://dx.doi.org/10.11648/j.ajmp.20140304.12.</ref> :<math>~ \mathbb{Q}^i= \int{ \left( B^{i0}+ U^{i0} +W^{i0}+P^{i0} \right) dV }. </math> Vanishing of the integral vector allows us to explain the 4/3 problem, according to which the mass-energy of field in the momentum of field of the moving system in 4/3 more than in the field energy of fixed system. On the other hand, according to, <ref name="gen"/> the generalized Poynting theorem and the integral vector should be considered differently inside the matter and beyond its limits. As a result, the occurrence of the 4/3 problem is associated with the fact that the time components of the stress-energy tensors do not form four-vectors, and therefore they cannot define the same mass in the fields’ energy and momentum in principle. == Relativistic mechanics == As in relativistic mechanics, and in [[general relativity]] (GR), the acceleration stress-energy tensor is not used. Instead it uses the so-called stress-energy tensor of matter, which in the simplest case has the following form: <math>~ \phi_{ n \beta }= \rho_0 u_n u_\beta </math>. In GR, the tensor <math>~ \phi_{ n \beta }</math> is substituted into the equation for the metric and its covariant derivative gives the following: :<math>~ \nabla^\beta \phi _{n \beta} = \nabla^\beta (\rho_0 u_n u_\beta) = u_n \nabla^\beta J_\beta + \rho_0 u_\beta \nabla^\beta u_n . </math> In GR it is assumed that there is the [[w:continuity equation |continuity equation]] in the form <math>~ \nabla^\beta J_\beta =0 .</math> Then, using the operator of proper-time-derivative the covariant derivative of the tensor <math>~ \phi_{ n \beta }</math> gives the product of the mass density and [[four-acceleration]], i.e. the density of 4-force: :<math>~ \nabla^\beta \phi _{n \beta} = \rho_0 u_\beta \nabla^\beta u_n = \rho_0 \frac {Du_n}{D \tau }. \qquad (4)</math> However, the continuity equation is valid only in the special theory of relativity as <math>~ \partial^\beta J_\beta = \partial_\beta J^\beta =0 .</math> In curved space-time instead would have to be the equation <math>~ \nabla^\beta J_\beta =0 </math>, but instead of zero on the right side of this equation there appears an additional non-zero term with [[w:Riemann curvature tensor | Riemann curvature tensor]]. <ref name="f"/> Consequently, (4 ) is not an exact expression, and tensor <math>~ \phi_{ n \beta }</math> determines the properties of the matter only in the special theory of relativity. In contrast, in the covariant theory of gravitation equation (3) is written in covariant form, so that the acceleration stress-energy tensor <math>~ B_{ n \beta }</math> describes well the acceleration field of matter particles in curved Riemannian space-time. == See also == * [[Acceleration field]] * [[Gravitational stress-energy tensor]] * [[Pressure stress-energy tensor]] * [[Dissipation stress-energy tensor]] * [[w:Electromagnetic stress-energy tensor |Electromagnetic stress-energy tensor]] * [[Acceleration tensor]] * [[General field]] * [[Dissipation field]] * [[Pressure field]] == References == <references/> ==External links == * [http://www.wikiznanie.ru/ru-wz/index.php/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D0%B8-%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81%D0%B0_%D0%BF%D0%BE%D0%BB%D1%8F_%D1%83%D1%81%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D0%B8%D0%B9 Acceleration stress-energy tensor in Russian] [[Category:Theory of relativity]] [[Category:Tensors]] [[Category:Covariant theory of gravitation]] hr0slq5tx9hvn1tjuj6rkvqs3ofbxm7 0 162164 2693325 2668558 2024-12-26T18:55:23Z Tule-hog 2984180 /* Objectives */ bump stages 2693325 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[IC3|introductory computer]] and [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> mhylzl140ti2up780lmurvjfqpmoo4p 2693331 2693325 2024-12-26T18:58:06Z Tule-hog 2984180 /* Preparation */ isolate preceding cert 2693331 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> n404m9zg0uwisu57k57e6hocxu9v3l2 2693351 2693331 2024-12-26T19:27:16Z Tule-hog 2984180 /* Guides */ mk section 2693351 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) ==Guides== These are compiled resources to help familiarize learners with Network+ concepts. Feel free to add relevant material (with caution for copy violations - Wikipedia may be freely copied with attribution). * [[/Further reading|Further reading]] == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> jy0u47sube4e5lc297sv881ulm93jms 2693359 2693351 2024-12-26T19:37:54Z Tule-hog 2984180 /* Old guides */ mk section 2693359 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) ==Guides== These are compiled resources to help familiarize learners with Network+ concepts. Feel free to add relevant material (with caution for copy violations - Wikipedia may be freely copied with attribution). * [[/Further reading|Further reading]] ===Old guides=== These were made for a previous version of the exam, and may contain outdated material. * [[/Old guides/Routing|Routing]] == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> q5clyv0xkhcndqes9l6i04q7108aclt 2693408 2693359 2024-12-26T22:51:05Z Tule-hog 2984180 /* Guides */ add activities 2693408 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) ==Guides== These are compiled resources to help familiarize learners with Network+ concepts. Feel free to add relevant material (with caution for copy violations - Wikipedia may be freely copied with attribution). * [[/Further reading|Further reading]] * [[/Activities|Activities]] ===Old guides=== These were made for a previous version of the exam, and may contain outdated material. * [[/Old guides/Routing|Routing]] == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> 10uef6focmthsubkz72dhn7jz0tns32 2693412 2693408 2024-12-26T22:56:31Z Tule-hog 2984180 /* See also */ mv section to bottom 2693412 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) ==Guides== These are compiled resources to help familiarize learners with Network+ concepts. Feel free to add relevant material (with caution for copy violations - Wikipedia may be freely copied with attribution). * [[/Further reading|Further reading]] * [[/Activities|Activities]] ===Old guides=== These were made for a previous version of the exam, and may contain outdated material. * [[/Old guides/Routing|Routing]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> 58h5x9prumx91yho9t5sf38xkdsof5w 2693417 2693412 2024-12-26T23:11:45Z Tule-hog 2984180 /* Old guides */ add item 2693417 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) ==Guides== These are compiled resources to help familiarize learners with Network+ concepts. Feel free to add relevant material (with caution for copy violations - Wikipedia may be freely copied with attribution). * [[/Further reading|Further reading]] * [[/Activities|Activities]] ===Old guides=== These were made for a previous version of the exam, and may contain outdated material. * [[/Old guides/Routing|Routing]] * [[/Old guides/Networking tools|Networking tools]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> iy0tsqdck75hfvwl6lv79kdf1877elg 2693434 2693417 2024-12-26T23:24:05Z Tule-hog 2984180 /* Old guides */ add item 2693434 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) ==Guides== These are compiled resources to help familiarize learners with Network+ concepts. Feel free to add relevant material (with caution for copy violations - Wikipedia may be freely copied with attribution). * [[/Further reading|Further reading]] * [[/Activities|Activities]] ===Old guides=== These were made for a previous version of the exam, and may contain outdated material. * [[/Old guides/Routing|Routing]] * [[/Old guides/Networking tools|Networking tools]] * [[/Old guides/Ethernet|Ethernet]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> qdp9pxg6muwq2knp8yjkyxewu7w32b5 2693492 2693434 2024-12-27T00:27:05Z Tule-hog 2984180 /* Old guides */ add item 2693492 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) ==Guides== These are compiled resources to help familiarize learners with Network+ concepts. Feel free to add relevant material (with caution for copy violations - Wikipedia may be freely copied with attribution). * [[/Further reading|Further reading]] * [[/Activities|Activities]] ===Old guides=== These were made for a previous version of the exam, and may contain outdated material. * [[/Old guides/Routing|Routing]] * [[/Old guides/Networking tools|Networking tools]] * [[/Old guides/Ethernet|Ethernet]] * [[/Old guides/Network media|Network media]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> rij1x3agrpkgx31lm43kbhxptckcopz 2693510 2693492 2024-12-27T00:43:09Z Tule-hog 2984180 /* Old guides */ add item 2693510 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) ==Guides== These are compiled resources to help familiarize learners with Network+ concepts. Feel free to add relevant material (with caution for copy violations - Wikipedia may be freely copied with attribution). * [[/Further reading|Further reading]] * [[/Activities|Activities]] ===Old guides=== These were made for a previous version of the exam, and may contain outdated material. * [[/Old guides/Routing|Routing]] * [[/Old guides/Networking tools|Networking tools]] * [[/Old guides/Ethernet|Ethernet]] * [[/Old guides/Network media|Network media]] * [[/Old guides/OSI Model|OSI Model]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> 61zl7toaz1cye1wizudnsxfnon1j0f6 2693534 2693510 2024-12-27T00:51:50Z Tule-hog 2984180 /* Old guides */ item mv'd to activity 2693534 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) ==Guides== These are compiled resources to help familiarize learners with Network+ concepts. Feel free to add relevant material (with caution for copy violations - Wikipedia may be freely copied with attribution). * [[/Further reading|Further reading]] * [[/Activities|Activities]] ===Old guides=== These were made for a previous version of the exam, and may contain outdated material. * [[/Old guides/Routing|Routing]] * [[/Old guides/Ethernet|Ethernet]] * [[/Old guides/Network media|Network media]] * [[/Old guides/OSI Model|OSI Model]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> cicxa7fcq8k1ohg17qz6qf4mny3qwg0 2693547 2693534 2024-12-27T01:02:34Z Tule-hog 2984180 /* Old guides */ add item 2693547 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} '''Network+''' is a [[Wikipedia:CompTIA|CompTIA]] computer networking certification that includes computer network concepts, installation and configuration, media and topologies, management, and security. This [[wv:LR|resource]] features a combination of Wikipedia readings, YouTube videos, and hands-on learning activities as a [[:Category:Study guides|study guide]] to prepare for [[Wikipedia:CompTIA|CompTIA]] Network+ Certification. {{noprint | This entire Wikiversity course can be downloaded as a PDF by selecting Download Learning Guide in the sidebar (which will preserve [[:w:hyperlinks|hyperlinks]].)}} == Preparation == Learners would be best served by first approaching [[Computer Support|computer support]] concepts. ==Objectives== See [[Network+/Objectives|list of all objectives]]. For each domain: {{stages}} # [[Network+/Objectives/Networking Concepts|Networking Concepts]] {{stage|50%}} # [[Network+/Objectives/Network Implementation|Network Implementation]] {{stage|50%}} # [[Network+/Objectives/Network Operations|Network Operations]] {{stage|25%}} # [[Network+/Objectives/Network Security|Network Security]] {{stage|50%}} # [[Network+/Objectives/Network Troubleshooting|Network Troubleshooting]] {{stage|25%}} Also see [[Network+/Acronyms|list of acronyms]] {{stage|25%}} and [[Network+/Technologies|technologies]] {{stage|25%}}. ==Test Details== Exam description: <blockquote>The CompTIA Network+ certification verifies that the successful candidate has the knowledge and skills required to: * Establish network connectivity by deploying wired and wireless devices. * Understand and maintain network documentation. * Understand the purpose of network services. * Understand basic datacenter, cloud and virtual networking concepts. * Monitor network activity, identifying performance and availability issues. * Implement network hardening techniques. * Manage, configure, and troubleshoot network infrastructure.</blockquote> Number of questions: Maximum of 90 Length of test: 90 minutes Passing score: 720 (on 100-900 scale) Recommended experience: <blockquote>CompTIA A+ certification and a minimum of 9-12 months of hands-on experience working in a junior network administrator/network support technician job role.</blockquote> Exam code: N10-009 Languages: English, Japanese, German, Spanish, (Portuguese to follow) ==Guides== These are compiled resources to help familiarize learners with Network+ concepts. Feel free to add relevant material (with caution for copy violations - Wikipedia may be freely copied with attribution). * [[/Further reading|Further reading]] * [[/Activities|Activities]] ===Old guides=== These were made for a previous version of the exam, and may contain outdated material. * [[/Old guides/Routing|Routing]] * [[/Old guides/Ethernet|Ethernet]] * [[/Old guides/Network media|Network media]] * [[/Old guides/OSI Model|OSI Model]] * [[/Old guides/IP Model|IP Model]] == External Links == * [https://www.cybrary.it/course/comptia-network-plus/ Cybrary: CompTIA N10-009 Network+ Course] * [https://www.professormesser.com/network-plus/n10-009/n10-009-video/n10-009-training-course/ Professor Messer’s CompTIA N10-009 Network+ Training Course] == References == * [https://partners.comptia.org/docs/default-source/resources/comptia-network-n10-009-exam-objectives-(4-0) CompTIA: Network+ Certification Exam Objectives - Exam N10-009] <noinclude>===Further reading=== * {{cite book|title=CompTIA Network+ Study Guide: Exam N10-009 |edition=6 |isbn= 1394235607 |author1=Todd Lammle |author2=Jon Buhagiar |date=May 7, 2024 |publisher=Sybex}} * {{cite book |title=CompTIA Network+ N10-009 Exam Cram |edition=8 |isbn=0135340837 |date= September 9, 2024 |publisher=Pearson IT Certification |author=Emmett Dulaney}} == See Also == * [[Computer Skills]] * [[IC3|Internet and Computing Core Certification (IC³)]] * [[Exam 98-366: Networking Fundamentals]] * [[Computer Networks]] * [[Internet Protocol Analysis]] * [[Network Administration]] ===CompTIA=== {{Wikibooks|Network Plus Certification}} * [[IT Fundamentals]] * [[A+ Certification]] * [[Security+ Certification]] {{Hide|{{Information technology|theme=14}}}} [[Category:Computer Networks| ]] {{Hide|{{Tertiary|theme=14}}}} [[Category:Certifications]] {{BookCat}} </noinclude> qdqinkjl7e8nl1txscdi1b8eocdkoio 0 162165 2693551 2594432 2024-12-27T01:08:19Z Tule-hog 2984180 test rm old components 2693551 wikitext text/x-wiki {{noprint | __NOTOC__{{sidebar | title = [[{{Titleparts|1}}]] | navbar = none | topimage = [[File:Map marker icon – Nicolas Mollet – Metro Network – Industry – Simple +90.png | 84px | link={{Titleparts|1}}]] | headingstyle = | bodystyle = width:20em; | contentstyle = text-align:left; <!-- | content1 = :* [[{{Titleparts|1}}/Architecture|Architecture]] :* [[{{Titleparts|1}}/Operations|Operations]] :* [[{{Titleparts|1}}/Security|Security]] :* [[{{Titleparts|1}}/Troubleshooting|Troubleshooting]] :* [[{{Titleparts|1}}/Standards|Standards]] --> ---- :[[{{Titleparts|1}}/Collection | Download Learning Guide]] }}}} r4izpvkznoyam61jstpjs4kntu3qbcy 2693552 2693551 2024-12-27T01:11:18Z Tule-hog 2984180 test new components 2693552 wikitext text/x-wiki {{noprint | __NOTOC__{{sidebar | title = [[{{Titleparts|1}}]] | navbar = none | topimage = [[File:Map marker icon – Nicolas Mollet – Metro Network – Industry – Simple +90.png | 84px | link={{Titleparts|1}}]] | headingstyle = | bodystyle = width:20em; | contentstyle = text-align:left; | content1 = :* [[{{Titleparts|1}}/Objectives|Objectives]] :* [[{{Titleparts|1}}/Further reading|Further reading]] :* [[{{Titleparts|1}}/Activities|Activities]] :* [[{{Titleparts|1}}/Old guides|Old guides]] <!-- :* [[{{Titleparts|1}}/Architecture|Architecture]] :* [[{{Titleparts|1}}/Operations|Operations]] :* [[{{Titleparts|1}}/Security|Security]] :* [[{{Titleparts|1}}/Troubleshooting|Troubleshooting]] :* [[{{Titleparts|1}}/Standards|Standards]] --> ---- :[[{{Titleparts|1}}/Collection | Download Learning Guide]] }}}} n6ckeoe6jt8qhlcbr22w33tvft89e8o 2693553 2693552 2024-12-27T01:13:53Z Tule-hog 2984180 no point testing until old pgs rm'd 2693553 wikitext text/x-wiki {{noprint | __NOTOC__{{sidebar | title = [[{{Titleparts|1}}]] | navbar = none | topimage = [[File:Map marker icon – Nicolas Mollet – Metro Network – Industry – Simple +90.png | 84px | link={{Titleparts|1}}]] | headingstyle = | bodystyle = width:20em; | contentstyle = text-align:left; | content1 = :* [[{{Titleparts|1}}/Architecture|Architecture]] :* [[{{Titleparts|1}}/Operations|Operations]] :* [[{{Titleparts|1}}/Security|Security]] :* [[{{Titleparts|1}}/Troubleshooting|Troubleshooting]] :* [[{{Titleparts|1}}/Standards|Standards]] ---- :[[{{Titleparts|1}}/Collection | Download Learning Guide]] }}}} kdfp0vqscixywjyd3yp9un8aabh61k4 0 162166 2693561 1570899 2024-12-27T01:22:03Z Tule-hog 2984180 nominate speedy 2693561 wikitext text/x-wiki {{delete|Content merged into updated [[Network+]]. See [[User_talk:Dave_Braunschweig#IT_Security_update|permission from original author]] and [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]. ''Please delete all subpages as well''.}} {{:{{PAGENAME}}/Sidebar}} This section of the computer networks course covers the OSI model, theory and concepts, wireless, wired, policies and procedures, safety practices, equipment placement, change management, protocol concepts, and protocol usage. == Preparation == Learners should already be familiar with [[Network Fundamentals|network fundamentals]]. == Lessons == * [[/OSI Model/]] * [[/Theory and Concepts/]] * [[/Wireless/]] * [[/Wired/]] * [[/Policies and Procedures/]] * [[/Safety Practices/]] * [[/Equipment Placement/]] * [[/Change Management/]] * [[/Protocol Concepts/]] * [[/Protocol Usage/]] == See Also == * [[IC3/Network Fundamentals | IC³ - Network Fundamentals]] == References == * [http://certification.comptia.org/docs/default-source/exam-objectives/comptia-network-(n10-006)_examobjectives.pdf CompTIA: Network+ Certification Exam Objectives - Exam N10-006] [[Category:Computer Networks]] f19tof4nai7vo8tjc6bowqcmmsqyjlz 0 162176 2693503 2552126 2024-12-27T00:38:27Z Tule-hog 2984180 Tule-hog moved page [[Network+/Standards/OSI Model/OSI Components]] to [[Network+/Old guides/OSI Model/OSI Components]]: alter parent 2552126 wikitext text/x-wiki This table shows the [[OSI model]] layers and the components that operate at each layer. {| class="wikitable" |- ! OSI Layer !! Purpose !! TCP/IP Layer !! Protocol !! Packet Data Unit !! Address !! Device !! Troubleshoot |- | [[Wikipedia:Application layer|Application]] || Interface (API) || [[Wikipedia:Application layer|Application]] || HTTP, SMTP, etc. || Message || - || - || Wireshark |- | [[Wikipedia:Presentation layer|Presentation]] || Formatting, Encryption, Compression || [[Wikipedia:Application layer|Application]] || HTTP, SMTP, etc. || Message || - || - || Wireshark |- | [[Wikipedia:Session layer|Session]] || Authentication, Authorization || [[Wikipedia:Application layer|Application]] || HTTP, SMTP, etc. || Message || - || Gateway || NSLOOKUP, NBTSTAT, Wireshark |- | [[Wikipedia:Transport layer|Transport]] || Reliability || [[Wikipedia:Transport layer|Transport]] || TCP, UDP || Segment (TCP), Datagram (UDP)|| Port || Firewall || TELNET, NETSTAT, Wireshark |- | [[Wikipedia:Network layer|Network]] || Addressing, Routing || [[Wikipedia:Internet layer|Internet]] || IP, ICMP || Packet || IP Address || Router || IPCONFIG, PING, TRACERT, Wireshark |- | [[Wikipedia:Data link layer|Data Link]] || Logical Link Control, Media Access Control || [[Wikipedia:Link layer|Link]] || Ethernet, Wi-Fi, PPP, etc. || Frame || MAC Address || Switch, Bridge, Access Point || Lights on device, ARP, Wireshark |- | [[Wikipedia:Physical layer|Physical]] || Transmission || [[Wikipedia:Link layer|Link]] || CAT 5, RJ-45, etc. || Bit || - || Hub, NIC, Cable, Wireless || Lights on device |} One easy way to remember the OSI layer is to think:{{source}} A - All P - People S - Seem T - To N - Need D - Data P - Processing Another popular acrostic to remember OSI layers names is (inferring that it is required to attend classes to pass networking certification exams): A - Away P - Pizza S - Sausage T - Throw N - Not D - Do P - Please Suggestions from David Bombal (7 to 1): A - All P - People S - Sleeping T - Through N - Networking D - Don't P - Pass Going the opposite way (1-7): P - Please D - Do N - Not T - Teach S - Students P - Pointless A - Acronyms Going the opposite way, widely used in India Schools (1-7): P - Please D - Do N - Not T - Take S - Sales P - Person's A - Advice 7ga34zfylmqpypcew2wappgofw7fx5m 0 162313 2693554 1570796 2024-12-27T01:20:22Z Tule-hog 2984180 nominate speedy 2693554 wikitext text/x-wiki {{delete|Content merged into updated [[Network+]]. See [[User_talk:Dave_Braunschweig#IT_Security_update|permission from original author]] and [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]. ''Please delete all subpages as well''.}} {{:{{PAGENAME}}/Sidebar}} This section of the computer networks course covers devices, remote access, services, WAN technologies, media, topologies, infrastructure, addressing, routing, unified communications, virtualization and cloud, and implementation. == Preparation == Learners should already be familiar with [[Network Fundamentals|network fundamentals]]. == Lessons == * [[/Devices/]] * [[/Remote Access/]] * [[/Services/]] * [[/WAN Technologies/]] * [[/Media/]] * [[/Topologies/]] * [[/Infrastructure/]] * [[/Addressing/]] * [[/Routing/]] * [[/Unified Communications/]] * [[/Virtualization and Cloud/]] * [[/Implementation/]] == See Also == * [[IC3/Network Fundamentals | IC³ - Network Fundamentals]] == References == * [http://certification.comptia.org/docs/default-source/exam-objectives/comptia-network-(n10-006)_examobjectives.pdf CompTIA: Network+ Certification Exam Objectives - Exam N10-006] [[Category:Computer Networks]] 0eb7896y38cf0wa9gr1kqwq91g02vw9 0 162315 2693557 1570874 2024-12-27T01:20:59Z Tule-hog 2984180 nominate speedy 2693557 wikitext text/x-wiki {{delete|Content merged into updated [[Network+]]. See [[User_talk:Dave_Braunschweig#IT_Security_update|permission from original author]] and [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]].}} {{:{{PAGENAME}}/Sidebar}} This section of the computer networks course covers troubleshooting methodology, tools, wireless issues, copper issues, fiber issues, network issues, security issues, and WAN issues. == Preparation == Learners should already be familiar with [[../Architecture|network architecture]] and [[../Operations|network operations]]. == Lessons == * [[/Methodology/]] * [[/Tools/]] * [[/Wireless Issues/]] * [[/Copper Issues/]] * [[/Fiber Issues/]] * [[/Network Issues/]] * [[/Security Issues/]] * [[/WAN Issues/]] == See Also == * [[IC3/Network Fundamentals | IC³ - Network Fundamentals]] == References == * [http://certification.comptia.org/docs/default-source/exam-objectives/comptia-network-(n10-006)_examobjectives.pdf CompTIA: Network+ Certification Exam Objectives - Exam N10-006] [[Category:Computer Networks]] ny7yccpanm1shilcgqrsdng1b6fjcsl 2693560 2693557 2024-12-27T01:21:50Z Tule-hog 2984180 nominate speedy 2693560 wikitext text/x-wiki {{delete|Content merged into updated [[Network+]]. See [[User_talk:Dave_Braunschweig#IT_Security_update|permission from original author]] and [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]. ''Please delete all subpages as well''.}} {{:{{PAGENAME}}/Sidebar}} This section of the computer networks course covers troubleshooting methodology, tools, wireless issues, copper issues, fiber issues, network issues, security issues, and WAN issues. == Preparation == Learners should already be familiar with [[../Architecture|network architecture]] and [[../Operations|network operations]]. == Lessons == * [[/Methodology/]] * [[/Tools/]] * [[/Wireless Issues/]] * [[/Copper Issues/]] * [[/Fiber Issues/]] * [[/Network Issues/]] * [[/Security Issues/]] * [[/WAN Issues/]] == See Also == * [[IC3/Network Fundamentals | IC³ - Network Fundamentals]] == References == * [http://certification.comptia.org/docs/default-source/exam-objectives/comptia-network-(n10-006)_examobjectives.pdf CompTIA: Network+ Certification Exam Objectives - Exam N10-006] [[Category:Computer Networks]] 38zrnm430f8vagswu03lwcprx5ux11g 0 162317 2693555 1570830 2024-12-27T01:20:32Z Tule-hog 2984180 nominate speedy 2693555 wikitext text/x-wiki {{delete|Content merged into updated [[Network+]]. See [[User_talk:Dave_Braunschweig#IT_Security_update|permission from original author]] and [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]].}} {{:{{PAGENAME}}/Sidebar}} This section of the computer networks course covers monitoring, performance, configuration management, segmentation, updates, switching, and wireless. == Preparation == Learners should already be familiar with [[../Architecture|network architecture]]. == Lessons == * [[/Monitoring/]] * [[/Performance/]] * [[/Configuration Management/]] * [[/Segmentation/]] * [[/Updates/]] * [[/Switching/]] * [[/Wireless/]] == See Also == * [[IC3/Network Fundamentals | IC³ - Network Fundamentals]] == References == * [http://certification.comptia.org/docs/default-source/exam-objectives/comptia-network-(n10-006)_examobjectives.pdf CompTIA: Network+ Certification Exam Objectives - Exam N10-006] [[Category:Computer Networks]] t2j96v1f2iv3hlaahj2dinw1gg1fcwu 2693558 2693555 2024-12-27T01:21:20Z Tule-hog 2984180 clarify tag 2693558 wikitext text/x-wiki {{delete|Content merged into updated [[Network+]]. See [[User_talk:Dave_Braunschweig#IT_Security_update|permission from original author]] and [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]. ''Please delete all subpages as well''.}} {{:{{PAGENAME}}/Sidebar}} This section of the computer networks course covers monitoring, performance, configuration management, segmentation, updates, switching, and wireless. == Preparation == Learners should already be familiar with [[../Architecture|network architecture]]. == Lessons == * [[/Monitoring/]] * [[/Performance/]] * [[/Configuration Management/]] * [[/Segmentation/]] * [[/Updates/]] * [[/Switching/]] * [[/Wireless/]] == See Also == * [[IC3/Network Fundamentals | IC³ - Network Fundamentals]] == References == * [http://certification.comptia.org/docs/default-source/exam-objectives/comptia-network-(n10-006)_examobjectives.pdf CompTIA: Network+ Certification Exam Objectives - Exam N10-006] [[Category:Computer Networks]] ek63khdm6eunzdondrswc3zohaf5yrc 0 162319 2693556 1570848 2024-12-27T01:20:44Z Tule-hog 2984180 nominate speedy 2693556 wikitext text/x-wiki {{delete|Content merged into updated [[Network+]]. See [[User_talk:Dave_Braunschweig#IT_Security_update|permission from original author]] and [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]].}} {{:{{PAGENAME}}/Sidebar}} This section of the computer networks course covers risk, threats, hardening, physical security, firewalls, access control, and forensics. == Preparation == Learners should already be familiar with [[../Architecture|network architecture]] and [[../Operations|network operations]]. == Lessons == * [[/Risk/]] * [[/Threats/]] * [[/Hardening/]] * [[/Physical Security/]] * [[/Firewalls/]] * [[/Access Control/]] * [[/Forensics/]] == See Also == * [[IC3/Network Fundamentals | IC³ - Network Fundamentals]] == References == * [http://certification.comptia.org/docs/default-source/exam-objectives/comptia-network-(n10-006)_examobjectives.pdf CompTIA: Network+ Certification Exam Objectives - Exam N10-006] [[Category:Computer Networks]] owfg434nc5zrxz6ukp705vguwp8vh2j 2693559 2693556 2024-12-27T01:21:36Z Tule-hog 2984180 clarify tag 2693559 wikitext text/x-wiki {{delete|Content merged into updated [[Network+]]. See [[User_talk:Dave_Braunschweig#IT_Security_update|permission from original author]] and [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]. ''Please delete all subpages as well''.}} {{:{{PAGENAME}}/Sidebar}} This section of the computer networks course covers risk, threats, hardening, physical security, firewalls, access control, and forensics. == Preparation == Learners should already be familiar with [[../Architecture|network architecture]] and [[../Operations|network operations]]. == Lessons == * [[/Risk/]] * [[/Threats/]] * [[/Hardening/]] * [[/Physical Security/]] * [[/Firewalls/]] * [[/Access Control/]] * [[/Forensics/]] == See Also == * [[IC3/Network Fundamentals | IC³ - Network Fundamentals]] == References == * [http://certification.comptia.org/docs/default-source/exam-objectives/comptia-network-(n10-006)_examobjectives.pdf CompTIA: Network+ Certification Exam Objectives - Exam N10-006] [[Category:Computer Networks]] giqqh0btvvp8swwjb8ok9tch7je4012 14 162552 2693466 1280442 2024-12-27T00:03:22Z Tule-hog 2984180 mv introduction to resource, nominate prod of single resource cat 2693466 wikitext text/x-wiki {{prod|single resource category}} [[Category:Networks]] gal9oe0npp2ce7kir6ww62pen2ca2wp 2693474 2693466 2024-12-27T00:09:06Z Tule-hog 2984180 Bot: Replacing category Networks with [[:Category:Networking|Networking]] 2693474 wikitext text/x-wiki {{prod|single resource category}} [[Category:Networking]] tep97s1plbr3qpoe7em4sj8rb5f6oyg 0 162573 2693543 1571110 2024-12-27T00:58:57Z Tule-hog 2984180 fix target 2693543 wikitext text/x-wiki #REDIRECT [[Network+/Old guides/IP Model]] efflpmprqg0t5b57tkf20bn5le1w7jy 2693549 2693543 2024-12-27T01:03:46Z Tule-hog 2984180 add cat 2693549 wikitext text/x-wiki #REDIRECT [[Network+/Old guides/IP Model]] [[Category:TCP/IP Fundamentals| ]] koiddoyxixyez0gjrcfte296eaj4f4r 14 191708 2693533 1322045 2024-12-27T00:51:12Z Tule-hog 2984180 Bot: Replacing category Computer networks with [[:Category:Networking|Networking]] 2693533 wikitext text/x-wiki [[Category:Networking]] iu53wq4uh22j60xsw43mzeyp248lixq 0 204776 2693562 2652196 2024-12-27T01:26:51Z Tule-hog 2984180 clarify tag 2693562 wikitext text/x-wiki {{delete|reason=updating course with [[User_talk:Dave_Braunschweig#IT_Security_update|author's permission]] (see also [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]). ''Please delete subpages as well'' - they have '''not''' been marked with {{tlx|delete}}.}} {{:{{PAGENAME}}/Sidebar}} This section of the IT security course covers devices, network administration, network design, protocols, and wireless security. == Preparation == Learners should already be familiar with [[IC3|introductory computer concepts]], [[Computer Support|computer support]] concepts, and [[Computer Networks|computer networking]] concepts. == Lessons == * [[/Devices/]] * [[/Administration/]] * [[/Design/]] * [[/Protocols/]] * [[/Wireless/]] == See Also == * [[Computer Networks/Security]] == References == * [https://certification.comptia.org/docs/default-source/exam-objectives/comptia-security-sy0-401.pdf CompTIA: Security+ Certification Exam Objectives - Exam SY0-401] {{CourseCat}} orbrwi8h99ckjk7s4yeu6cixykelyxx 0 204784 2693563 2652197 2024-12-27T01:27:03Z Tule-hog 2984180 clarify tag 2693563 wikitext text/x-wiki {{delete|reason=updating course with [[User_talk:Dave_Braunschweig#IT_Security_update|author's permission]] (see also [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]). ''Please delete subpages as well'' - they have '''not''' been marked with {{tlx|delete}}.}} This section of the IT security course covers operational risk, integration, mitigation, forensics, incident response, training, physical security, best practices, and control. == Preparation == Learners should already be familiar with [[IC3|introductory computer concepts]], [[Computer Support|computer support]] concepts, and [[Computer Networks|computer networking]] concepts. == Lessons == * [[/Risk/]] * [[/Systems Integration/]] * [[/Risk Mitigation/]] * [[/Forensics/]] * [[/Incident Response/]] * [[/Training/]] * [[/Physical Security/]] * [[/Best Practices/]] * [[/Controls/]] == See Also == == References == * [https://certification.comptia.org/docs/default-source/exam-objectives/comptia-security-sy0-401.pdf CompTIA: Security+ Certification Exam Objectives - Exam SY0-401] {{CourseCat}} 17mqucewclkr5en55rb2rbd4wxjay88 2693564 2693563 2024-12-27T01:27:32Z Tule-hog 2984180 restore sidebar 2693564 wikitext text/x-wiki {{delete|reason=updating course with [[User_talk:Dave_Braunschweig#IT_Security_update|author's permission]] (see also [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]). ''Please delete subpages as well'' - they have '''not''' been marked with {{tlx|delete}}.}} {{:{{PAGENAME}}/Sidebar}} This section of the IT security course covers operational risk, integration, mitigation, forensics, incident response, training, physical security, best practices, and control. == Preparation == Learners should already be familiar with [[IC3|introductory computer concepts]], [[Computer Support|computer support]] concepts, and [[Computer Networks|computer networking]] concepts. == Lessons == * [[/Risk/]] * [[/Systems Integration/]] * [[/Risk Mitigation/]] * [[/Forensics/]] * [[/Incident Response/]] * [[/Training/]] * [[/Physical Security/]] * [[/Best Practices/]] * [[/Controls/]] == See Also == == References == * [https://certification.comptia.org/docs/default-source/exam-objectives/comptia-security-sy0-401.pdf CompTIA: Security+ Certification Exam Objectives - Exam SY0-401] {{CourseCat}} ixezji21xl21axd9tkyw5izdr327gg7 0 204785 2693565 2652198 2024-12-27T01:27:50Z Tule-hog 2984180 clarify tag 2693565 wikitext text/x-wiki {{delete|reason=updating course with [[User_talk:Dave_Braunschweig#IT_Security_update|author's permission]] (see also [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]). ''Please delete subpages as well'' - they have '''not''' been marked with {{tlx|delete}}.}} {{:{{PAGENAME}}/Sidebar}} This section of the IT security course covers threats, including malware, attacks, social engineering, wireless, application, mitigation, tools, and testing. == Preparation == Learners should already be familiar with [[IC3|introductory computer concepts]], [[Computer Support|computer support]] concepts, and [[Computer Networks|computer networking]] concepts. == Lessons == * [[/Malware/]] * [[/Attacks/]] * [[/Social Engineering/]] * [[/Wireless Attacks/]] * [[/Application Attacks/]] * [[/Threat Mitigation/]] * [[/Tools/]] * [[/Testing/]] == See Also == == References == * [https://certification.comptia.org/docs/default-source/exam-objectives/comptia-security-sy0-401.pdf CompTIA: Security+ Certification Exam Objectives - Exam SY0-401] {{CourseCat}} ivlhe3npu5ezeqwhid0qelm4z70tzcw 0 204786 2693566 2652199 2024-12-27T01:28:02Z Tule-hog 2984180 clarify tag 2693566 wikitext text/x-wiki {{delete|reason=updating course with [[User_talk:Dave_Braunschweig#IT_Security_update|author's permission]] (see also [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]). ''Please delete subpages as well'' - they have '''not''' been marked with {{tlx|delete}}.}} {{:{{PAGENAME}}/Sidebar}} This section of the IT security course covers host security, including application, mobile, host, data, and environmental mitigation. == Preparation == Learners should already be familiar with [[IC3|introductory computer concepts]], [[Computer Support|computer support]] concepts, and [[Computer Networks|computer networking]] concepts. == Lessons == * [[/Application/]] * [[/Mobile/]] * [[/Host/]] * [[/Data/]] * [[/Environmental Mitigation/]] == See Also == == References == * [https://certification.comptia.org/docs/default-source/exam-objectives/comptia-security-sy0-401.pdf CompTIA: Security+ Certification Exam Objectives - Exam SY0-401] {{CourseCat}} d921a2m7qxnoasz5qegjkztdlgr59ac 0 204787 2693567 2652200 2024-12-27T01:28:20Z Tule-hog 2984180 clarify tag 2693567 wikitext text/x-wiki {{delete|reason=updating course with [[User_talk:Dave_Braunschweig#IT_Security_update|author's permission]] (see also [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]). ''Please delete subpages as well'' - they have '''not''' been marked with {{tlx|delete}}.}} {{:{{PAGENAME}}/Sidebar}} This section of the IT security course covers access control, including Authentication, Authorization, and Accounting (AAA). == Preparation == Learners should already be familiar with [[IC3|introductory computer concepts]], [[Computer Support|computer support]] concepts, and [[Computer Networks|computer networking]] concepts. == Lessons == * [[/Authentication Services/]] * [[/Authentication and Authorization/]] * [[/Account Management/]] == See Also == * [[IAM]] Identity and Access Management == References == * [https://certification.comptia.org/docs/default-source/exam-objectives/comptia-security-sy0-401.pdf CompTIA: Security+ Certification Exam Objectives - Exam SY0-401] {{CourseCat}} 10i6jt0gyyhyvklrls40ffnrp6p0u1r 0 204788 2693568 2652201 2024-12-27T01:28:30Z Tule-hog 2984180 clarify tag 2693568 wikitext text/x-wiki {{delete|reason=updating course with [[User_talk:Dave_Braunschweig#IT_Security_update|author's permission]] (see also [[User_talk:Tule-hog#IT_Security/Network|clarification of improvements]]). ''Please delete subpages as well'' - they have '''not''' been marked with {{tlx|delete}}.}} {{:{{PAGENAME}}/Sidebar}} This section of the IT security course covers cryptography, including concepts, methods, and PKI. == Preparation == Learners should already be familiar with [[IC3|introductory computer concepts]], [[Computer Support|computer support]] concepts, and [[Computer Networks|computer networking]] concepts. == Lessons == * [[/Concepts/]] * [[/Methods/]] * [[/PKI/]] == See Also == == References == * [https://certification.comptia.org/docs/default-source/exam-objectives/comptia-security-sy0-401.pdf CompTIA: Security+ Certification Exam Objectives - Exam SY0-401] {{CourseCat}} pex9ghjh9ufs5j2h6kdzhmluqykiz6u 0 205671 2693507 1571108 2024-12-27T00:39:24Z Tule-hog 2984180 fix target 2693507 wikitext text/x-wiki #REDIRECT [[Network+/Old guides/OSI Model/OSI Components]] mbrosb3e0sk1s9o6vhlo49a6buqss21 0 206822 2693491 2118936 2024-12-27T00:26:29Z Tule-hog 2984180 alter lk 2693491 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} This lesson introduces computer network media. == Objectives and Skills == Objectives and skills for the media portion of Network+ certification include:<ref>[http://certification.comptia.org/docs/default-source/exam-objectives/comptia-network-(n10-006)_examobjectives.pdf CompTIA: Network+ Certification Exam Objectives - Exam N10-006]</ref> {{colbegin}} * Install and properly terminate various cable types and connectors using appropriate tools ** Copper cables *** Shielded vs unshielded *** CAT3, CAT5, CAT5e, CAT6, CAT6a *** PVC vs plenum *** RG-59 *** RG-6 *** Straight-through vs crossover vs rollover ** Copper connectors *** RJ-11 *** RJ-45 *** RJ-48C *** DB-9/RS-232 *** DB-25 *** UTP coupler *** BNC coupler *** BNC *** F-connector *** 110 block *** 66 block ** Fiber cables *** Single mode *** Multimode *** APC vs UPC ** Fiber connectors *** ST *** SC *** LC *** MTRJ *** FC *** Fiber coupler ** Media converters *** Single mode fiber to Ethernet *** Multimode fiber to Ethernet *** Fiber to coaxial *** Single mode to multimode fiber ** Tools *** Cable crimpers *** Punch down tool *** Wire strippers *** Snips *** OTDR *** Cable certifier * Given a scenario, deploy the appropriate wired connectivity standard ** Wiring standards *** EIA/TIA 568A/568B * Given a scenario, troubleshoot and resolve common copper cable issues ** Shorts ** Opens ** Incorrect termination (mismatched standards) *** Straight-through *** Crossover ** Cross-talk *** Near end *** Far end ** EMI/RFI ** Distance limitations ** Attenuation/Db loss ** Bad connector ** Bad wiring ** Split pairs ** Tx/Rx reverse ** Cable placement ** Bad SFP/GBIC - cable or transceiver * Given a scenario, troubleshoot and resolve common fiber cable issues ** Attenuation/Db loss ** SFP/GBIC - cable mismatch ** Bad SFP/GBIC - cable or transceiver ** Wavelength mismatch ** Fiber type mismatch ** Dirty connectors ** Connector mismatch ** Bend radius limitations ** Distance limitations * Given a scenario, install and configure equipment in the appropriate location using best practices ** Intermediate distribution frame ** Main distribution frame ** Cable management *** Patch panels ** Power management *** Power converters *** Circuits *** UPS *** Inverters *** Power redundancy ** Device placement ** Air flow ** Cable trays ** Rack systems *** Server rail racks *** Two-post racks *** Four-post racks *** Free-standing racks ** Labeling *** Port labeling *** System labeling *** Circuit labeling *** Naming conventions *** Patch panel labeling ** Rack monitoring ** Rack security {{colend}} == Readings == # [[Network+/Old guides/Network media|Media Introduction]] # [[Wikipedia: Networking cables]] # [[Wikipedia: Twisted pair]] # [[Wikipedia: Coaxial cable]] # [[Wikipedia: Optical fiber cable]] # [[Wikipedia: Plenum cable]] # [[Wikipedia: Registered jack]] # [[Wikipedia: D-subminiature]] # [[Wikipedia: BNC connector]] # [[Wikipedia: F connector]] # [[Wikipedia: Patch panel]] # [[Wikipedia: 110 block]] # [[Wikipedia: 66 block]] # [[Wikipedia: Optical fiber connector]] # [[Wikipedia: Fiber media converter]] # [[Wikipedia: Broadband over power lines]] # [[Wikipedia: Crimp connection]] # [[Wikipedia: Punch down tool]] # [[Wikipedia: Wire stripper]] # [[Wikipedia: Optical time-domain reflectometer]] # [[Wikipedia: Cable tester]] # [[Wikipedia: TIA/EIA-568]] == Multimedia == # [https://www.youtube.com/watch?v=-i7aFSO_iic YouTube: Copper Cabling - CompTIA Network+ N10-006 - 1.5] # [https://www.youtube.com/watch?v=y5364wkR1RQ YouTube: Copper Connectors- CompTIA Network+ N10-006 - 1.5] # [https://www.youtube.com/watch?v=jv60qUVNz4c YouTube: Straight-Through, Crossover, and Rollover Cables - CompTIA Network+ N10-006 - 1.5] # [https://www.youtube.com/watch?v=5ft-iZ8yUM8 YouTube: Fiber Cables - CompTIA Network+ N10-006 - 1.5] # [https://www.youtube.com/watch?v=98YziMWPAak YouTube: Fiber Connectors - CompTIA Network+ N10-006 - 1.5] # [https://www.youtube.com/watch?v=DcsjJOKXt-Y YouTube: Media Converters - CompTIA Network+ N10-006 - 1.5] # [https://www.youtube.com/watch?v=VtR6XUg4cL0 YouTube: Network Cabling Tools - CompTIA Network+ N10-006 - 1.5] # [https://www.youtube.com/watch?v=lullzS740wI YouTube: How To Make RJ45 Network Patch Cables - Cat 5E and Cat 6] # [https://www.youtube.com/watch?v=cCFQJV0Kr7U YouTube: Tools Comparison: Terminating Ethernet Cable with Different Punchdown Tools] # [https://www.youtube.com/watch?v=oZjmVnMaFHk YouTube: Troubleshooting Copper Cables - CompTIA Network+ N10-006 - 4.4] # [https://www.youtube.com/watch?v=NhUKCbkJ0_c YouTube: Troubleshooting Signal Loss - CompTIA Network+ N10-006 - 4.4] # [https://www.youtube.com/watch?v=lRv6hP5x6XU YouTube: Troubleshooting Network Cabling - CompTIA Network+ N10-006 - 4.4] # [https://www.youtube.com/watch?v=5tTaRJYZ77c YouTube: Troubleshooting Fiber Issues - CompTIA Network+ N10-006 - 4.5] # [https://www.youtube.com/watch?v=f9bT_hmWfHA YouTube: MDF and IDF - CompTIA Network+ N10-006 - 5.7] # [https://www.youtube.com/watch?v=jHtJKRFVin8 YouTube: Cable and Power Management - CompTIA Network+ N10-006 - 5.7] # [https://www.youtube.com/watch?v=0XhkTTDzKro YouTube: Rack Systems - CompTIA Network+ N10-006 - 5.7] # [https://www.youtube.com/watch?v=uXtZefFa36k YouTube: Labeling and Documentation - CompTIA Network+ N10-006 - 5.7] == Activities == # Review [[Wikipedia: Category 5 cable]] and [[Wikipedia: Category 6 cable]]. Examine a network cable you have available or attached to your computer or switch/router. Check for labeling on the cable to identify whether it is a Category 5 cable or Category 6 cable. Then check the wiring pattern on each end. Is it wired with the 568A standard or the 568B standard? Are both ends wired to the same standard, or is it a cross-over cable? # Review [https://www.warehousecables.com/how-to-make-a-cat5-patch-cable Warehouse Cables: How to Make a Cat5 Patch Cable]. Using available Category 5 or Category 6 cable, some RJ-45 connectors, and a cable crimper, create your own patch cable. Test the cable with a cable tester if you have one available, or connect it to your computer and switch/router and test the connection. # Review [[Wikipedia: Punch down tool]]. Using available Category 5 or Category 6 cable and an RJ-45 jack or 110 block, practice terminating network cables. # Compare [https://www.youtube.com/watch?v=_fsyYYWDxk4 YouTube: Scary Network Wiring and Cabling Fails in the IT Closet] and [https://www.youtube.com/watch?v=FY1XB0rrYes YouTube: Cat 6 amazing dressing and termination]. Then request a tour of your school or organization's wiring closet, computer room, or data center. Which video does your organization's wiring most resemble? == Lesson Summary == == Key Terms == == See Also == == References == {{Reflist}} {{CourseCat}} oekmhcca9gqkmq2tr0tx2nx9afp5yfn 0 211721 2693550 1570797 2024-12-27T01:07:12Z Tule-hog 2984180 nominate speedy unused redirect 2693550 wikitext text/x-wiki {{speedy|C10}} #REDIRECT [[Network+/Architecture]] b46yxn0b7w0r5fr0za7d2s1h1xy3hm7 0 231043 2693386 1871532 2024-12-26T21:26:31Z OhanaUnited 18921 note about backlog 2693386 wikitext text/x-wiki {{WikiJHum_top_menu}} {{red|'''Please note that there is a substantial backlog of submissions to WikiJournal of Humanities.'''}} {{WikiJHum_right_menu}} __NOTOC____NOEDITSECTION__ {{WikiJournal_Preprints}} <!-- Transclusion of WikiJournal Preprints page to unify submission system --> [[Category:WikiJournal of Humanities]] 8uvzoms7nnqsuomvgewd1h2rc4kniey 10 232671 2693447 2456964 2024-12-26T23:36:04Z Tule-hog 2984180 nominate prod 2693447 wikitext text/x-wiki {{prod|Unused, duplicate of [[Template:BookCat]]?}} <includeonly>Category:Book:{{{book|{{safesubst:<noinclude/>BOOKNAME|{{{1|{{safesubst:<noinclude/>FULLPAGENAME}}}}}}}}}}</includeonly><noinclude> {{Documentation}} </noinclude> 0i7rne88eagkq2o8hi8jidjpealhpmz 0 233564 2693505 1825099 2024-12-27T00:39:01Z Tule-hog 2984180 fix target 2693505 wikitext text/x-wiki #REDIRECT [[Network+/Old guides/OSI Model/OSI Components]] mbrosb3e0sk1s9o6vhlo49a6buqss21 0 252361 2693436 2459858 2024-12-26T23:25:31Z 97.104.167.161 /* Economies of Scope */ 2693436 wikitext text/x-wiki == Firms == === What is a firm? === A firm is a business organisation (which can take form as a corporation, partnership, limited liability company (LLC) among others) which transforms inputs into outputs for profit. This can be in the form of goods (such as laptops or french fries), services (such as gardening or cleaning), or both (think restaurants where you pay for the food you order, but also for the experience). Firms typically embody some kind of institutional structure with the management of the firm having both a set of objectives and a strategy with the goal of maximising profits. <ref>{{Cite web|url=http://www.businessdictionary.com/definition/business-firm.html|title=What is business firm? definition and meaning|website=BusinessDictionary.com|language=en|access-date=2019-08-16}}</ref>. Furthermore, adding value to the final product, such as when a person buys different parts of a bike and builds the bike, then sells it on the market. The example before is slightly different when the person buys a product and immediately sells it off in the market, which could be a grey area. As previously mentioned, with an institutional structure, firms will start to employ and form an organisational structure; a simple structure could be a top manager being in charge of a senior associate overlooking the juniors of the firm. Additionally, in the eyes of the government, to constitute as a firm, they can pay taxes, which is an income, by registering their firm with authorities. Most firms, in the way we talk about in economics, are assumed that what a firm does maximises its profits. In general, that's true - it is in most legislation, and it is also required by law that a public traded company, which has shareholders, has an obligation to their shareholders who expect this firm would act in the best interest of the shareholders. Otherwise, the firm is not fulfilling that obligation. However, there are also some other types of firms: * ''Social enterprises -'' Social enterprises are firms that exist for the purpose of maximising well-being and social impact, rather than profit. These firms may still be involved in buying and selling, as well as transforming inputs to outputs. In the business process, social enterprises try to achieve different purposes. For example, social enterprises may try to employ people from disadvantaged backgrounds who have difficulties in finding jobs otherwise, so they are paying back society through employment, or social enterprises may use their profits to help the society. In a sense, social enterprises are a bit like charities which don't get donations but instead use a firm as a vehicle to generate profits which are then donated to people in need. Given that a social enterprise is set up to maximise it's social impact, which it typically uses it's profits to drive, social enterprises are often still profit maximising firms. * ''Government-owned companies -'' In general, they serve the public and also have certain obligations like the Universal Service Obligation<ref>{{Cite web|url=https://www.communications.gov.au/what-we-do/phone/phone-services/universal-service-obligation|title=Universal Service Obligation|last=Arts|first=Department of Communications and the|date=2017-12-14|website=www.communications.gov.au|language=en|access-date=2019-09-09}}</ref>. Even if it is not profitable to serve certain persons because, for instance, they live very far and the outback, it might be required for those government-owned companies to serve those people, which they wouldn't necessarily do so if they were private companies due to the lack of profitability. But in general, we deal with firms that maximise, or at least try to maximise their profits. As profits should be maximising at a long-term vision which can't be seen from now, expectations are needed to estimate the prospects of firms. Market prices of publicly traded firms are expected to reflect those long-term visions, and people can deduce that if the share price of a company goes up, the firm is doing well in terms of its long-term profitability. === Why do firms exist? === Firms exist to simplify and reduce the transactional costs of coordinating economic activities (Ronald Coase "The Nature of the Firm" 1937). By utilising the principles of economies of scale and scope, firms are able to reduce the transactional costs of operating within the market. Larger firms reduce costs by more efficiently satisfying 3 major factors required in economic activities: #''Search & Information'' - Firms can minimise search costs regarding things like marketing and advertising (e.g. it's easier for a university than individual lecturers to find at their best price the lecture halls, students, etc, as the university is doing these things for all lectures and all degrees throughout the university) . #'' Bargaining & Decision making'' - Firms can use enterprise bargaining to set a price for everyone compared to freelancers negotiating at different prices with different people. #'' Policing & enforcement'' - Firms have strong policies in place to maintain quality. An example of this is a model working freelance who has to do all of their own advertising, marketing and management of finances. If the model worked under a booking agent, their jobs would be set up, transport organised and payment collected by the firm. Therefore, it would take the pressure and stress of the model to perform their job better. To further explain, firms are needed to set a price and create a market. By creating a market a firm is able to consolidate demands of a certain good and produce it altogether to achieve economies of scale. Firms might even be able to use their resources to aid in the production of other in-demand products, which then becomes a form of economies of scope. Industries formed by different firms competing in the same market may face disruption due to the rise of a new technology which helps eliminate transaction costs and consequently reduces the need for firms. Examples: * Person-to-person car sharing: Where people's idle cars are temporarily made available to people who need transport. This results in a significantly lower demand for car rental agencies as any person can make their vehicles available through the application and can avoid the logistics required by a large firm (people might get rid of the firms that are normally organising these activities, and they might have individual trades with each other without the firms' getting percentage cuts). * Airbnb: Airbnb is disrupting the hotel industry through the use of new technology such as applications to connect homeowners with travellers. *Video streaming: online video streaming like YouTube and Netflix are disrupting Television company, which changes the way of people watching programs. *Coursera: Coursera is disrupting universities as it provides massive free online learning courses which allow learners to be more flexible in their learning. However, these new technologies can help further lower transaction cost which is a benefit for consumers. Over time, some conventional firms in the industry might be eliminated, while some might learn from these new technologies and further improve their industry standards. == Horizontal Boundaries of Firms == Horizontal boundaries refer to the quantity (Economies of Scale) and variety (Economies of Scope) of products that a firm produces. Economies of Scale and Scope exist whenever large-scale operations provide a cost advantage over smaller ones, such that the average variable cost per unit reduces as the quantity of output of a single product, or a variety of products produced in a single plant increases. Production processes that are Capital Intensive generally are more likely to display Economies of Scale or Scope than Labour or Resource Intensive processes. Capital Intensive processes tend to have a higher fixed to variable cost ratio which benefits more from improving production techniques. === '''Economies of Scale''' === Economies of scale are the cost advantages that firms obtain based on their scale of operation, with the cost per unit of output decreasing as the scale of production increases. However, some of the companies will not take advantage of economies of scale, preferring differentiation over cost leadership. When a market is producing at a level of economies of scale, allocative efficiency within the market is achieved. This means the market is producing at perfect competition, reducing costs and profit. Economies of Scale exists if there are: # '''Indivisibilities in production:''' Indivisibilities exist when the minimum level of production is significantly larger for new entrants to be economically viable. This occurs when there are high setup costs, long-run fixed costs, and volumetric returns to scale or a combination of all three. Larger firms can take advantage of indivisibilities by spreading costs over a greater volume of production as well as having better access to capital markets (assuming imperfect access to firms). Indivisibilities exist when it is possible to do things on a large scale that cannot be done on a small scale. Some inputs cannot be scaled down below a certain minimum size, even when the level of output is minimal. In general term, there is minimum expenditure a firm must incur in order to commence production (e.g. with a small backyard farm, a tractor is still needed to reduce labour intensity because it is not possible to purchase 0.01 of a tractor) # '''Large volumes of input purchases:''' Firms that purchase relatively greater quantities of inputs may obtain discounts from suppliers. Reasons for this include lower negotiating costs with a single supplier as opposed to multiple suppliers, suppliers benefiting from an association with reputable firms purchasing the inputs, and the security of confidentiality dealing with a single supplier can all induce discounting. Additionally, suppliers that rely on large purchases from a few firms are more inclined to discount, as they are risk-averse to losing the firms they supply. # '''Marketing costs:''' Advertising has a certain fixed cost to all firms; therefore larger firms are able to spread this cost over a relatively larger number of potential customers and can better adapt production to changes in demand from advertising campaigns in comparison to smaller firms. These fixed costs are similar for national firms as they are for regional firms. Firms with a greater scope of product offerings benefit from umbrella branding, which influences customer perception for all of the brand’s products despite a campaign focussing on a single offering. Other sources of Economies of Scale include labour specialisation, more efficient inventory management due to predictable customer demand and industries that encounter the cube square rule – where processes are volume related but costs are area related. A reduction in per-unit costs occurs in the short run when fixed costs are spread over increased production through better utilisation of a production plant’s given capacity. In the long run this is represented by improvements in technology or increases in a plant’s total production capacity, altering the dynamic of a firm’s fixed to variable cost ratio. ==== Two types of Economies of scale ==== # Internal- Internal economies are factors and capabilities that are unique to and can be controlled by an organization that at minimal costs, can produce in large quantities. The big operational and financial size of an organization usually means they can take advantage of internal economies. # External- External economies result from advantageous conditions coming from outside the organization or, within an entire industry or economy. External economies mean that as an industry or sector grows, the average cost of doing business falls. ==== The 4 major sources of Economies of Scale and Scope ==== 1. '''Indivisibilities and the Spreading of Fixed Cost''' One of the common sources of Economies of Scale is the spreading of fixed costs over an even larger volume of output produced. Indivisibilities refer to the minimum level at which any element of production requires to operate. Some inputs of production cannot be scaled down to a minimum size regardless of how small the output level be. Take, for example, if you have a very tiny garden and require a tractor, it is impossible to use 0.5 of a tractor. Therefore, the first unit produced requires a significantly higher level of investment than the subsequent units, with the increase in subsequent units produced, it allows for costs to be spread out, allowing for Economies of Scale. If the indivisible input is not overly specialised, the firm can diversify its line of products at a lower cost as opposed to the total cost of individual specialised enterprises. Indivisibilities also promote economies of scope. For example when airlines add new routes, they utilise conveyancing. 2. '''Specialisation''' Specialisation occurs when workers assigned to specific production tasks, increase in productivity and efficiency over time, allowing them to benefit from the lower average costs per increased in output. In order to conduct specialization, firms must be ready to make substantial investments, however, reluctance for firms will occur unless the present or forecasted demand justifies the volume to utilize specialization. From Adam Smith’ theorem, it has stated that the division of labour is restricted to the span of the market (1) As the markets increase in size, economies of scale will enable the utilization of specialisation in productions, (2)The larger markets with volume advantages will support an arrangement of specialised activities. 3. '''Inventories''' By carrying inventories, firms who conduct high volumes of business are able to maintain a lower ratio of inventory to sales. By buying in bulk, moving and storing big volumes of inventory reduces the overall cost per unit, hence Economies of Scale. Additionally, consolidation of inventories reduces costs associated with stock-outs and lost sales. There are various incentives for firms to possess inventories (1) Avoid stock-outs and lost sales (Safety stock is essential due to the uncertainty in the forecasts of sales. The added accuracy to the forecast, the fewer safety stock is required) (2)Avoid adversely influencing customer commitment, (3)Assuring no setbacks occur in the production process. However, by taking onto excess inventories, there will be consequences attached to such action as (1) Opportunity cost of cashflow restricted in inventory, (2) Rent, depreciation, insurance needed for inventory storage (3) Cost of deterioration and obsolescence of the inventories 4. '''The Cube Square Rule''' Many processes are dealt with in volume but their costs are associated with an area (i.e. Storage). As the volume of a vessel increases by a given proportion, the surface area then increases by less than this proportion. In various production processes, production capacity is to be found proportional to the volume of the production vessel while the total cost of producing at capacity is proportional to the available surface area of the vessel. This concludes that as capacity increases, the average cost of producing at capacity will decrease because the ratio of surface area to volume decreases. For example, shipping of chairs in a shipping container. By stacking the chairs on top of each other, the capacity of chairs increases, hence decreasing the cost of shipping per surface area, achieving Economies of Scale. <ref> Besanko, D., Dranove, D. and Schaefer, S. (2012). Economics of Strategy. 6th ed. Wiley Global Education, pp.61-97. </ref> ==== Short-Run Economies of Scale ==== The reductions in unit costs are related to spreading fixed costs for a firm of a given size. Short-run economies of scale occur because of firms utilising a plant of a given capacity. For short-run economies of scale, it is assumed that there are fixed costs and the short term average cost curve has a U-Shape <ref>[https://courses.lumenlearning.com/wmopen-microeconomics/chapter/economies-of-scale/]Economies of scale, lumen learning Microeconomics.</ref>. The average cost in the short run is calculated by taking the total cost and dividing by output at each different level of output. Average cost shows that firms can earn profits given the market price. ==== Long-Run Economies of Scale ==== The reductions in unit costs are caused by a firm switching from a low fixed/high variable cost plant to a high fixed/low variable cost plant. This happens when new technology is adopted by firms or when plant sizes are increased. For the long-run economies of scale, the average cost curve us more downward-sloping and it assumes that all factors/variables of production could change <ref>[https://courses.lumenlearning.com/wmopen-microeconomics/chapter/economies-of-scale/)], Economies of scale, Lumen microeconomics </ref>. === Diseconomies of Scale === The opposite are diseconomies of scale, where production costs increase as the firm produces more units. The law of diminishing returns impacts large firms because after a certain point, the amount of benefits gained is smaller than the additional investment eventually resulting in negative returns There are several major sources of diseconomies of scale: '''Reduction of incentives and growth of bureaucracy''' As a firm becomes bigger, it inevitably becomes more compartmentalized. A potential source of diseconomies of scale comes from managers who are not fully incentivised by making the firm as profitable as possible, and may practice ‘Office Politics’, making decisions that do not benefit the company but are advantageous to their career. '''Difficulty to monitor performance''' The more workers a firm employs, the less able it is to ensure all workers are performing optimally. '''Non pecuniary benefits of working at a smaller firm''' There are several non financial reasons why working at a smaller firm might be preferable. They tend to be more personal, flexible, and upwardly mobile. Therefore, larger firms may need to pay workers a premium to work for them and not smaller competitors '''Lack of specialized resources''' Diseconomies of scale can occur when a firm is reliant on very specialized resources, human or otherwise. For example, a company that needs very skilled engineers might experience diseconomies of scale when increasing production, because they may not be able to find enough engineers with the required skillset to keep production at the previous rate. === Sources of scale economies === '''Procurement''' * When making larger purchases, a firm will be able to exploit economies of scale and scope. As large firms buy in bulk, this reduces transaction costs from the seller, in turn giving discounts to the large firms. These provide a large firm with a cost advantage over smaller size rivals. Furthermore, the large firm usually carries a reputation which could benefit the other party going into business with the large firm making use of the large firm's reputation. Additionally, the indivisibility of equipment is another key element which may achieve the economies of scale for firms. It is compulsory to have a range of equipment to commence operation for some industries (eg. manufacturing industry, mining industry). This is the fixed costs that incur the minimum inputs meant to be spread by purchasing all of the equipment and raw materials to start the production process which will achieve the economies of scale as well. '''Cube-Square Rule''' * The mathematical concept which can be addressed to explain economies of scale. The Cube-Square Rule expresses the relationship between volume and the surface area. It implies that an increase in volume will incur an increase in surface area proportionally. This is another source of economies of scale. For many manufacturing processes, the capability of the machine to produce is related to the volume of the production vessel, and the total cost of production is closely associated with the surface area of the vessel. It is likely to have low average cost per unit by increasing the capacity of production of the plant and decrease the ratio between the surface area and the volume of the production vessel. '''Marketing''' * Advertising usually requires a fixed cost to be incurred by the firm. If two firms that are drastically different in size make identical advertising campaigns, they are most likely charged the same price. Economies of scale arise when the cost of the marketing campaign is spread out among their total potential consumers. A larger firm with more potential consumers will have a wider reach as compared to a smaller firm. Cost per potential consumers is lower for the large firm showing economies of scale as they are able to spread their fixed cost over a greater unit of output. This is in part due to indivisibilities, a smaller firm cannot purchase half an advertisement or hire one quarter of a film crew. If due to budget restrictions a smaller firm chooses to create shorter commercials, aired at off-peak hours, or shown on less viewed networks, they will not have the same reach as a larger firm. The procurement advantages of economies of scale also impacts larger firms' marketing advantage, as they can acquire more material at a lower cost. An example of this is that the cost per brochure is lower when ordering 500,000 than it is for 1000. Another approach to be discussed will be the "Umbrella Branding" effect. This approach is particularly efficient while consumers adopt the message projected in a commercial regarding one product, and to create crossing thoughts on another product under the common brand name. This effect will then cause a reduction in advertising costs per effective image as mentioned above. The Umbrella Branding approach works for companies such as Apple inc. and P&G as they have a positive brand equity and products manufactured under their company are similar to one another. '''Research and Development (R&D)''' * R&D is a significant source of economies of scope. Approaches and ideas generated from one project may be able to accommodate the development of another project which is also known as a positive spillover. In economic terms, cost advantage will be procured from the reductions in unit cost from the spreading out of R&D expenses between multiple projects. For instance, R&D departments will demand a minimum number of researchers where their labour is indivisible. Consequently, with the increasing number of output from the research department, the R&D costs per unit will fall. Also, with the reality that there is a substantial indivisible investment subjected to R&D, it suggests that average fixed costs will diminish in a faster manner as the output volume increases. While firms of scale & scope have a cost advantage when participating in R&D, larger firms also typically have more income that they can allocate to research & development, and can assign a higher proportion of their overall income. Additionally, if a firm requires financing to participate in R&D projects, firms of scale and scope have an easier time obtaining credit. '''Strategic Fit''' * This source will be associated with the concept of complementarities. Strategic fit is defined as the degree to which an organization is matching its resources and capabilities with the opportunities in the external environment (Grant 2014). Therefore, it is a complementarity action that allows economies of scope. It is the extent to which the actions from the various functional parts of a business operating synchronically in a manner that complements one another to attain a competitive advantage. For instance, AirAsia attempts for the speediest turnaround of any airline, usually turning around in most airports within a 25 minutes period limit, alongside an operating block time of 12 hours a day as compared to 8 hours a day by other airlines<ref>https://ir.airasia.com/what_lcc.html</ref>. To do so, it uses several complementary practices that were decided by its business strategy decision. === '''Network effects''' === The network effect is a unique source of Economies of Scale, which arises when customers experience greater benefit from using a product as a result of more people using it. For instance, Facebook provide the same value as a diary without the social function of interacting with other users. The utility and value of Facebook is higher than a diary is due to having increasingly high volume of users.The resulting ‘demand-side’ of Economies of Scale has a network effect if it benefits other adopters of the product (total effect) and incentivises others to adopt the product (marginal effect). '''Advantages:''' Effects may be localised if current customers only benefit if the adopters are people that are known to the customer such as family and friends. However, not all network effects are advantages. '''Disadvantages:''' Adoption of a good or platform by telemarketers or spammers may lower the incentive for potential adopters to adopt the good and networks that are supposed to be niche may suffer if too many adopters deemed as "outsiders" join. '''Lock-in Effect:''' Additionally, it is very common for network effects to cause a "lock-in" effect on certain goods, further enhancing the value of the good while effectively blocking out any competition. This is not always positive, the QWERTY keyboard being the most common example of a sub optimal lock-in effect for an industry. Even though the layout is not the most efficient in terms of typing, after the industry had "locked-in" to QWERTY it became so common and has such a high switching cost it has become impractical to change. Broadly there are two kinds of network effects: * '''Direct network effects''' '''-''' It is also known as same-side effects. An increase in the adoption of a product has a direct increase in the value of that product to the users, which in turn attract more people to adopt it. For example, telephone systems, fax machines, and social networks all imply direct contact among users. A direct network effect is called a same-side network effect. Another example is that players of an online game benefit from other players joining and populating the game. For example, in the telephone network, telephone's value is 0 when there is only one person owns it as they cannot do anything with the network. However, when there is a second person owning it, the first person is able to call the second person and there is value generated. If everyone owns a phone, the network is vital to all users. Therefore, users gain more from widespread adoption, and the larger the telephone networks with more users, the more attractive to non-users contemplating adoption. * '''Indirect network effects -''' The increasing usage of a product also has a flow-on increase in the value of a complementary product or network, which can, in turn, increase the value of the original. Examples of complementary goods include software (such as an Office suite for operating systems) and DVDs (for DVD players). This is also called a cross-side network effect. Windows and Linux might compete not just for users, but for software developers. Most two-sided markets (or platform-mediated markets) are characterised by indirect network effect; For instance, users of hardware may gain more as more users joining the network, not only due to direct effect, but also due to the inducing the improvement of the quality and quantity of software. Taking Uber as an example, more business opportunities are given to the drivers when there are more riders join the platform. Reversely, when there are more drivers join the network, the waiting time will be shortened and there will be more locations available for the riders. Therefore, the network valuable. The major differences between direct and indirect network is about the type of users who join. According to the Uber example, there will no additional value attributed to the driver. Moreover, when there are a rider join Uber, it will increase the value of all Uber drivers. <nowiki>There are also, in addition, two sources of economic value that are relevant when analysing products that display network effects:</nowiki> * '''Network value -''' I derive value from other people's use of the product * '''Inherent value -''' I derive value from my use of the product === '''Economies of Scope''' === Economies of scope happen when manufacturing one good causes the reduction of the production cost of another related product. As a result, which the marginal cost or the long-run average of a company decreases due to the production of complementary goods and services. Consequently, economies of scope are described by variety. Economies of scope can be achieved when the cost of producing two different products together is less costly when a single firm produces them instead of two separate firms. (ie. C(q₁,q₂) <  C(q₁,0) + C(0,q₂) where q₁ the production level of good one and q₂ is the production level of good two). This may occur when two products are complementary in their use to each other, when they have complementary production processes or when they share the same inputs to production. '''Learning economies''' - There is a learning economy where costs fall with experience. The learning economy can not directly expand the size of the company, but it can contribute to the success of the company. This stems from the fact that over time, managers and employees become more efficient in tasks. while managers become better at allocating resources and scheduling production processes. === '''Economies of Scale and Scope''' === Economies of both Scale '''and''' Scope present whenever large-scale production, distribution, or retail processes provide a cost advantage over smaller processes. In general, capital intensive production processes are more inclined to demonstrate economies of scale and scope as compared to other labour or materials intensive processes. By allowing cost advantages, economies of scale and scope will not only influence the magnitude of firms and the structure of markets, but they will, too, configure critical business strategy arrangements, with an example of the possibility of the merging of independent firms and the likelihood of a firm achieving a long-term cost advantage. ==Economies and Scale and Scope== Economies of scale and scope are used to help cut a firm’s operational costs. They occur when a firm experiences a cost advantage from implementing large-scale production over smaller processes. Economies of scope deal with average total cost of production of multiple goods while economies of scale are concerned with cost advantage that occurs due to the increased production of a single good. Economies of scope occur if it is possible for a firm to produce more than one good with the same resources, to increase the range of products they produce, while saving money on production costs, as opposed to producing the same amount of output with different resources. Economies of scale only occur with the indivisibilities, or the ability to manufacture products on a large scale that can’t be manufactured on a smaller scale. Indivisibilities include returns to scale, long-run fixed costs and setup costs, costs that would be too expensive to maintain production if only a small volume of output was being produced. === Differences between Economies of Scope and Economies of Scale === The economy of scope and economy of scale are two different concepts used to help cut a company's costs. Economies of scope focus on the average total cost of production of a variety of goods, whereas economies of scale focus on the cost advantage that arises when there is a higher level of production of one good. Economies of scale are reductions in average costs because production volume increases; whereas, economies of scope are reductions in average costs because the number of good produced increases <ref>[https://www.encyclopedia.com/management/encyclopedias-almanacs-transcripts-and-maps/economies-scale-and-economies-scope], September 2019, Economies of scale and economies of scope, Encyclopedia.com. </ref>. Differences: 1)Economies of scale - firms reach a point of production where the cost of it no longer increases (bulk production). It is an old concept used in business and economics. This reduces the cost of one product. It consists in producing one type of product in bulk. The strategy behind is the standardization of the product. It uses a large number of resources because of bulk production. 2)Economies of scope - firms produce a variety of products and their cost of production gets reduced. It is a new term in business economics. This reduces the cost of multiple products. It consists of producing multiple products under the same operation. The strategy behind economies of scope is the diversification of products. It uses fewer resources because firms produce multiple products under one operation. === Horizontal Mergers === Horizontal mergers have very high potential to have anticompetitive effects. This is because the total number of firms is reduced by one. Any potential increase in market power (ability of firm to raise prices above marginal cost) of a single firm must be balanced against any socially beneficial cost savings. Real world mergers can be very complex and require a number of steps to assess their viability. 1. Market Definition - this can be defined by the product, geography, product function, customers etc. Another way is the SSNIP test which refers to a 'small but significant non-transitory increase in price', this method helps define the market a firm operates in by assessing its market power. 2. Safe Harbours - mergers are significantly less likely to have negative effects on competition if post-merger market concentration is low. Market concentration can be determined by the Herfindahl index (HHI) 3. Effect of Merger on Existing Competition - evaluation of the competitive nature of the market, taking into account: the type of competition (price, quantity, fixed capacities), conduct of firms (coordinated?) and product differentiation 4. Effect of Merger on Potential Competition - possibility of entry deterrence/predation with or without merger, supplier relations and alternative technologies/networks 5. Other Competition Factors - changes in market powers of buyers and suppliers, scope for efficiency defence ==Vertical Boundaries of the Firm== '''Vertical boundaries of the firm''' refers to how much control the firm has over its industry operations, such as the production and distribution of their good or service. '''Vertical integration''' can be divided into two streams – forward integration and backward integration. In forward integration, companies will control their downstream counterparts, in order to increase control over the supply chain. For example, a gas mining company may own an energy power plant. Backward integration refers to when companies seek to control their upstream counterparts, in order to increase control over the final product. For example, a chocolate manufacturer may seek to own cocoa farms. ==='''Why not use the market for supplying inputs?'''=== ====Benefits of using the market==== * Firms can achieve '''economies of scale''' that in-house departments producing for their own needs cannot - specialised firms will typically produce more than an in house department will *'''Discipline of the market''' forces efficiencies on firms. That is, competition tends to promote effectiveness and quality. Relying on an in house department that meets the bare minimum requirements, will not have the same level of innovation & quality as an external firm. ====Costs of using the market==== # '''Hold-up problem''': Is an issue of imperfect contracts - that is where negotiations/changes in circumstances can result in time delays or increased costs. This raises the costs of transacting market exchanges. #* It is argued that the possibility of hold-up can lead to underinvestment in relationship-specific investments and hence to inefficiency. For example, one supplier has an exclusive contract to supply body parts for the cars of General Motors. The supplier can hold up General Motors by increasing the price for the additional parts produced if exceeding demands occur. #* It can lead to difficult contract negotiations and more frequent renegotiations #* It can lead to distrust between corporations # '''Difficulties in coordination''': External firms are harder to control than internal departments. This in turn can raise costs with bottlenecks in the production flow. The failure of one firm to deliver supplies on time can lead to another factory being shut down. # '''Security of private information''': Private information may be leaked when using the market. Leakages can result in firms' competitive advantage being compromised. An example of this is a patent or special know-how. # '''Transaction costs in contracting''': Cost incurred during the process of purchasing and selling goods or services. ===Vertical Separation=== Some firms may decide to develop looser relationships then complete full vertical integration. That is, instead of fully moving all production in-house, they will utilise a balance between internal departments and the market. '''Advantage of a looser relationship over full vertical integration''' #Preservation of firm’s independence #Avoidance of costs that may be associated with full vertical integration '''Examples of looser relationships:''' * Franchising – involves a specific contractual relationship/arrangement between franchiser & franchiseeE.g. McDonald's, Hungry Jacks, 7-Eleven * Networks of independent firms that are linked vertically & establish nonexclusive contracts or relationships with one another E.g. Grocery retailers and Metcash (grocery wholesaler) '''Other alternatives to vertical integration:''' *'''Tapered integration''': A mix of vertical integration and market exchange, making some inputs and buying the remaining portion from independent firms. Example: BMW uses some external market research along with in house market research. The advantage of Tapered Integration: Producing part of the production requested materials and input the rest of the materials from other companies in exists in the market. This will reduce the initial cost of capital and reduce the cost of misunderstanding the market price. Manufacturing some of the demand whist purchasing the rest from the market will not only increase the bargaining power of the company itself, and also threat the external suppliers to discipline the supply process and quality of the supplies. Disadvantage of Tapered Integration: The company may not achieve the economies of scale, because of the sufficiency of production will need both internal production and external suppliers to coordinate. Other than the loss of economies of scale, the tapered integration may incur higher coordination costs and freight in and out costs due to purchasing supplies from external suppliers. The efficiency of production process will also be a problem if coordination of supplies and internal production process does not collaborate. *'''Joint Venture''': Where two or more parties decide to work together by pooling their resources with the goal of achieving a specific task or completing a certain business activity. However, the venture is separate from the other business interests and the two companies operate as one in the venture. Example: the creation of google earth was as a result of a joint venture between Google and NASA. *'''Strategic Alliance''': A strategic alliance is where two or more firms work together to increase each other's performance. They operate in the same way as a joint venture however what makes them different is they operate as separate companies and don’t require a legal contract. Example: ApplePay and Mastercard; Mastercard was the first to offer ApplePay this alliance means they benefit from sharing their users. *'''Long term collaborative relationships''': At least two parties who agree to share resources, such as finance, knowledge and people to accomplish a mutual goal. Example: business relationships. *'''Implicit contracts between firms:''' Are a non-binding agreement voluntarily entered into in regard to future exchanges of goods and services. Example: an employer continues to offer employment given the employee remains sincere in not looking for another job and continues their duties *'''Recently in Western countries:''' This strategy forester the vertical disintegration and concentrate on create core competencies for companies, aiming to outperform other companies in within the market. ===Franchising=== Franchising is both a marketing and business model strategy that establishes the fundamental basis of the company. When implemented, a Franchisor licenses its business 'know-how' and produces intellectual property of that knowledge, its business model and brand name that is then to be sold to a Franchisee, who runs the day to day operations of that branch. In return of the Franchisee pays certain fees and agrees to comply with obligations laid out by the franchisor, such as a franchisee restaurant being obligated to use the business's own plates, own restaurant design outlook, own tissues papers and branded goods. '''Advantages:''' # '''It avoids the hold-up problem''' as the franchisees have the same incentive to perform well that the franchiser does. # '''Reduces transaction costs''' for Franchisees as any relationships with suppliers and buyers have already been already established by the company themselves. # It facilitates '''rapid business expansion''' as establishing costs are incurred by the Franchisee # '''Reduces managerial lag''' as the franchiser is not involved in the day to day operations. The franchisee is incentivised to do well and so the franchiser does not have to micro manage them in a traditional sense. # The Franchisee gains business based on '''brand recognition'''. Consumers are more likely to purchase from that particular franchise. '''Disadvantage:''' # '''The franchise may harm the firm's reputation''', any bad press for one franchise hurts the entire firms' reputation and so recruitment process needs to be thorough. # '''It lowers profits for the firm''' seeing as the company has to share these profits with the investors. This leads to relatively lower profits than if the company had opened its own stores. Additional '''advantages''' of franchising include: - Franchisee not required to be an expert in marketing additionally to the franchise - Training and learning through the business operations provided - Tactics and strategies training for the franchisee’s staff - Support and advice from fellow franchisees - Value addition in existing product line from suppliers Additional '''disadvantages''' of franchising include: - Lack of independence to completely innovate - Risk of franchisee business failure is not eliminated - Risk of franchisor choosing unsuccessful marketing - Additional costs for training franchisee’s staff or may not be available - Franchisee may lack freedom of choice in suppliers - Franchisor may receive rebates on franchisee purchases A '''SWOT''' analysis of the merits and demerits of franchising has been conducted below. '''Strengths:''' - Easy setup - Brand recognition - Lower risks for failure - Ready customer portfolio - Easy to find financial support '''Weaknesses:''' - High initial and ongoing costs - Dependency - Strict rules '''Opportunities:''' - Offers some market opportunities like discovering and exploitation - Entrepreneurs have the chance to become their own boss '''Threats:''' - Other new franchise competitors entering the market place - Decline of branding in market - Publication of new business models - Continuing growth of existing franchised competitors (Nisar, 2017)<ref>Nisar, A. (2017). The main benefits and disadvantages of franchising. Pakistan & Gulf Economist, 25-29.</ref> ===Entrepreneurship as experimentation=== The knowledge required to be a successful entrepreneur cannot be learned in advance, it needs to be gained through a process of trial and error, otherwise known as creative destruction (Schumpeter, 1943). For an entrepreneur, it is impossible to guess whether or not a certain technology or product would be successful until one decides to invest in the idea. As Hayek (1948) put it, “the solution of the economic problem of society is . . . always a voyage of exploration into the unknown.”<ref>"Entrepreneurship as Experimentation"[https://pdfs.semanticscholar.org/e4cf/4b946114d7ef05728bd34223c7ddf27daa99.pdf .https://pdfs.semanticscholar.org/e4cf/4b946114d7ef05728bd34223c7ddf27daa99.pdf]. Retrieved 2019-10-14</ref> Barriers to entrepreneurship have been lessened due to technological advancements. * There are low costs of investing in an experiment such as open-source software and cloud computing. * Costs and constraints on the ability to experiment alter the type of organisational form surrounding innovation. ** Lean Start-up methodology, for instance, focuses on developing “minimal viable products” (MVPs). ** Ability to terminate projects. For example: *** Sunk cost fallacy (throwing good money after bad money) - An inability to stop investing in a product even though the failure rate is high after the process of product development has started. The continuation of risky behaviour due to prior investments that are perceived by the firms as being "in too deep" into the process to quit. An example would be a business. When you pay salaries to your employees or workers, you don't expect to get back that money. *** Large organisations start projects that are less experimental in anticipation of the inability to terminate failures and the added reputation cost of these failures. Ending those failing experiments can be more difficult for larger companies due to corporate bureaucracy and as a result, they preemptively anticipate these difficulties, which in turn leads to them not taking the risk in the first place. *** Reputation cost of failure - Large firms might be better off waiting for an alternative product to grow and then buying it up == Summary == The below section aims at briefly summarising the key takeaways from this topic: '''What is a firm?''' A firm is an organisation who transforms inputs into outputs with the desire to maximise profit (generally). Firms exist to reduce transaction costs of always having to go to the market. This is done by reducing search costs, bargaining and decision making, policing and enforcement. The need for firms is being disrupted by technologies allowing for peer-to-peer trading which creates a platform to remove these transaction costs. '''Economies of Scale''': Advantages from large scale production allow for a reduction in average cost as output increases. This is more common with capital intensive production than labour or material intensive processes. Some advantages include: * Splitting of fixed costs * Volumetric returns to scale * Network effects '''Diseconomies of Scale''': As a firm grows they are more likely to face management inefficiencies. Additionally, as a firm grows the likelihood of competition increases. '''Economies of Scope''': Splitting or sharing production costs amongst multiple products to reduce costs as compared to just on one good. '''Vertical Boundaries of a firm''': Sometimes better to vertical integrate and sometimes it is better to vertical separate. It is better to separate if you cannot achieve the same level of specialisation or not enough incentive to innovate and remain efficient for a given element of the vertical chain. The benefits of integrating prevent holdup, difficulties in coordination, leaking of private information and the reduction/avoidance of transaction costs. '''Franchising''': Involves a specific contractual relationship between franchiser & franchisee. Independently owned firms have motivation to maximise profits and therefore are easier to manage. Pros – avoid hold up, reduce transaction cost, facilitates business expansion, reduces managerial lag. Cons – might harm firms' reputation and potentially could reduce profits. '''Other benefits of franchising''': (Harold, 1981). 1. Reduction of the risk due to the know-how provided. Therefore, the company have the certainty the business will work. 2. Managing a business that is already proven to have an appealing reputation to consumers. 3. The franchisor may provide expert advice and negotiating tips as well. '''Other forms of Vertical Integration''': -      Tapered integration – make some and buy the rest -      Joint Ventures and strategic alliances -      Collaborative relationships An example of collaborative relationships would be Japanese and Korean industrial firms, they organised a vertical chain rather than an arm's length transaction.Japanese manufacturers maintain close, informal, long term relationship with their network of subcontractors - the typical relationship between a manufacturer and a subcontractor involves far more asset specificity in Japan than in the West (i.e. in Japan subcontractors invest more in relationship-specific assets and routines). -      Implicit contracts – e.g. grand father clauses Definition: Unstated understanding between firms in a business relationship. Longstanding relationships causes both firms to behave cooperatively amongst each other without a formal contract. It is however, not enforceable by law and the threat of losing a business partner is enough of a barrier to deter a business from any opportunistic actions. '''Sunk Cost Fallacy''': Occurs when a firm incurs irretrievable costs which will never be recovered, such as time or money invested. The firm may continue with an inefficient decision to justify the sunk cost of the investment. According to Arkes and Blumer (1985), this fallacy which results in ongoing commitment is linked to status quo bias and loss aversion. Moreover, the adaptability and potential development in the market of a firm depends highly on the magnitude of the sunk cost. Studies have proven that the degree in of effectiveness in which a company competes in the market is determined by the level of investment they put into the sunk capital. For instances, the lower the investment the higher will be the contestability of the firm (García-Díaz, Witteloostuijn & Péli, 2015). Example 1: Purchasing new software, discovering it doesn't have it's desired benefit (such as improving performance), yet continuing to run the software anyway due to the cost. Example 2: Continuing to wait in line purely to justify the time already spent waiting in the queue. == Reference/s == https://www.profolus.com/topics/types-and-sources-of-economies-of-scale/ <ref>Arkes, H. R., & Blumer, C. (1985). The psychology of sunk cost. Organizational Behavior and Human Decision Processes, 35(1), 124-140. http://dx.doi.org/10.1016/0749-5978(85)90049-4</ref> García-Díaz, César, van Witteloostuijn, A., & Péli, G.L. (2015). Micro-Level Adaptation, Macro-Level Selection, and the Dynamics of Market Partitioning. PLoS One, urn:issn:1932–6203. Brown, Harold. (1981). The benefits of franchising. Commercial Law Journal, 86(2). [[Category:Economics]] 92s7hcnr1vbx41vqf98v7e03mv3dbf5 0 252752 2693493 2060335 2024-12-27T00:31:17Z Tule-hog 2984180 nominate speedy 2693493 wikitext text/x-wiki {{delete|Moved to [[Network+/Activities]]}} [[w:Reverse proxy]] is a type of [[proxy server]] that retrieves resources on behalf of a client from one or more servers. === Activities === # [[/Review list of available proxy servers/]] == See also == * [[Proxy Server]] [[Category:Networking]] kimjtld3crcl391l5jm97jttad00ttk 0 252753 2693494 2046817 2024-12-27T00:31:41Z Tule-hog 2984180 Tule-hog moved page [[Network+/Architecture/Services/Reverse Proxy/Review list of available proxy servers]] to [[Network+/Activities/Review list of available proxy servers]]: alter parent 2046817 wikitext text/x-wiki * [[aiCache]] is a commercial reverse proxy and a caching reverse proxy. * [http://www.ergon.ch/en/airlock/ Airlock], a Web Application Firewall developed and marketed by the Swiss company Ergon Informatik AG. It offers SSL termination, upstream authentication, blacklist and white-list filtering as well as load balancing capabilities. * [[Apache HTTP Server]] may be extended with [[mod_proxy]] to be used as a reverse proxy; a caching reverse proxy server may be configured using the mod_cache module in conjunction with mod_proxy.<ref name="apache-mod-proxy">{{cite web|url=http://httpd.apache.org/docs/2.0/mod/mod_proxy.html|title=Apache Module mod_proxy|publisher=The Apache Software Foundation|accessdate=9 February 2011}}</ref> * Apache [[Traffic Server]], an open source, high-performance routing and caching server. * [[ApplianSys#CACHEbox|ApplianSys CACHEbox]] is a high-performance HTTP/HTTPS/FTP caching proxy appliance supporting reverse- as well as forward deployment modes. * [[Arahe SiteCelerate]] is a commercial high performance reverse proxy with caching and compression. It offers image and text compression. * [[Armorlogic]] Profense, an advanced reverse proxy (with web application firewall module) and content load balancer. * [[Blue Coat Systems]] ProxySG, a forward proxy that can also be used as a reverse proxy. * [[F5 Networks]] [[BIG-IP]] can be used as a reverse proxy with load balancing capabilities and has an optional application security module (ASM) to protect against attacks. * [[Cherokee_(webserver)|Cherokee]] can be used as a reverse proxy as well as a web server and load balancer. * [[HAProxy]] is a free, very fast and reliable solution offering high availability, load balancing, and proxying for TCP and HTTP-based applications. * [[Hiawatha (web server)|Hiawatha]] can be used as a reverse proxy which is also a secure, easy to use, lightweight web server. * [[LBL&reg;LoadBalancer]], new generation applications high availability mission-critical, business-critical load balancer. LBL&reg;LoadBalancer is a module of LBL&reg;A.A.I.(Application Availability Infrastructure). * [[Linoma_Software|GoAnywhere Gateway]], an enhanced reverse proxy that allows FTP, FTPS, SFTP and HTTP services without exposing sensitive files in the [[DMZ (computing)|DMZ]] or opening incoming ports into the internal network. * [[Internet Information Services]] 7.0 with URL Rewrite v2 and Application Request Routing can act as a reverse proxy.<ref>{{cite web|url=http://learn.iis.net/page.aspx/659/reverse-proxy-with-url-rewrite-v2-and-application-request-routing/|title=Reverse Proxy with URL Rewrite v2 and Application Request Routing|publisher=Microsoft Corporation|date=July 16, 2009|accessdate=9 February 2011}}</ref> * [[Lighttpd]] can be used as a reverse proxy with load balancing capabilities. * [[LiteSpeed Technologies Inc.|LiteSpeed Web Server]] can be used as a transparent reverse proxy server running in front of any web server or application server that supports the HTTP protocol. * [[Secure_Computing#Web_Security_products|McAfee Web Gateway]] is a product that can act as a reverse proxy. It also provides SSL decryption, caching, [[anti-virus]], [[anti-spam]] and other threat detection features. * [[Microsoft Forefront Threat Management Gateway]] (Forefront TMG), formerly known as Microsoft Internet Security and Acceleration Server (ISA Server), is a commercial proxy, firewall and caching solution by Microsoft. * [[Microsoft Forefront Unified Access Gateway]] (Forefront UAG). * [[Citrix Systems]] [[Netscaler]] ADC, A hardware and software solution providing advanced application and service delivery. Netscaler is a reverse-proxy with high-speed load balancing and content switching, data compression, content caching, SSL acceleration, network optimization, application visibility and application security on a single platform.[http://www.citrix.com/English/ps2/products/feature.asp?contentID=2300357 Citrix Netscaler ADC] * [http://www.adnovum.ch/en/products/index.php?page=secprod&subpage=nevis nevisProxy] is a secure reverse proxy with integrated web application firewall (WAF). Developed and marketed by the Swiss company AdNovum Informatik AG. * [[Nginx]] is a web- and reverse proxy server. * [[Novell Access Manager]] is a commercial security solution which includes a reverse proxy, a policy-based access manager, and SSL [[VPN]]. All components use an [[LDAP]]-like directory or federation with Liberty and others. * [[Perlbal]] is a [[Perl]]-based reverse proxy load balancer and web server. * [http://sourceforge.net/p/portfusion/ PortFusion] is an open-source, cross-platform (Windows, Linux, OS X, FreeBSD), tiny, multi-protocol, distributed reverse / forward proxy and tunneling solution for all types of TCP-based traffic. Developed at the [[University of Heidelberg]] for remote administration and web service routing. Its focus is on maximum throughput, small binary and source code size and easy configuration from the command line. * [[Pound (networking)|Pound]] is a reverse proxy, load balancer and HTTPS front-end for Web server(s). * [http://www.united-security-providers.com/en/it-security-solutions/protection-for-web-applications/ USP Secure Entry Server™], a Reverse Proxy developed and marketed by Switzerland's United Security Providers AG. It offers SSL termination, filtering, quality of application, integration engine as well as secure login service with a wide range of authentication protocols. * [http://help.sap.com/saphelp_nw04s/helpdata/en/e9/3bb7f8f6ea4e938ef0b9687cbb6c14/content.htm SAP Web Dispatcher] is a reverse proxy and load balancer. * [[Squid (software)|Squid]] is a proxy server that may be installed in a reverse proxy configuration.Squid is free software, released under the [[ GNU ]] General Public License. * [[Stunnel]] can be used as a local SSL reverse proxy. * [[Sun Java System Web Server]] includes a reverse proxy module with load-balancing capabilities. * [[Tinyproxy]] is a minimalistic HTTP proxy which can be configured to work as a reverse proxy. * [[Tivoli Access Manager for eBusiness, WebSEAL]] is one of IBM's security products with WebSEAL being the reverse proxy. * [[Varnish cache|Varnish Cache]] is a performance-focused, open source reverse proxy. It has a policy configuration language to allow for extension. It features [[Edge Side Includes|ESI]], SaintMode, DNS director, built-in Load Balancing and native support for Varnish Modules written in C. * [[WinGate (computing)|WinGate]] supports reverse-proxying with SSL, authentication, and multiple virtual hosts. * [[Zeus Web Server|Zeus]] is a product that can function as both a forward and reverse proxy, as well as content load balancer. [[Category:Computer Networks]] h32afdfi23yr07o2as6m7qxarb44s59 0 263813 2693444 2693234 2024-12-26T23:34:41Z Scogdill 1331941 2693444 wikitext text/x-wiki ==Also Known As== * Family name: Bourke [pronounced ''burk'']<ref name=":62">{{Cite journal|date=2024-05-07|title=Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Mayo&oldid=1222668659|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Earl_of_Mayo.</ref> * The Hon. Algernon Bourke * Mrs. Guendoline Bourke * Lady Florence Bourke * See also the [[Social Victorians/People/Mayo|page for the Earl of Mayo]], the Hon. Algernon Bourke's father. == Overview == Although the Hon. Algernon Henry Bourke was born in Dublin in 1854 and came from a family whose title is in the Peerage of Ireland,<ref name=":6">1911 England Census.</ref> he seems to have spent much of his adult life generally in England and especially in London. Mrs. Guendoline Bourke was a noted horsewoman and an excellent shot, exhibited at dog shows successfully and was "an appreciative listener to good music."<ref>"Vanity Fair." ''Lady of the House'' 15 June 1899, Thursday: 4 [of 44], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004836/18990615/019/0004.</ref> She was reported as attending many social events without her husband, usually with a quick description of what she wore. The Hon. Algernon Bourke and Mr. Algernon Bourke, depending on the newspaper article, were the same person. Calling him Mr. Bourke in the newspapers, especially when considered as a businessman or (potential) member of Parliament, does not rule out the son of an earl, who would normally be accorded the honorific of ''Honorable''. == Acquaintances, Friends and Enemies == === Algernon Bourke === * [[Social Victorians/People/Montrose|Marcus Henry Milner]], "one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys"<ref name=":8">"Metropolitan Notes." ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 2a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.</ref> * Caroline, Duchess of Montrose — her "legal advisor" on the day of her marriage to Marcus Henry Milner<ref>"Metropolitan Notes." ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 1b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.</ref> === Guendoline Bourke === * Lord and Lady Alington, Belvedere House, Scarborough == Organizations == === Guendoline Bourke === * Member, the Ladies Committee for the Prince's Skating Club, which also included [[Social Victorians/People/Princess Louise|Princess Louise]] (Duchess of Argyll), the [[Social Victorians/People/Portland|Duchess of Portland]], [[Social Victorians/People/Londonderry|Lady Londonderry]], [[Social Victorians/People/Campbell|Lady Archibald Campbell]], [[Social Victorians/People/Ribblesdale|Lady Ribblesdale]], and [[Social Victorians/People/Asquith|Mrs. Asquith]]<ref name=":11">"What the 'World' Says." ''Northwich Guardian'' 01 November 1902, Saturday: 6 [of 8], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001975/19021101/134/0006. Print title: The ''Guardian'', p. 6.</ref> (1902, at least) === Algernon Bourke === * Eton * Cambridge University, Trinity College, 1873, Michaelmas term<ref name=":7">Cambridge University Alumni, 1261–1900. Via Ancestry.</ref> * Conservative Party * 1879: Appointed a Poor Law Inspector in Ireland, Relief of Distress Act * 1885: Office of the 7th Surrey Rifles Regiment<ref>"7th Surrey Rifles." ''South London Press'' 08 August 1885, Saturday: 12 [of 16], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18850808/165/0012. Print p. 12.</ref> * Special Correspondent of The ''Times'' for the Zulu War, accompanying Lord Chelmsford * Head, Messrs. Bourke and Sandys, "that well-known firm of stockbrokers"<ref name=":8" /> ( – 1901 [at least]) * White's gentleman's club, St. James's,<ref>{{Cite journal|date=2024-10-09|title=White's|url=https://en.wikipedia.org/wiki/White's|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/White%27s.</ref> Manager (1897)<ref>"Side Lights on Drinking." ''Waterford Standard'' 28 April 1897, Wednesday: 3 [of 4], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001678/18970428/053/0003.</ref> * Willis's Rooms<blockquote>... the Hon. Algernon Burke [sic], son of the 6th Earl of Mayo, has turned the place into a smart restaurant where choice dinners are served and eaten while a stringed band discourses music. Willis's Rooms are now the favourite dining place for ladies who have no club of their own, or for gentlemen who are debarred by rules from inviting ladies to one of their own clubs. The same gentleman runs a hotel in Brighton, and has promoted several clubs. He has a special faculty for organising places of the kind, without which such projects end in failure.<ref>"Lenten Dullness." ''Cheltenham Looker-On'' 23 March 1895, Saturday: 11 [of 24], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000226/18950323/004/0011. Print p. 275.</ref></blockquote> ==== Boards of Directors ==== *1883: One of the directors, the Franco-English Tunisian Esparto Fibre Supply Company, Ltd.<ref>''Money Market Review'', 20 Jan 1883 (Vol 46): 124.</ref> *1891: One of the founders, the Discount Banking Company, Ltd., which says Algernon Bourke is a director of District Messenger Services and News Company, Ltd.<ref>"Public Company." ''Nottingham Journal'' 31 October 1891, Saturday: 4 [of 8], Col. 8a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001896/18911031/099/0004. Print title: ''The Nottingham Daily Express'', p. 4.</ref> *1894: One of the directors, the Frozen Lake, Ltd., with Admiral Maxse, Lord [[Social Victorians/People/Beresford|Marcus Beresford]], [[Social Victorians/People/Williams|Hwfa Williams]]<ref>"The Frozen Lake, Limited." ''St James's Gazette'' 08 June 1894, Friday: 15 [of 16], Col. 4a [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18940608/085/0015. Print p. 15.</ref> ==== Committees ==== *Member, Men's Committee of the Prince's Skating Club, which also included Lord Edward Cecil, Lord Redesdale, Mr. [[Social Victorians/People/Lyttelton|Alfred Lyttelton]], Sir Edgar Vincent, Sir William Hart Dyke, and Mr. [[Social Victorians/People/Grenfell|W. H. Grenfell]]<ref name=":11" /> (1902, at least) *[[Social Victorians/Timeline/1896#25 March 1896, Wednesday|The Sala Memorial Fund]], member of the committee (from 25 March 1896) * Member of an "influential committee" headed by the Lord Mayor "to restore salmon to the Thames" (June 1899)<ref>"Salmon in the Thames." ''Berks and Oxon Advertiser'' 30 June 1899, Friday: 5 [of 8], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002298/18990630/079/0005. Print n.p.</ref> == Timeline == '''1872 February 8''', Richard Bourke, 6th Earl of Mayo was assassinated while inspecting a "convict settlement at Port Blair in the Andaman Islands ... by Sher Ali Afridi, a former Afghan soldier."<ref>{{Cite journal|date=2024-12-01|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo.</ref> The Hon. Algernon's brother Dermot became the 7th Earl at 19 years old. '''1876 November 24, Friday''', the Hon. Algernon Bourke was one of 6 men (2 students, one of whom was Bourke; 2 doctors; a tutor and another man) from Cambridge who gave evidence as witnesses in an inquest about the death from falling off a horse of a student.<ref>"The Fatal Accident to a Sheffield Student at Cambridge." ''Sheffield Independent'' 25 November 1876, Saturday: 7 [of 12], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000181/18761125/040/0007. Print title: ''Sheffield and Rotherham Independent'', n. p.</ref> '''1884 May 3, Saturday''', the "Rochester Conservatives" announced that they would "bring forward the Hon. Algernon Bourke, brother of Lord Mayo, as their second candidate,"<ref>"Election Intelligence." ''Yorkshire Gazette'' 03 May 1884, Saturday: 4 [of 12], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18840503/011/0004.</ref> but because he could not be the first candidate, Bourke declined.<ref>"Rochester." London ''Daily Chronicle'' 09 May 1884, Friday: 3 [of 8], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/18840509/049/0003.</ref> '''1884 June 18, Wednesday''', Mr. Algernon Bourke was on a committee to watch a [[Social Victorians/Timeline/1884#18 June 1884, Wednesday|Mr. Bishop's "thought-reading" experiment]], which was based on a challenge by Henry Labourchere made the year before. This "experiment" took place before a fashionable audience. '''1885 October 3, Saturday''', the Hon. Algernon Bourke was named as the Conservative candidate for Clapham in the Battersea and Clapham borough after the Redistribution Bill determined the electoral districts for South London.<ref>"South London Candidates." ''South London Press'' 03 October 1885, Saturday: 9 [of 16], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18851003/096/0009. Print p. 9.</ref> The Liberal candidate, who won, was Mr. J. F. Moulton. '''1886 July 27, Tuesday''', Algernon Bourke attended a service honoring a memorial at St. Paul's for his father, who had been assassinated.<ref>"Memorial to the Late Earl of Mayo." ''Northern Whig'' 28 July 1886, Wednesday: 6 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000434/18860728/143/0006. Print p. 6.</ref> '''1886 September 2, Thursday''', Mr. Algernon Bourke was part of a group of mostly aristocratic men taking part in [[Social Victorians/Timeline/1886#8 September 1886, Wednesday|a "trial-rehearsal" as part of Augustus Harris's production]] ''A Run of Luck'', about sports. '''1886 October 2, Saturday''', the Duke of Beaufort and the Hon. Algernon Bourke arrived in Yougal: "His grace has taken a residence at Lismore for a few weeks, to enjoy some salmon fishing on the Blackwater before the close of the season."<ref>"Chippenham." ''Wilts and Gloucestershire Standard'' 02 October 1886, Saturday: 8 [of 8], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001955/18861002/142/0008. Print p. 8.</ref> '''1887 December 15''', Hon. Algernon Bourke and Guendoline Stanley were married at St. Paul's, Knightsbridge, by Bourke's uncle the Hon. and Rev. George Bourke. Only family members attended because of "the recent death of a near relative of the bride."<ref>"Court Circular." ''Morning Post'' 16 December 1887, Friday: 5 [of 8], Col. 7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18871216/066/0005.</ref> '''1888 July 26''', [[Social Victorians/People/Montrose|Caroline Graham Stirling-Crawford]] (known as Mr. Manton for her horse-breeding and -racing operations) and Marcus Henry Milner married.<ref name=":12">"Hon. Caroline Agnes Horsley-Beresford." {{Cite web|url=https://thepeerage.com/p6863.htm#i68622|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> According to the ''Nottingham Evening Post'' of 31 July 1888,<blockquote>LONDON GOSSIP. (From the ''World''.) The marriage of "Mr. Manton" was the surprise as well the sensation of last week. Although some wise people noticed a certain amount of youthful ardour in the attentions paid by Mr. Marcus Henry Milner to Caroline Duchess of Montrose at '''Mrs. Oppenheim's ball''', nobody was prepared for the sudden ''dénouement''; '''and it''' were not for the accidental and unseen presence [[Social Victorians/People/Mildmay|a well-known musical amateur]] who had received permission to practice on the organ, the ceremony performed at half-past nine on Thursday morning at St. Andrew's, Fulham, by the Rev. Mr. Propert, would possibly have remained a secret for some time to come. Although the evergreen Duchess attains this year the limit of age prescribed the Psalmist, the bridegroom was only born in 1864. Mr. "Harry" Milner (familiarly known in the City as "Millions") was one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys, and Mr. Algernon Bourke, the head of the house (who, of course, takes a fatherly interest in the match) went down to Fulham to give away the Duchess. The ceremony was followed by a ''partie carrée'' luncheon at the Bristol, and the honeymoon began with a visit to the Jockey Club box at Sandown. Mr. Milner and the Duchess of Montrose have now gone to Newmarket. The marriage causes a curious reshuffling of the cards of affinity. Mr. Milner is now the stepfather of the [[Social Victorians/People/Montrose|Duke of Montrose]], his senior by twelve years; he is also the father-in-law of [[Social Victorians/People/Lady Violet Greville|Lord Greville]], Mr. Murray of Polnaise, and [[Social Victorians/People/Breadalbane|Lord Breadalbane]].<ref name=":8" /></blockquote>'''1888 December 1st week''', according to "Society Gossip" from the ''World'', the Hon. Algernon Bourke was suffering from malaria, presumably which he caught when he was in South Africa:<blockquote>I am sorry to hear that Mr. Algernon Bourke, who married Miss Sloane-Stanley a short time ago, has been very dangerously ill. Certain complications followed an attack of malarian fever, and last week his mother, the Dowager Lady Mayo, and his brother, Lord Mayo, were hastily summoned to Brighton. Since then a change for the better has taken place, and he is now out of danger.<ref>"Society Gossip. What the ''World'' Says." ''Hampshire Advertiser'' 08 December 1888, Saturday: 2 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18881208/037/0002. Print title: ''The Hampshire Advertiser County Newspaper''; print p. 2.</ref></blockquote>'''1889 – 1899 January 1''', the Hon. Algernon Bourke was "proprietor" of White's Club, St. James's Street.<ref name=":9">"The Hon. Algernon Bourke's Affairs." ''Eastern Morning News'' 19 October 1899, Thursday: 6 [of 8], Col. 7c [of7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001152/18991019/139/0006. Print p. 6.</ref> '''1889 June 8, Saturday''', the Hon. Algernon Bourke contributed some art he owned to the collection of the Royal Institute of Painters in Water-Colours' [[Social Victorians/Timeline/1889#8 June 1889, Saturday|exhibition of "the works of the 'English Humourists in Art.'"]] '''1892''', the Hon. Algernon Bourke privately published his ''The History of White's'', the exclusive gentleman's club. '''1893 February 11, Tuesday''', Algernon Bourke opened Willis's Restaurant:<blockquote>Mr. Algernon Bourke has in his time done many things, and has generally done them well. His recently published history of White's Club is now a standard work. White's Club itself was a few years ago in its agony when Mr. Bourke stepped in and gave it a renewed lease of life. Under Mr. Bourke's auspices "Willis's Restaurant" opened its doors to the public on Tuesday last in a portion of the premises formerly so well known as Willis's Rooms. This new venture is to rival the Amphitryon in the matter of cuisine and wines; but it is not, like the Amphitryon, a club, but open to the public generally. Besides the restaurant proper, there are several ''cabinets particuliers'', and these are decorated with the very best of taste, and contain some fine portraits of the Georges.<ref>"Marmaduke." "Letter from the Linkman." ''Truth'' 20 April 1893, Thursday: 25 [of 56], Col. 1a [of 2]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025# https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025]. Print p. 855.</ref></blockquote>'''1893 April 1, Saturday''', Algernon Bourke published a letter to the editor of the ''Times'', reprinted in the ''Kildare Observer'', arguing against Gladstone's Home Rule bill on the grounds that Ireland would not be able to take out a loan on its own behalf because of its obligations to the U.K., including what was called its share of the national debt.<ref>"Irish Unionist Alliance." ''Kildare Observer and Eastern Counties Advertiser'' 01 April 1893, Saturday: 6 [of 8], Col. 4c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001870/18930401/062/0006. Print: The ''Kildare Observer'', n.p.</ref> '''1893 November 30, Thursday''', with Sir Walter Gilbey the Hon. Algernon Bourke "assisted" in "forming [a] collection" of engravings by George Morland that was exhibited at Messrs. J. and W. Vokins’s, Great Portland-street.<ref>"The George Morland Exhibition at Vokins's." ''Sporting Life'' 30 November 1893, Thursday: 4 [of 4], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000893/18931130/058/0004.</ref> '''1895 February 23, Saturday''', the Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1895#23 February 1895, Saturday|fashionable wedding of Laurence Currie and Edith Sibyl Mary Finch]]. '''1895 August 24, Saturday''', "Marmaduke" in the Graphic says that Algernon Bourke "opened a cyclists' club in Chelsea."<ref>"Marmaduke." "Court and Club." The ''Graphic'' 24 August 1895, Saturday: 11 [of 32], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18950824/017/0011. Print p. 223.</ref> '''1895 October''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#24 October 1902, Friday|opened the Prince's ice-skating rink for the season]]. '''1896 June 29, Monday''', Algernon Bourke published a letter to the editor of the ''Daily Telegraph'':<blockquote>To the Editor of “The Daily Telegraph.” Sir — Permit me to make my bow to the public. I am the manager of the Summer Club, which on two occasions bas been the subject of Ministerial interpellation in Parliament. The Summer Club is a small combination, which conceived the idea of attempting to make life more pleasant in London by organising breakfast, luncheon, and teas in Kensington Gardens for its members. This appears to have given offence in some way to Dr. Tanner, with the result that the catering arrangements of the club are now "by order" thrown open to the public. No one is more pleased than I am at the result of the doctor's intervention, for it shows that the idea the Summer Club had of using the parks for something more than mere right of way bas been favourably received. In order, however, that the great British public may not be disappointed, should they all come to lunch at once, I think it necessary to explain that the kitchen, which by courtesy of the lessee of the kiosk our cook was permitted to use, is only 10ft by 5ft; it has also to serve as a scullery and pantry, and the larder, from which our luxurious viands are drawn, is a four-wheeled cab, which comes up every day with the food and returns after lunch with the scraps. Nevertheless, the Summer Club says to the British public — What we have we will share with you, though it don't amount to very much — I am, Sir, your obedient servant, ALGERNON BOURKE. White's Club, June 27<ref>"The Summer Club." ''Daily Telegraph & Courier'' (London) 29 June 1896, Monday: 8 [of 12], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18960629/072/0008. Print title: ''Daily Telegraph'', p. 8.</ref></blockquote>'''1896 July 4, Saturday''', "Marmaduke" in the ''Graphic'' took Bourke's side on the Summer Club in Kensington Park:<blockquote>Most of us have noticed that if we read in the newspapers the account of some matter which we are personally acquainted with the account will generally contain several errors. I have also noticed that when a question is asked in the House of Commons regarding some matter about which I know all the facts the question and the official answer to it frequently contain serious errors. Last week Mr. Akers-Douglas was asked in the House to explain how it was that Mr. Algernon Bourke obtained permission to open the "Summer Club" in Kensington Gardens, and he was questioned upon other particulars connected with the same matter. Both the questions and the official reply showed considerable ignorance of the facts. There has been from time immemorial a refreshment kiosk in Kensington Gardens. Mr. Bourke obtained from the tenant of this permission to use the kitchen for the benefit of the "Summer Club," and to supply the members of the latter with refreshments. It was a purely commercial transaction. Mr. Bourke then established some wicker seats, a few tables, a tent, and a small hut upon a lawn in the neighbourhood of the kiosk. To do this he must have obtained the permission of Mr. Akers-Douglas, as obviously he would otherwise have been immediately ordered to remove them. Mr. Akers-Douglas equally obviously would not have given his sanction unless he had been previously informed of the objects which Mr. Bourke had in view — to wit, that the latter intended to establish a club there. That being the case, it is difficult to understand for what reason Mr. Akers-Douglas has now decided that any member of the public can use the chairs, tables, and tent belonging to the "Summer Club," can insist upon the club servants attending upon him, and can compel them to supply him with refreshments. Mr. Akers-Douglas should have thought of the consequences before he granted the permission.<ref>"Marmaduke." "Court and Club." The ''Graphic'' 04 July 1896, Saturday: 14 [of 32], Col. 1b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18960704/029/0014. Print p. 14.<blockquote></blockquote></ref></blockquote>'''1896 August 10, Monday''', the Morning Leader reported that the Hon. Algernon Bourke, for the Foreign Office, received Li Hung Chang at St. Paul's:<blockquote>At St. Paul's Li Hung was received by Field-Marshal Simmons, Colonel Lane, the Hon. Algernon Bourke, of the Foreign Office (who made the necessary arrangements for the visit) and Canon Newbolt, on behalf of the Dean and Chapter. A crowd greeted Li with a cheer as he drove up in Lord Lonsdale’s striking equipage, and his Excellency was carried up the steps in an invalid chair by two stalwart constables. He walked through the centre door with his suite, and was immediately conducted by Canon Newbolt to General Gordon’s tomb in the north aisle, where a detachment of boys from the Gordon Home received him as a guard of honor. Li inspected the monument with marked interest, and drew the attention of his suite to the remarkable likeness to the dead hero. He laid a handsome wreath of royal purple asters, lilies, maidenhair fern, and laurel, tied with a broad band of purple silk, on the tomb. The visit was not one of inspection of the building, but on passing the middle aisle the interpreter called the attention of His Excellency to the exquisite architecture and decoration of the chancel. Li shook hands in hearty English fashion with Canon Newbolt and the other gentlemen who had received him, and, assisted by his two sons, walked down the steps to his carriage. He returned with his suite to Carlton House-terrace by way of St. Paul’s Churchyard, Cannon-st., Queen Victoria-st., and the Embankment.<ref>"At St. Paul's." ''Morning Leader'' 10 August 1896, Monday: 7 [of 12], Col. 2b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18960810/134/0007. Print p. 7.</ref></blockquote>'''1896 August 19, Wednesday''', the ''Edinburgh Evening News'' reported on the catering that White's Club and Mr Algernon Bourke arranged for the visiting Li Hung Chang:<blockquote>It is probably not generally known (says the "Chef") that Mr Algernon Bourke, manager of White's Club, London, has undertaken to the whole of the catering for our illustrious visitor front the Flowery Land. Li Hung Chang has five native cooks in his retinue, and the greatest good fellowship exists between them and their English ''confreres'', although considerable difficulty is experienced in conversation in understanding one another's meaning. There are between 40 and and 50 to cater for daily, besides a staff about 30; that Mr Lemaire finds his time fully occupied. The dishes for his Excellency are varied and miscellaneous, and from 14 to 20 courses are served at each meal. The bills of fare contain such items as bird's-nest soup, pigs' kidneys stewed in cream, boiled ducks and green ginger, sharks' fins, shrinips and prawns stewed with leeks and muscatel grapes, fat pork saute with peas and kidney beans. The meal usually winds with fruit and sponge cake, and freshly-picked green tea as liqueur.<ref>"Li Hung Chang's Diet." ''Edinburgh Evening News'' 19 August 1896, Wednesday: 3 [of 4], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18960819/057/0003.</ref></blockquote> '''1896 November 6, Friday''', Algernon Bourke was on the committee for [[Social Victorians/Timeline/1896#1896 November 6, Friday|the Prince's Club ice-skating rink, which opened on this day]].<p> '''1896 November 25, Wednesday''', Mr. and Mrs. Algernon Bouke attended [[Social Victorians/Timeline/1896#23 November 1896, Monday23 November 1896, Monday|Lord and Lady Burton's party for Derby Day]].<p> '''1896 December 4, Friday''', the Orleans Club at Brighton was robbed:<blockquote>The old building of the Orleans Club at Brighton, which opens its new club house at 33, Brunswick-terrace to-day, was the scene of a very ingenious burglary during the small hours of yesterday morning. The greater portion of the club property had already been removed to the new premises, but Mr Algernon Bourke, his private secretary, and some of the officials of the club, still occupied bed-rooms at the house in the King’s-road. The corner shop of the street front is occupied by Mr. Marx, a jeweller in a large way of business, and upon his manager arriving at nine o'clock he discovered that the place had been entered through hole in the ceiling, and a great part of a very valuable stock of jewelry extracted. An examination of the morning rooms of the club, which runs over Mr. Marx's establishment reveal a singularly neat specimen of the burglar's art. A piece of the flooring about 15in square had been removed by a series of holes bored side by side with a centre-bit, at a spot where access to the lofty shop was rendered easy by a tall showcase which stood convemently near. A massive iron girder had been avoided by a quarter of an inch, and this circumstance and the general finish of the operation point to an artist in his profession, who had acquired an intimate knowledge of the premises. The club doors were all found locked yesterday morning, and the means of egress adopted by the thief are at present a mystery.<ref>"Burglary at Brighton." ''Daily Telegraph & Courier'' (London) 05 December 1896, Saturday: 5 [of 12], Col. 7a [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18961205/090/0005. Print title: ''Daily Telegraph''; p. 5.</ref></blockquote> '''1896 December 10, Thursday''', Guendoline Bourke was present to help staff a stall at the [[Social Victorians/Timeline/1896#10 December 1896, Thursday|Irish Industries Exhibition and Sale, Brighton]]. '''1897 July 2, Friday''', the Hon. A. and Mrs. A. Bourke and Mr. and Mrs. Bourke attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House. '''1897 July 11–16, week of''', a dog of Guendoline Bourke's won a prize at the [[Social Victorians/Timeline/1897#11–16 July 1897, Week Of|Ladies' Kennel Association show in the Royal Botanic Gardens in Regent's Park]].<p> '''23 July 1897 — or 30 July 1897 – Friday''', Guendonline Bourke attended [[Social Victorians/Timeline/1897#23 July 1897, Friday|Lady Burton's party at Chesterfield House]]. <blockquote>Far the prettiest women in the room were Lady Henry Bentinck (who looked perfectly lovely in pale yellow, with a Iong blue sash; and Mrs. Algernon Bourke, who was as smart as possible in pink, with pink and white ruchings on her sleeves and a tall pink feather in her hair.<ref>"Lady Burton's Party at Chesterfield House." ''Belper & Alfreton Chronicle'' 30 July 1897, Friday: 7 [of 8], Col. 1c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004151/18970730/162/0007. Print title: ''Belper and Alfreton Chronicle''; n.p.</ref></blockquote> '''1897 October 30, Saturday''', ''Black and White'' published J.P.B.'s "The Case of Mrs. Elliott,"<ref name=":13">J.P.B. "The Case of Mrs. Elliott." ''Black & White'' 30 October 1897, Saturday: 12 [of 34], Cols. 1a–2b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004617/18971030/036/0012. Print title ''Black and White'', p. 542.</ref> an odd short short story in which the Honourable Algernon Bourke Herriott is "rude to Mrs. Elliott."<ref name=":13" />{{rp|Col. 2b}} J.P.B. echoes the biographical Algernon Bourke's career in the stock market when Mrs. Christine Elliott does not even simulate interest in her husband's bicycling: "a soul is a grievous burthen for a stockbroker's wife."<ref name=":13" />{{rp|Col. 2a}} The Hon. Algy<blockquote>was a senior member of several junior clubs. A woman had dubbed him once "a rip with a taste for verses." The description was severe, but not unwarranted. His was a pretty pagan sensualism, though, singing from a wine palate to Church music. For the rest, he had just imagination enough to despise mediocrity.<ref name=":13" />{{rp|Col. 2a}}</blockquote> '''1898 January 5, Wednesday''', the ''Irish Independent'' reported that "Mr Algernon Bourke, the aristocratic stock broker ... was mainly responsible for the living pictures at the Blenheim Palace entertainment.<ref>"Mr Algernon Bourke ...." ''Irish Independent'' 05 January 1898, Wednesday: 6 [of 8], Col. 2c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001985/18980105/115/0006.</ref><p> '''1899 January 10, Tuesday''', the Brighton Championship Dog Show opened:<blockquote>Princess of Wales a Winner at the Ladies’ Kennel Club Show. [Exclusive to "The Leader.") The Brighton Championship Dog Show opened in the Dome and Corn Exchange yesterday, and was very well patronised by visitors and exhibitors. Among the latter was H.R.H. the Princess of Wales, who did very well; and others included Princess Sophie Duleep Singh, Countess De Grey, Sir Edgar Boehm, the Hon Mrs. Algernon Bourke, Lady Cathcart, Lady Reid, Mr. Shirley (chairman of the Kennel Club), and the Rev. Hans Hamiiton (president of the Kennel Club). The entry of bloodhounds is one of the best seen for some time; the Great Danes are another stronyg lot; deerhounds are a fine entry, all good dogs, and most of the best kennels represented; borzois are another very stylish lot. The bigger dogs are, as usual, in the Corn Exchange and the "toy" dogs in the Dome. To everyone's satsfaction the Princess of Wales carried off two first prizes with Alex in the borzois class.<ref>"Dogs at Brighton." ''Morning Leader'' 11 January 1899, Wednesday: 8 [of 12], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18990111/142/0008. Print p. 8.</ref></blockquote>'''1899 January 11, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1899#11 January 1899, Wednesday|a luncheon Stanfield-hall, home of Mr. and Mrs. Basil Montogomery, for Princess Henry of Battenberg]], that also included the Countess of Dudley (sister of Mrs. Montgomery), General Oliphant, and the Mayor and Mayoress of Romsey. '''1899 February 7, Tuesday''', Guendoline Bourke was a member of the very high-ranking committee organizing a [[Social Victorians/Timeline/1899#1899 February 7, Tuesday|ball at the Hotel Cecil on 7 February 1899]]. '''1899 June 1, Thursday''', the Hon. Algernon and Guendoline Bourke attended the wedding of her brother, Sloane Stanley and Countess Cairns at Holy Trinity Church, Brompton.<ref>"Marriage of Mr. Sloane Stanley and Countess Cairns." ''Hampshire Advertiser'' 03 June 1899, Saturday: 6 [of 8], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18990603/049/0006. Print p. 6.</ref> '''1899 July 1, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1899#1 July 1899, Saturday|meeting in London at the Duke of Westminster's Grosvenor House]] about preserving Killarney as part of the National Trust and seems to have been acting for someone who wanted to purchase the Muckross Estate.<p> '''1899 October 19, Thursday''', the Hon. Algernon Bourke had a bankruptcy hearing:<blockquote>The public examination of the Hon. Algernon Bourke was held before Mr Registrar Giffard yesterday, at the London Bankruptcy Court. The debtor, described as proprietor of a St. James's-street club, furnished a statement of affairs showing unsecured debts £13,694 and debts fully secured £12,800, with assets which are estimated at £4,489 [?]. He stated, in reply to the Official Receiver, that he was formerly a member of the Stock Exchange, but had nothing to do with the firm of which he was a member during the last ten years. He severed his connection with the firm in May last, and believed he was indebted to them to the extent of £2,000 or £3,000. He repudiated a claim which they now made for £37,300. In 1889 he became proprietor of White's Club, St. James's-street, and carried it on until January 1st last, when he transferred it to a company called Recreations, Limited. One of the objects of the company was to raise money on debentures. The examination was formally adjourned.<ref name=":9" /></blockquote>'''1899 November 8, Wednesday''', the Hon. Algernon Bourke's bankruptcy case came up again:<blockquote>At Bankruptcy Court, yesterday, the case the Hon. Algernon Bourke again came on for hearing before Mr. Registrar Giffard, and the examination was concluded. The debtor has at various times been proprietor of White’s Club, St. James’s-street, and the Orleans’ Club, Brighton, and also of Willis's Restaurant, King-street, St. James's. He attributed his failure to losses sustained by the conversion of White’s Club and the Orleans' Club into limited companies, to the payment of excessive Interest on borrowed money, and other causes. The liabilities amount to £26,590, of which £13,694 are stated to be unsecured, and assets £4,409.<ref>"Affairs of the Hon. A. Bourke." ''Globe'' 09 November 1899, Thursday: 2 [of 8], Col. 1c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18991109/020/0002. Print p. 2.</ref></blockquote> '''1899 December 23, Saturday''', "Mr. Algernon Bourke has departed for a tour in Africa, being at present the guest of his brother in Tunis."<ref>"The Society Pages." ''Walsall Advertiser'' 23 December 1899, Saturday: 7 [of 8], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001028/18991223/143/0007. Print p. 7.</ref> '''1900 February 15, Thursday''', Miss Daphne Bourke, the four-year-old daughter of the Hon. Algernon and Mrs. Bourke was a bridesmaid in the wedding of Enid Wilson and the Earl of Chesterfield, so presumably her parents were present as well.<ref>"London Day by Day." ''Daily Telegraph'' 15 February 1900, Thursday: 8 [of 12], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/19000215/175/0008. Name in British Newspaper Archive: ''Daily Telegraph & Courier'' (London). Print p. 8.</ref> '''1900 September 16''', the Hon. Algernon Bourke became the heir presumptive to the Earldom of Mayo when his older brother Captain Hon. Sir Maurice Archibald Bourke died. '''1900 October 06, Saturday''', the ''Weekly Irish Times'' says that Mr. Algernon Bourke, now heir presumptive to the earldom of Mayo, "has been for some months lately staying with Mr. Terence Bourke in Morocco."<ref>"Society Gossip." ''Weekly Irish Times'' 06 October 1900, Saturday: 14 [of 20], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001684/19001006/121/0014. Print p. 14.</ref><p> '''1901 May 30, Thursday''', the Hon. Mrs. Algernon Bourke attended the fashionable [[Social Victorians/Timeline/1900s#1901 May 30, Thursday|Ladies' Kennel Association Dog Show at the Botanic Garden]].<p> '''1901 July 4, Thursday''', Guendoline and Daphne Bourke attended a children's party hosted by the Countess of Yarborough:<blockquote>The Countess of Yarborough gave a charming children's party on Thursday (4th) afternoon at her beautiful house in Arlington Street. The spacious ballroom was quite filled with little guests and their mothers. Each little guest received a lovely present from their kind hostess. The Duchess of Beaufort, in grey, and with a large black picture hat, brought her two lovely baby girls, Lady Blanche and Lady Diana Somerset, both in filmy cream [Col. 2b–3a] lace frocks. Lady Gertrude Corbett came with her children, and Ellen Lady Inchiquin with hers. Lady Southampton, in black, with lovely gold embroideries on her bodice, brought her children, as also did Lady Heneage and Mr. and Lady Beatrice Kaye. Lady Blanche Conyngham, in écru lace, over silk, and small straw hat, was there; also Mrs. Smith Barry, in a lovely gown of black and white lace. The Countess of Kilmorey, in a smart grey and white muslin, brought little Lady Cynthia Needham, in white; Mrs. Arthur James, in black and white muslin; and the Countess of Powys, in mauve silk with much white lace; Lady Sassoon, in black and white foulard; Victoria Countess of Yarborough, came on from hearing Mdme. Réjane at Mrs. Wernher's party at Bath House; and there were also present Lord Henry Vane-Tempest, the Earl of Yarborough, Lady Naylor-Leyland's little boys; the pretty children of Lady Constance Combe, Lady Florence Astley and her children, and Lady Meysey Thompson (very smart in mauve and white muslin) with her children; also Hon. Mrs. Algernon Bourke, in pale grey, with her pretty little girl.<ref>"The Countess of Yarborough ...." ''Gentlewoman'' 13 July 1901, Saturday: 76 [of 84], Col. 2b, 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010713/381/0076. Print p. xxxvi.</ref></blockquote>'''1901 July 20, Saturday''', the ''Gentlewoman'' published the Hon. Mrs. Algernon Bourke's portrait (identified with "Perthshire") in its 3rd series of "The Great County Sale at Earl's Court. Portraits of Stallholders."<ref>"The Great County Sale at Earl's Court. Portraits of Stallholders." ''Gentlewoman'' 20 July 1901, Saturday: 31 [of 60], Col. 4b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010720/141/0031. Print n.p.</ref> Their daughter Daphne appears in the portrait as well.<p> '''1901 September 12, Thursday''', Mrs. Guendoline Bourke's name is listed as Gwendolen Bourke, but the spelling is not what she objected to:<blockquote>Mr. Underhill, the Conservative agent, mentioned to the Revising Barrister (Mr. William F. Webster) that the name of the Hon. Mrs. Gwendolen Bourke was on the list in respect of the house, 75, Gloucester-place. The lady had written to him to say that she was the Hon. Mrs. Algernon Bourke and that she wished that name to appear on the register. In reply to the Revising Barrister, Mr. Underhill said that “Algernon” was the '''name the lady’s husband'''. Mr. Cooke, the rate-collector, said that Mrs. Bourke had asked to be addressed Mrs. Algernon Bourke, but that the Town Clerk thought the address was not a correct one. The lady signed her cheques Gwendolen.” Mr. Underhill said the agents frequently had indignant letters from ladies because they were not addressed by their husband’s Christian name. The Revising Barrister — lf a lady gave me the name of Mrs. John Smith I should say I had not got the voter’s name. The name Gwendolen must remain.<ref>"Ladies’ Names." ''Morning Post'' 12 September 1901, Thursday: 7 [of 10], Col. 3a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19010912/130/0007. Print p. 7.</ref></blockquote> '''1902 September 4, Thursday''', the ''Daily Express'' reported that "Mrs. Algernon Bourke is staying with Lord and Lady Alington at Scarborough."<ref>"Onlooker." "My Social Diary." "Where People Are." ''Daily Express'' 04 September 1902, Thursday: 5 [of 8], Col. 1b? [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004848/19020904/099/0005. Print p. 4, Col. 7b [of 7].</ref><p> '''1902 October 24, Friday''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#24 October 1902, Friday|opened the Prince's ice-skating rink for the season]], which he had been doing since 1895.<p> '''1902 October 31, Friday''', the [[Social Victorians/Timeline/1900s#31 October 1902, Friday|7th opening of the Prince's Skating Club]]. Guendoline Bourke was on the Women's Committee and Algernon Bourke was on the Men's.<p> '''1902 December 9, Tuesday''', Guendonline Bourke attended [[Social Victorians/Timeline/1900s#9 December 1902, Tuesday|Lady Eva Wyndham-Quin's "at home," held at the Welch Industrial depot]] for the sale Welsh-made Christmas gifts and cards. Bourke wore "a fur coat and a black picture hat."<ref>"A Lady Correspondent." "Society in London." ''South Wales Daily News'' 11 December 1902, Thursday: 4 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000919/19021211/082/0004. Print p. 4.</ref><p> '''1903 March 17, Tuesday''', Guendoline Bourke staffed a booth at a [[Social Victorians/Timeline/1900s#1903 March 17, Tuesday|sale of the Irish Industries Association]] on St. Patrick's Day with [[Social Victorians/People/Mayo|Lady Mayo]], [[Social Victorians/People/Dudley|Georgina Lady Dudley]] and [[Social Victorians/People/Beresford|Miss Beresford]]. A number of other aristocratic women were also present at the sale in other booths, including [[Social Victorians/People/Londonderry|Lady Londonderry]] and [[Social Victorians/People/Lucan|Lady Lucan]].<p> '''1903 June 23, Tuesday''', Guendoline and Daphne Bourke were invited to a [[Social Victorians/Timeline/1900s#1903 June 23, Tuesday|children's party at Buckingham Palace for Prince Eddie's birthday]].<p> '''1905 February 17, Friday''', the Dundee ''Evening Post'' reported that Algernon Bourke "set up a shop in Venice for the sale of art treasures and old furniture."<ref>"Social News." Dundee ''Evening Post'' 17 February 1905, Friday: 6 [of 6], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000582/19050217/105/0006. Print p. 6.</ref><p> '''1905, last week of July''', Guendoline Bourke and daughter Daphne Bourke — who was 10 years old — attended [[Social Victorians/Timeline/1900s#Last week of July, 1905|Lady Cadogan's children's party at Chelsea House]]. Daphne was "One of loveliest little girls present."<ref>"Court and Social News." ''Belfast News-Letter'' 01 August 1905, Tuesday: 7 [of 10], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/19050801/157/0007. Print p. 7.</ref><p> '''1913 May 7, Wednesday''', Guendoline Bourke presented her daughter Daphne Bourke at court:<blockquote>Mrs. Algernon Bourke presented her daughter, and wore blue and gold broché with a gold lace train.<ref>"Social and Personal." London ''Daily Chronicle'' 08 May 1913, Thursday: 6 [of 12], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/19130508/120/0006. Print p. 6.</ref></blockquote> The ''Pall Mall Gazette'' has a description of Daphne Bourke's dress, but what exactly "chiffon [[Social Victorians/Terminology#Hoops|paniers]]" means in 1913 is not clear:<blockquote>Court dressmakers appear to have surpassed all previous records in their efforts to make the dresses for to-night’s Court as beautiful as possible. Noticeable among these is the dainty presentation gown to be worn by Miss Bourke, who will be presented by her mother, the Hon. Mrs. Algernon Bourke. This has a skirt of soft white satin draped with chiffon [[Social Victorians/Terminology#Hoops|paniers]] and a bodice veiled with chiffon and trimmed with diamanté and crystal embroidery. Miss Bourke’s train, gracefully hung from the shoulders, is of white satin lined with pale rose pink chiffon and embroidered with crystal and diamanté.<ref>"Fashion Day by Day. Lovely Gowns for To-night's Court." ''Pall Mall Gazette'' 07 May 1913, Wednesday: 13 [of 18], Col. 1a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/19130507/199/0013. Print n.p.</ref></blockquote> '''1904 September 15, Thursday''', according to what was at the time called the ''Irish Daily Independent and Nation'', Algernon Bourke was living in Venice and not in the UK at this point:<blockquote>Algernon Bourke, who usually lives in Venice, has spent some time in England during the present summer, and has now gone on a fishing expedition to Sweden, accompanied by his brother, Lord Mayo. Lady Mayo has been staying meanwhile in Ireland, and has had a visit from her mother, Lady Maria Ponsonby, who is a sister of Lend Obventry.<ref name=":10">"Society Notes." ''Irish Independent'' 15 September 1904, Thursday: 4 [of 8], Col. 5b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001986/19040915/131/0004. Print title: ''Irish Daily Independent and Nation'', p. 4.</ref></blockquote>'''1909 May 22, Saturday''', Algernon Bourke appears to have been living in Pisa. A columnist for the ''Queen'' reported on the Royal School of Art Needlework:<blockquote>Lady Leconfield [?] was there, also her sister-in-law, the Dowager Lady Mayo, only just back from her winter on the Continent, when she spent most of the time at Pisa, where her son Mr Algernon Bourke has also been staying. The latter is a great connoisseur as regards [art?] notably in what is really good in the way of old Italian sculpture and carving. He and his handsome wife have a place near to Putney, and this winter again Mr Bourke, as the result of his Italian travels, has been sending home such relics of the old Italian palace gardens as as stone and marble carved vases, garden seats, and what-not of the kind — not all for himself and his own gardens by any means, I fancy; but his friends, relying on his knowledge in such matters, get him when abroad to choose for [them?] the adornment of their English terraces and gardens.<ref>"My Social Diary." The ''Queen'' 22 May 1909, Saturday: 31 [of 86], Col. 1b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/19090522/203/0031. Print p. 871.</ref></blockquote> == Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == According to both the ''Morning Post'' and the ''Times'', the Hon. Algernon Bourke was among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#"Oriental" Procession|"Oriental" procession]] at the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].<ref name=":2" /><ref name=":3" /> Based on the people they were dressed as, Guendonine Bourke was probably in this procession but it seems unlikely that Algernone Bourke was. [[File:Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a headdress and a very large fan|Hon. Guendoline Bourke as Salammbô. ©National Portrait Gallery, London.]] === Hon. Guendoline Bourke === [[File:Alfons Mucha - 1896 - Salammbô.jpg|thumb|left|alt=Highly stylized orange-and-yellow painting of a bare-chested woman with a man playing a harp at her feet|Alfons Mucha's 1896 ''Salammbô''.]] Lafayette's portrait (right) of "Guendoline Irene Emily Bourke (née Sloane-Stanley) as Salammbô" in costume is photogravure #128 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":4">"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Hon. Mrs. Algernon Bourke as Salammbo."<ref>"Mrs. Algernon Bourke as Salammbo." ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158491/Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.</ref> ==== Newspaper Accounts ==== The Hon. Mrs. A. Bourke was dressed as * Salambo in the Oriental procession.<ref name=":2">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref><ref name=":3">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref> * "(Egyptian Princess), drapery gown of white and silver gauze, covered with embroidery of lotus flowers; the top of gown appliqué with old green satin embroidered blue turquoise and gold, studded rubies; train of old green broché."<ref>“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. 40, Col. 3a}} *"Mrs. A. Bourke, as an Egyptian Princess, with the Salambo coiffure, wore a flowing gown of white and silver gauze covered with embroidery of lotus flowers. The top of the gown was ornamented with old green satin embroidered with blue turquoise and gold, and studded with rubies. The train was of old green broché with sides of orange and gold embroidery, and from the ceinture depended long bullion fringe and an embroidered ibis."<ref>“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 3b}} ==== Salammbô ==== Salammbô is the eponymous protagonist in Gustave Flaubert's 1862 novel.<ref name=":5">{{Cite journal|date=2024-04-29|title=Salammbô|url=https://en.wikipedia.org/w/index.php?title=Salammb%C3%B4&oldid=1221352216|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Salammb%C3%B4.</ref> Ernest Reyer's opera ''Salammbô'' was based on Flaubert's novel and published in Paris in 1890 and performed in 1892<ref>{{Cite journal|date=2024-04-11|title=Ernest Reyer|url=https://en.wikipedia.org/w/index.php?title=Ernest_Reyer&oldid=1218353215|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Ernest_Reyer.</ref> (both Modest Mussorgsky and Sergei Rachmaninoff had attempted but not completed operas based on the novel as well<ref name=":5" />). Alfons Mucha's 1896 lithograph of Salammbô was published in 1896, the year before the ball (above left).[[File:Algernon Henry Bourke Vanity Fair 20 January 1898.jpg|thumb|alt=Old colored drawing of an elegant elderly man dressed in a 19th-century tuxedo with a cloak, top hat and formal pointed shoes with bows, standing facing 1/4 to his right|''Algy'' — Algernon Henry Bourke — by "Spy," ''Vanity Fair'' 20 January 1898]] === Hon. Algernon Bourke === [[File:Hon-Algernon-Henry-Bourke-as-Izaak-Walton.jpg|thumb|left|alt=Black-and-white photograph of a man richly dressed in an historical costume sitting in a fireplace that does not have a fire and holding a tankard|Hon. Algernon Henry Bourke as Izaak Walton. ©National Portrait Gallery, London.]] '''Lafayette's portrait''' (left) of "Hon. Algernon Henry Bourke as Izaak Walton" in costume is photogravure #129 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":4" /> The printing on the portrait says, "The Hon. Algernon Bourke as Izaak Walton."<ref>"Hon. Algernon Bourke as Izaak Walton." ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158492/Hon-Algernon-Henry-Bourke-as-Izaak-Walton.</ref> This portrait is amazing and unusual: Algernon Bourke is not using a photographer's set with theatrical flats and props, certainly not one used by anyone else at the ball itself. Isaak Walton (baptised 21 September 1593 – 15 December 1683) wrote ''The Compleat Angler''.<ref>{{Cite journal|date=2021-09-15|title=Izaak Walton|url=https://en.wikipedia.org/w/index.php?title=Izaak_Walton&oldid=1044447858|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Izaak_Walton.</ref> A cottage Walton lived in and willed to the people of Stafford was photographed in 1888, suggesting that its relationship to Walton was known in 1897, raising a question about whether Bourke could have used the fireplace in the cottage for his portrait. (This same cottage still exists, as the [https://www.staffordbc.gov.uk/izaak-waltons-cottage Isaak Walton Cottage] museum.) A caricature portrait (right) of the Hon. Algernon Bourke, called "Algy," by Leslie Ward ("Spy") was published in the 20 January 1898 issue of ''Vanity Fair'' as Number 702 in its "Men of the Day" series,<ref>{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).</ref> giving an indication of what he looked like out of costume. === Mr. and Mrs. Bourke === The ''Times'' made a distinction between the Hon. Mr. and Mrs. A. Bourke and Mr. and Mrs. Bourke, including both in the article.<ref name=":3" /> Occasionally this same article mentions the same people more than once in different contexts and parts of the article, so they may be the same couple. (See [[Social Victorians/People/Bourke#Notes and Question|Notes and Question]] #2, below.) == Demographics == *Nationality: Anglo-Irish<ref>{{Cite journal|date=2020-11-14|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Richard_Bourke,_6th_Earl_of_Mayo&oldid=988654078|journal=Wikipedia|language=en}}</ref> *Occupation: journalist. 1895: restaurant, hotel and club owner and manager<ref>''Cheltenham Looker-On'', 23 March 1895. Via Ancestry but taken from the BNA.</ref> === Residences === *Ireland: 1873: Palmerston House, Straffan, Co. Kildare.<ref name=":7" /> Not Co. Mayo? *1888–1891: 33 Cadogan Terrace, S.W., Kensington and Chelsea, a dwelling house<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970, Register of Voters, 1891.</ref> *1894: 181 Pavilion Road, Kensington and Chelsea<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1894. Via Ancestry.</ref> *1900: 181 Pavilion Road, Kensington and Chelsea<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1900. Via Ancestry.</ref> *1904: Algernon Bourke was "usually liv[ing] in Venice"<ref name=":10" /> *1911: 1911 Fulham, London<ref name=":6" /> *20 Eaton Square, S.W. (in 1897)<ref name=":0">{{Cite book|url=https://books.google.com/books?id=Pl0oAAAAYAAJ|title=Who's who|date=1897|publisher=A. & C. Black|language=en}} 712, Col. 1b.</ref> (London home of the [[Social Victorians/People/Mayo|Earl of Mayo]]) == Family == *Hon. Algernon Henry Bourke (31 December 1854 – 7 April 1922)<ref>"Hon. Algernon Henry Bourke." {{Cite web|url=https://www.thepeerage.com/p29657.htm#i296561|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}</ref> *Guendoline Irene Emily Sloane-Stanley Bourke (c. 1869 – 30 December 1967)<ref name=":1">"Guendoline Irene Emily Stanley." {{Cite web|url=https://www.thepeerage.com/p51525.htm#i515247|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}</ref> #Daphne Marjory Bourke (5 April 1895 – 22 May 1962) === Relations === *Hon. Algernon Henry Bourke (the 3rd son of the [[Social Victorians/People/Mayo|6th Earl of Mayo]]) was the older brother of Lady Florence Bourke.<ref name=":0" /> ==== Other Bourkes ==== *Hubert Edward Madden Bourke (after 1925, Bourke-Borrowes)<ref>"Hubert Edward Madden Bourke-Borrowes." {{Cite web|url=https://www.thepeerage.com/p52401.htm#i524004|title=Person Page|website=www.thepeerage.com|access-date=2021-08-25}} https://www.thepeerage.com/p52401.htm#i524004.</ref> *Lady Eva Constance Aline Bourke, who married [[Social Victorians/People/Dunraven|Windham Henry Wyndham-Quin]] on 7 July 1885;<ref>"Lady Eva Constance Aline Bourke." {{Cite web|url=https://www.thepeerage.com/p2575.htm#i25747|title=Person Page|website=www.thepeerage.com|access-date=2020-12-02}} https://www.thepeerage.com/p2575.htm#i25747.</ref> he became 5th Earl of Dunraven and Mount-Earl on 14 June 1926. == Writings, Memoirs, Biographies, Papers == === Writings === * Bourke, the Hon. Algernon. ''The History of White's''. London: Algernon Bourke [privately published], 1892. * Bourke, the Hon. Algernon, ed., "with a brief Memoir." ''Correspondence of Mr Joseph Jekyll with His Sister-in-Law, Lady Gertrude Sloane Stanley, 1818–1838''. John Murray, 1893. * Bourke, the Hon. Algernon, ed. ''Correspondence of Mr Joseph Jekyll''. John Murray, 1894. === Papers === * Where are the papers for the Earl of Mayo family? Are Algernon Bourke's papers with them? == Notes and Questions == #The portrait of Algernon Bourke in costume as Isaac Walton is really an amazing portrait with a very interesting setting, far more specific than any of the other Lafayette portraits of these people in their costumes. Where was it shot? Lafayette is given credit, but it's not one of his usual backdrops. If this portrait was taken the night of the ball, then this fireplace was in Devonshire House; if not, then whose fireplace is it? #The ''Times'' lists Hon. A. Bourke (at 325) and Hon. Mrs. A. Bourke (at 236) as members of a the "Oriental" procession, Mr. and Mrs. A. Bourke (in the general list of attendees), and then a small distance down Mr. and Mrs. Bourke (now at 511 and 512, respectively). This last couple with no honorifics is also mentioned in the report in the London ''Evening Standard'', which means the Hon. Mrs. A. Bourke, so the ''Times'' may have repeated the Bourkes, who otherwise are not obviously anyone recognizable. If they are not the Hon. Mr. and Mrs. A. Bourke, then they are unidentified. It seems likely that they are the same, however, as the newspapers were not perfectly consistent in naming people with their honorifics, even in a single story, especially a very long and detailed one in which people could be named more than once. #Three slightly difficult-to-identify men were among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#"Oriental" Procession|"Oriental" procession]]: [[Social Victorians/People/Halifax|Gordon Wood]], [[Social Victorians/People/Portman|Arthur B. Portman]] and [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Wilfred Wilson]]. The identification of Gordon Wood and Wilfred Wilson is high because of contemporary newspaper accounts. The Hon. Algernon Bourke, who was also in the Suite of Men, is not difficult to identify at all. Arthur Portman appears in a number of similar newspaper accounts, but none of them mentions his family of origin. #[http://thepeerage.com The Peerage] has no other Algernon Bourkes. #The Hon Algernon Bourke is #235 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; the Hon. Guendoline Bourke is #236; a Mr. Bourke is #703; a Mrs. Bourke is #704. == Footnotes == {{reflist}} 0hglez9ddv1nk03i6sqh00fxnaab7ju 0 264286 2693445 2691822 2024-12-26T23:34:46Z Scogdill 1331941 2693445 wikitext text/x-wiki [[Social Victorians/Timeline/1850s | 1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] [[Social Victorians/Timeline/1880s | 1880s Headlines]] [[Social Victorians/Timeline/1890s | 1890s Headlines]] [[Social Victorians/Timeline/1890 | 1890]] [[Social Victorians/Timeline/1891 | 1891]] [[Social Victorians/Timeline/1892 | 1892]] [[Social Victorians/Timeline/1893 | 1893]] [[Social Victorians/Timeline/1894 | 1894]] [[Social Victorians/Timeline/1895 | 1895]] [[Social Victorians/Timeline/1896 | 1896]] 1897 [[Social Victorians/Timeline/1898 | 1898]] [[Social Victorians/Timeline/1899 | 1899]] [[Social Victorians/Timeline/1900s|1900s]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]] ==Sometime in 1897== The year 1897 was the year of Victoria's Diamond Jubilee. ==January 1897== ===1 January 1897, Friday, New Year's Day=== ===1 January 1897, Wednesday=== [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], along with [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Stanley Wilson]], attended the [[Social Victorians/People/Warwick|Warwickshire]] Hunt Ball.<ref>"Warwickshire Hunt Ball." ''Leamington Courier'' 16 January 1897, Saturday: 5 [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18970116/027/0005 (accessed July 2019).</ref> === 30 January 1897, Saturday === The ''Queen'' reports<blockquote>A successful B<small>AL</small> P<small>OUDRÉ</small> was given recently by M<small>RS</small>. J<small>OHN</small> L<small>ANE</small> S<small>HRUBB</small>, at B<small>OLDRE</small> G<small>RANGE</small>, Lymington, Hants. There were over 140 guests, and dancing was kept up with spirit until nearly 4 a.m. The flowers were lovely both in the house and conservatory, and the general effect of the ladies' ''poudré'' and the gentlemen in uniform and hunt coats was particularly pleasing, and added brilliancy to the scene. The band of the Royal Marine Light infantry played with their usual skill and perfection. The house party included Capt. and Mrs [[Social Victorians/People/Hughes-Onslow|Hughes Onslow]], Mr Philip Crossley, Mr Bontram, Mr Ross Johnson, Mr and Mrs Breverley Shrubb, Mr [[Social Victorians/People/Hughes-Onslow|Somerset Hughes Onslow]]. The hostess looked particularly well in dark green velvet and white satin, trimmed with Genoese guipure antique, and carried a bouquet of Solfetaire roses. Her two daughters were dressed alike, in white satin with jewelled lace, and carried bouquets of pink carnations.<ref>"Entertainments, Balls, &c." The ''Queen'' 30 January 1897 Saturday: 36 [of 84], Col. 3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18970130/234/0036.</ref></blockquote> ==February 1897== 1897 February, [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] was part of a group who visited Dublin Castle to visit the Lord Lieutenant and Countess Cadogan:<blockquote>The following, among other guests, have arrived at Dublin Castle on a visit to the Lord Lieutenant and Countess Cadogan: — The Earl and Countess of Arran, the Countess of Dunraven and the Ladies Wyndham-Quin, the Earl of Portarlington, Viscount and Viscountess Duncannon, Lady Rossmore, Sir Richard and Lady Magdelen / William Bulkeley, Colonel the Hon. Charles and Miss Chrichton, Mrs. Menzies and [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Mr. Mildmay, M.P., and Miss Mildmay, the Hon. Thomas Egerton, and Mr. Portman. On Tuesday night their Excellencies gave a dinner, followed by a ball in St. Patrick’s Hall, and to-day there will be another dinner and ball. Lord and Lady Cadogan and their guests intended to be present at Lady Roberts’s ball at the royal Hospital last night.<ref>"Court and Personal." ''Public Opinion: A Weekly Review of Current Thought and Activity'' 5 February 1897: 177, Col. 1c–2a. ''Google Books'' https://books.google.com/books?id=Y7JEAQAAMAAJ (accessed July 2019).</ref></blockquote> ===7 February 1897, Sunday=== Probably the second week of February: "… in addition to his own concerts [Dolmetsch] took part in William Poel's ''Twelfth Night'' production at the Hall of the Middle Temple, where the play had been performed in 1601. A very distinguished audience were gathered together, among them [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]], sitting as a Bencher of the Inn, [[Social Victorians/People/Princess Louise|Princess Louise]] and the Duke of Teck."<ref>Campbell, Margaret. ''Dolmetsch: The Man and His Work''. U of Washington Press, 1975: 112.</ref> ===10 February 1897, Wednesday=== The [[Social Victorians/People/Louisa Montagu Cavendish|Duchess of Devonshire]]'s second reception for the season, at Devonshire House, Picadilly. ===End of February 1897=== <blockquote>The fun at the end of last week was at Melton where … and Lady Huntington was to entertain Mrs. Hwfa Williams, [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Stanley Wilson]] and [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Lord Stavordale, and Mr. Marjoribanks.<ref name=":2">"Society Gossip." ''Weston-super-Mare Gazette'' 3 March 1897, Wednesday: 3 [of 4], Col. 5a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001444/18970303/038/0003 (accessed July 2019).</ref></blockquote>Hard to tell if there was one or several parties, including at least one ball:<blockquote>… Lady Dudley stayed with lady Gerard for the Hunt Ball. Everyone said it was the smartest country ball they had seen for a long time. It was, indeed, quite like a very smart London one, only much cheerier and brighter-looking on account of al the men’s red coats. All the hunting world had parties for it, and all the women wore their best frocks and their diamonds too. … The two prettiest girls in the room were Miss Enid Wilson in white and Miss Muriel Wilson in white and silver with a soft blue sash.<ref name=":2" /></blockquote> ==March 1897== Sometime in March 1897, Wynn Westcott resigned from the [[Social Victorians/Golden Dawn|Golden Dawn]]. Sometime in March 1897, Wynn Westcott of the [[Social Victorians/Golden Dawn|Golden Dawn]] wrote Frederick Gardner, telling him to ask Florence Farr to "choose a gentleman adept friend" to act as intermediary -- but not W. A. Ayton.<ref>Howe 169.</ref> ==April 1897== ===16 April 1897, Friday=== Good Friday ===18 April 1897, Sunday=== Easter Sunday ==May 1897== Some William Rothenstein drawings made in May 1897 are now at Jesses, [[Social Victorians/Haslemere|Haslemere]].<ref>Campell 133.</ref> ===3 May 1897, Monday=== "Inaugural performance of the New Century Theatre, a rival to the independent, established by William Archer, Elizabeth Robins, H. W. Massingham, and Alfred Sutro, on order to promote experimental drama."<ref>Gibbs, Anthony Matthew. ''A Bernard Shaw Chronology''. ''Author Chronologies'', ed. Norman Page. Palgrave, 2001: 131.</ref> On '''Monday 3 or 10 May 1897''', a dance in London for 450 people to benefit the Italian Hospital. Benoist catered.<blockquote>Under the patronage of the King and Queen of Italy, the Prince and Princess of Naples, the Duke and Duchess of Saxe-Coburg and Gotha, the Marchioness of Lorne, the Duchess of Teck, the Duke and Duchess of Aosta, General Ferrero, and many others, a most successful dance was held on Monday, of last week, at the Institute of Painters in Water Colours, Piccadilly, in aid of the Italian Hospital, at Queen's-square. The guests, who were received by Mme. Ortelei, Lady Seymour, Mrs. Hoffnung-Goldsmith, and Mme. Allatini, included the Duke and Duchess San Germano di Calabrito, General Ferrero, Marchioness Caesar di Sain, Lady Carew, Lady Colin Campbell, Count di Casa Valencia, Lady Jessel, Count di Vaglio, Mr. S. Hoffnung-Goldsmith, Sir Joseph Sebag Montefiore, Count Herschell, Sir Halliday and Lady Macartney, Comm. John Ortelei, Signor and Signorina Bebigani, Signor Strazza, the Messrs. Simmons, Count Sielern, [[Social Victorians/People/Hadik|Count Hadik]], Mr. and Mrs. Silverston. Many handsome gowns were worn. Mme. Ortelei was most becomingly attired in grey, Lady Seymour in black, Lady Colin Campbell also in black, and Miss Hoffnung-Goldsmith in white, Mme. de Rinn was in pale blue, and Mme. Allatini in grey brocade. Benoist catered, as usual, perfectly, and some 450 guests supped at the myriad flower-decked little tables.<ref>"Under the Patronage." The ''Gentlewoman'' 15 May 1897 Saturday: 48 [of 72], Col. 1c [of 3]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970515/248/0048.</ref></blockquote> === 4 May 1897, Tuesday''', beginning at 4:20 p.m.''' === A disastrous fire in Paris broke out in a charity bazaar run and frequented by celebrities and aristocrats that killed and seriously injured many people. The fire had started in the cinematograph next door, whose ether supply for the light flame seems to have exploded, killing and injuring people in that as well. The ''Sheffield Independent'' published eye-witness accounts the next day.<ref>"The Paris Horror. Further Details. Searching for the Dead. Agonising Scenes. Traces of the Duchesse d'Alencon. Narrow Escape of an English Lady. Personal Narrative." ''Sheffield Independent'' 06 May 1897 Thursday: 5 [of 8], Col. 5a-8c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000181/18970506/110/0005.</ref> ===5 May 1897, Wednesday=== Florence Farr called a meeting of the [[Social Victorians/Golden Dawn|Golden Dawn]] (or the Inner Order?), which was held at 62 Oakley Square.<ref>Howe 126.</ref> ===6 May 1897, Thursday=== About the [[Social Victorians/Golden Dawn|Golden Dawn]]: H. C. Morris got Edward Berridge's "pamphlet" with the footnote about Annie Horniman and the handwritten "doggerrel."<ref>Howe 173-74.</ref> === 8 May 1897, Saturday === I believe the big funerals in Paris after the fire on Tuesday must have been held on Saturday 8 May, rather than on 15 May, when the story about them appeared in the ''Illustrated Police News''.<ref>"The Awful Catastrophe at a Charitable Bazaar in Paris. Over a Hundred Noble Ladies Burned to Death. The French Aristocracy Thrown into Mourning. Full Details of the Heart-Rending Calamity." ''Illustrated Police News'' 15 May 1897 Saturday: 3 [of 8], Cols. 1a–5c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000072/18970515/009/0003.</ref> ===9 May 1897, Sunday=== Annie Horniman was living at H. I. Montague Mansions, Portman Square, London.<ref>Horniman, Annie Elizabeth Fredericka. Typescript.</ref> ===15 May 1897, Saturday=== '''1897 May 15–17''', [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] was at a weekend house party at [[Social Victorians/People/Warwick|Warwick Castle]]: "The most interesting Saturday-to-Monday house party this year was at Warwick Castle, where Mr Balfour was the fellow-guests with Lord Rosebery and Mr Asquith. [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]] and Mr Buckle, editor of ‘The Times’ have been among the guests."<ref>''Hull Daily Mail'' 17 May 1897, Monday: 4 [of 6], Col. 5c [of 7]. ''British Newspaper Archive'' (accessed July 1897).</ref> Same house-party at Warwick Castle:<blockquote>The Countess of Warwick entertained the following distinguished house party at Warwick Castle at the end of last week: — The Portuguese Minister, the Earl of Rosebery, Earl of Crewe, Lady Randolph Churchill, Lord and Lady Algernon Gordon Lennox, the Right Hon. A. J. Balfour, the Right Hon. H. and Mrs. Asquith, the Hon. H. Lady Feodorowna Sturt, Mr. and the Hon. Mrs. Maguire, Mr. and Mrs. W. H. Grenfell, Mr. and Mrs. J. Menzies, Miss Muriel Wilson, Lord Kenyon, Lady Gerard, Major-General Arthur Ellis, the Hon. John Baring, the Hon. Sidney Greville, Major Wynn Finch, Mr. Buckle (the “Times”), Mr. Cecil Grenfell, Mr. Warrender, and Mr. T. Byard. The distinguished guests arrived at the Castle on Saturday afternoon. The Earl of Rosebery reached Warwick at four o’clock, and was driven to the Castle by Lady Warwick in her carriage and pair. Mr. Arthur Balfour come down to Leamington by the Zulu express, and rode on this bicycle to the Castle. He looked bronzed and healthy, although he has only recently recovered from an illness. On Sunday morning a number of the guests attended Divine service at St. Mary’s Church. Lord Warwick was present with the Corporation, as Mayor of the borough; and Lady Warwick was accompanied by Lord Rosebery. Lord Crewe, Mrs. Asquith, Mr. and the Hon. Mrs. Rochefort Maguire, Lord Algernon Gordon Lennox, Miss Muriel Wilson, Lady Marjorie Greville, and Miss Hamilton. The Rev. A. C. Irvine, M.A., was the preacher. Lord Rosebery was at once recognized as he left the church with the Castle party. The appearance of the Countess of Warwick at the meeting of the Birmingham Lifeboat Saturday Committee, held at the Council House, Birmingham last week, caused a big flutter among the ladies present. Her ladyship was attired in a striking costume of navy blue serge, faced with military braid, with lappels of the bodice trimmed with yellow silk. The Countess had travelled from London to preside at the meeting, and had to rush away before the proceedings were over in order to get to Warwick, where she was to preside over another gathering. The pupils of Warwick School of Art have been invited by the Countess of Warwick to prepare designs for the cover of the handbook of the Education Section of the Victorian Era Exhibition, of which her ladyship is president. The Earl and Countess of Warwick will, as stated in last week’s “Courier” entertain all the day and Sunday School children at the Castle on the 1st of June. They number about two thousand five hundred. On the 3rd of June the leading burgesses of the town, to the number of about thousand are invited to a garden party. The Lord Mayor and Lady Mayoress of London, the Lord Mayor and Lady Mayoress of Birmingham, the Mayors and Mayoresses of Coventry, Stratford, Leamington, and Sutton Coldfield, will be amongst the guests. The next day the farm tenantry on the estates of the Earl and Countess in Warwickshire, Staffordshire, Somersetshire, Northamptonshire, and Essex, will be entertained, being conveyed to and from Warwick by special trains. Among those who lunched with the Countess of Warwick at the Castle on Wednesday were the Bishop of Worcester, the Hon. T. A. Brassey the Rev. M. Hare, the Rev. Dr. Newman Hall, the Rev. W. J. and Mrs. Mathams, and the Directors of the British and Foreign Sailors' Society.<ref>"Distinguished Guests at Warwick Castle." ''Leamington'' Courier 22 May 1897, Saturday: 7 [of 8], Col. 4b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18970522/032/0007 (accessed July 2019).</ref></blockquote> ===25 May 1897, Tuesday=== Arthur Sullivan's ballet ''Victoria and Merrie England'' opened at the Alhambra Theatre, Leicester Square; Sullivan conducted at least the first performance.<ref name=":0" /> Mrs. [[Social Victorians/People/Oppenheim|Oppenheim]] hosted a "flower ball," which meant that women's costumes represented a flower:<blockquote>THE WORLD OF WOMEN. MRS. OPPENHEIM’S FLOWER BALL. NOW leafy June, the sweet month of roses, has set in right jubilantly in London, it is nice to dwell in imagination on the delicious Flower Ball Mrs. Henry Oppenheim gave on May 25 at her beautiful townhouse in Bruton Street. Is it not captivating to the fancy to learn that the balcony of the magnificent ball-room was arched with evergreens, studded with lovely flowers; that the handsome hostess looked radiant in a Poppy dress with bodice of basket-work in gold embroidery; that Lilian, Duchess of Marlborough, was appropriately attired in lily costume; and that roses garlanded the Duchess of Leeds’s superb brocade and tulle dress; while Mrs. Asquith, the wife of the ex-Home Secretary, also chose English roses? I am glad to know that the dear old Shamrock of Erin was not forgotten by an Irish beauty.<ref>"The World of Women. Mrs. Oppenheim's Flower Ball." ''Penny Illustrated Paper'' 05 June 1897, Saturday: 12 [of 16], Col. 2a, b [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000693/18970605/088/0012.</ref></blockquote> ===27 May 1897, Thursday=== Thursday, 27 May 1897, 2:30 p.m. On Friday, 28 May 1897, in "Court Circular" the London ''Times'' reported on the funeral of [[Social Victorians/People/Augustus Wollaston Franks|Sir Wollaston Franks]]:<blockquote>The funeral of [[Social Victorians/People/Augustus Wollaston Franks|Sir Augustus Wollaston Franks]], F.R.S., president of the Society of Antiquaries, took place yesterday at Kensal-green Cemetery. The service was held at St. Andrew's Church, Ashley-place, the Rev. H. E. Hall, nephew of Sir Augustus Franks, officiating. The chief mourners were Miss Franks, a sister, Mrs. Nesbitt, a sister, the Misses Hall, nieces, Mr. Frederick Franks and Mr. Amyard Hall, nephews, Mr. T. L. Murray Browne, and Mr. C. H. Read, F.R.S. A large number of the members of the Society of Antiquaries and the Royal Society and others attended, among them being Sir E. J. Poynter, P.R.A., Sir Clements Markham (president of the Royal Geographical Society), Sir John Evans, the Bishop of Stepney, Sir Henry Howorth, M.P., Sir J. C. Robinson, Sir Frederic W. Burton, Sir E. Maunde Thompson (librarian, British Museum), the Earl of Crawford, Viscount Dillon, Professor R. K. Douglas, Mr. J. Luard Pattisson, C.B., Mr. B. V. Head, Mr. E. Freshfield, LL.D. (treasurer of the Society of Antiquaries), Mr. Stanley Leighton, M.P., Mr. J. Leighton, Mr. W. Foster (secretary, Hakluyt Society), Mr. E. A. Bond, C.B., Mr. F. G. Hilton Price (director of the Society of Antiquaries), Mr. F. A. Eason (secretary of the Royal Academy), Mr. Philip [Col. 1a/Col. 2b] Norman, Mr. Willis Bund, Mr. H. O. [?] Maxwell Lyte, C.B., Mr. H. B. Wheatley, Mr. C. Purdon Clarke, Mr. W. de G. Birch, and Dr. Hicks. Assembled at the graveside were also Major-General Sir John Donnelly, head of the Science and Art Department, South Kensington, Mr. Everard Green (Herald's College), Mr. Charles Welch (Guildhall library), and Mr. T. Armstrong (Science and Art Department).<ref>"Court Circular." ''Times'' [London, England] 28 May 1897: 12, Col. 1c–2a. ''The Times Digital Archive''. Web. 2 May 2013.)</ref></blockquote> On Friday, 28 May 1897, in "The Queen's Reign" the London ''Times'' reported the following "[[Social Victorians/19thC Freemasonry|masonic service]]":<blockquote>A masonic service was held yesterday, at evensong, in the Collegiate Church of St. Saviour, Southwark, to celebrate the record reign of her Majesty and to assist the restoration fund of the church. By special dispensation of the Grand Master the brethren were permitted to attend the service in full masonic attire, and a very impressive scene was thus witnessed by the congregation. Among those present were Lord Lathom (pro-Grand Master), Lord Llangattock, Lord Connemara, Lord Harlech, Mr. Justice Bruce, and also the following brethren:— Mr. W. L. Jackson, M.P., Sir Offley Wakeman, Mr. Causton, M.P., Mr. H. Bancroft, Mr. E. Terry, Mr. Lionel Brough, Colonel A. B. Cook, Mr. R. Eve, Mr. R. Loveland Loveland (president of the Board of General Purposes), the Rev. Dr. Currie, Mr. Letchworth (grand secretary), and Mr. W. Lake (assistant grand secretary). Before the service the following voluntaries were rendered by organ and orchestra:— "Largo" (Handel), "Idyll" (Battison Haynes), and "occasional" Overture (Handel). Immediately preceding the service a procession was formed, in which the grand officers walked from the Ladye Chapel down the north, to the west end, thence up the nave to the reserved seats under the tower and the east end of the nave. The provincial grand officers proceeded from the parochial offices, and occupied the reserved seats at the east end of the nave. Clerical brethren included in the foregoing procession retired to the vestry on the north side of the choir, and joined the procession of the choir, clergy, and chapter to the choir seats and stalls. Clerical brethren with robes — not grand officers or provincial grand officers — after entering the church proceeded to the north choir aisle, where they waited and joined in the last-mentioned procession. The service was intoned by Archdeacon Sinclair (P.G.C.) and Canon Thompson. The opening hymn was "O Jerusalem the blissful," and the proper psalms followed. The special lessons (Haggai ii., 4 to 10 and 1. Cor. iii., 9 to 18) were read by the Rev. Dr. Childe and the Archdeacon of Essex. The Magnificat and Nunc Dimittis were sung to music by Gadsby in C and the anthem was "Lift up your heads" (Handel). The sermon was preached by the Very Rev. Dr. Hole, Dean of Rochester, grand chaplain, from Acts viii., 26, "Sirs, ye are brethren." The financial results of the service were a contribution towards the £7,000 required of about £2,340, including £1,000 from Lord Llangattock, £600 from Mr. Alfred Bevan, over £400 subscribed by the committee, and £320 collected at the service.<ref>"The Queen's Reign." ''Times'' [London, England] 28 May 1897: 12. The Times Digital Archive. Web. 2 May 2013.</ref></blockquote> ==June 1897== ===1 June 1897, Tuesday=== The Dowager Duchess of Marlborough and Lady Sarah Wilson hosted a dinner party and dance:<blockquote>The Dowager Duchess of Marlborough and Lady Sarah Wilson entertained at dinner yesterday evening, at her Grace’s house in Grosvenor-square, the Duke and Duchess of Roxburghe, the Marquis and Marchioness of Londonderry and Lady Helen Stewart, the Earl and Countess of Derby and Lady Isabel Stanley, the Earl of Chesterfield, the Earl of Stradbroke, the Earl of Essex, Lady Georgiana Curzon, the Ladies Margaret and Victoria Innes-Ker, Lady Lilian S. Churchill, Viscount Chrichton, Lord and Lady Wolverton, Lady Gerard, Lord Trevor, Lord Elcho, Sir Samuel and Lady Sophie Scott, Sir Edward and Lady Colebrooke, Lady de Trafford, the Hon. Dudley Marjoribanks, the Hon. Charles Willoughby, the Hon. Claud Willoughby, the Hon. John Baring, the Hon. Seymour Fortescue, Mr. and Mrs. W. H. Grenfell, Mr. and Mrs. J. Menzies, and Miss [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], Captain Ricardo, and Mr. Wilfred Wilson. There was a dance afterwards.<ref>"Court Circular." ''Times'', 2 June 1897, p. 12. ''The Times Digital Archive'', http://tinyurl.galegroup.com/tinyurl/AHQuW0. Accessed 20 June 2019.</ref></blockquote> ===2 June 1897, Wednesday=== Derby Day at Epsom Downs, so the [[Social Victorians/People/Louisa Montagu Cavendish|Luise Friederike Auguste Montagu]], Duchess of [[Social Victorians/People/Devonshire|Devonshire]], hosted a ball at Devonshire House that night? ===5 June 1897, Saturday=== 1897 June 5?, Mrs. Oppenheim's fancy-dress "flower ball": women came dressed as flowers or decorated with particular flowers:<blockquote>Of course you want to know all about Mrs. Oppenheim’s ball, which was undoubtedly very successful. It would very difficult to say who looked best, or who was the best-dressed person there. The hostess herself looked about twenty-five in a poppy gown with golden basket bodice, and Miss Oppenheim, as a harebell, looked very handsome. Lady De Grey, in red roses, looked magnificent, and Lady Kilmorey, as La France rose, was very artistic. Mrs. Walker, as a pansy, Lady Sarah Wilson, as tiger tulip, Lady Newtown-Butler, as an iris, all wore short dresses, and gave one the impression that they had left part of their costume behind. Miss Brassey’s gown was a heap of roses; Lady Lilian Churchill's was covered with forget-me-nots (and was very pretty); Lady Norah Churchill, with a little hat on her head and her short skirt, looked like a little Dresden Shepherdess. Lady Beatrice Butler, as a pimpernel, was lovely; Lady Vivian’s daughters, as violet and sweet peas, were very fresh and pretty. Miss Norah Bourke and Mrs. Lindsay both looked well. Lilian Duchess of Marlborough wore a white satin dress covered with large lilies, and Mrs. Jack Leslie's gown, with a tall flower growing out of a red velvet flower-pot, was very original. Miss [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], as a dandelion, wore, I think, the whole the most successful gown there, and she looked very handsome.<ref>"Society Gossip." ''Weston-super-Mare Gazette'' 5 June 1897, Saturday: 9 [of 12], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001444/18970605/122/0009 (accessed July 2019).</ref></blockquote>A week later, in the reporting of the Duchess of Devonshire's 2 July 1897 fancy-dress ball, is a story that brings Mrs. Oppenheim's ball to show what Louisa Duchess of Devonshire did not do: <blockquote>Mrs. Oppenheim, wife of the well-known financier, gave a flower party, of which great things were expected, and fairly fulfilled. Every lady personated a flower, and got herself up so far as possible to resemble one, or so decked her dress, with [Col. 1C–2A] simulated blossoms as clearly indicate her preference. Society talked flowers for a fall which was a great extension of the proverbial nine days' wonder. When all was said and done the great world discarded flowers, and decided to have no more of them for personal adornment, and as few as possible for tables and reception rooms. So ungrateful can the pampered world become for nature's prodigality.<ref>Cheltenham Looker-On 1897-06-12.</ref></blockquote> ===6 June 1897, Sunday=== Whit Sunday ===11 June 1897, Friday=== [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] attended a house party at Chatsworth House, the country house of the [[Social Victorians/People/Devonshire|Duke and Duchess of Devonshire]] (details on 12 June 1897 and on the page reporting gossip about the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July fancy-dress ball]]). On Saturday, 12 June, the fact that people were talking about the Duchess of Devonshire's upcoming ball was part of the story in the newspaper. === 12 June 1897, Saturday === Two parties took place on this day; according to newspaper reports, costumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]] were discussed. People at the house party at Chatsworth House, the country house of the [[Social Victorians/People/Devonshire|Duke and Duchess of Devonshire]], were talking about the upcoming party:<blockquote>The house party at Chatsworth this week included the Earl and Countess of Mar and Kellie, Lord Charles Montagu, Lord and Lady Gosford, Lord Elcho, the Right Hon. Arthur James Balfour, M.P., Count Mensdorf, of the Austrian Embassy; Miss [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] (Tranby Croft), and Mrs. Menzies. The Daily Mail says it is impossible not to talk about the Duchess of Devonshire's grand ball, for people will discuss scarcely anything else, and although each woman can keep the secret of her own intentions fairly well, she invariably betrays the confidences of her dearest friends; while the men, who are less hopeful of making a sensation, frankly discuss the difficulties in their way, and ask for advice or practical assistance from each of their lady friends. [[Social Victorians/People/Ripon|Lady de Grey]] is going as Zenobia, and is getting her dress from Doucet, I hear, while Worth also is making a great many costumes; but the greatest number are being made in England. The [[Social Victorians/People/Portland|Duchess of Portland]], the [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Duchess of Hamilton]], [[Social Victorians/People/Mar and Kellie|Lady Mar and Kellie]], and Miss [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] are all going to the costumier in Soho-square, and Alias has also been summoned to Marlborough House for a consultation. As to what the different people will wear people seem to change their minds every day, but according to the present report the [[Social Victorians/People/Marlborough|Duke of Marlborough]] will be dressed as Louis Seize, and the [[Social Victorians/People/Spencer Compton Cavendish|Duke of Devonshire]] will probably represent a portrait of Charles V., while [[Social Victorians/People/Gosford|Lady Gosford]], who was to have been Minerva, has now half decided to be a lady of his Court. Mr. Caryl Craven, who is so clever in such matters, is helping the [[Social Victorians/People/Leeds|Duchess of Leeds]] with her dress; in fact, everyone seems pressed into the service, and the result will be one of the most brilliant sights that ever was seen. [[Social Victorians/People/Adderley|Father Adderley]] (the Hon and Rev J Adderley), who always brings his religion up to date, has already denounced the ball from his pulpit, in imitation of an American divine; but he is probably very far wrong in estimating the cost of any one dress at £2,000! It is certain, however, that the ball, what with one thing and another, will run into enormous sums of money, and some ladies are actually having their jewels altered and reset to suit the costume of a single night. There is a Venetian quadrille, a poudré quadrille, two Empire quadrilles, and last, not least, some of the beauties will be dancing an Oriental measure in Eastern dress with floating scarves, and this will be the prettiest and most picturesque feature of the night.<ref>Derbyshire Times and Chesterfield Herald 1897-06-12.</ref></blockquote>The ''Reading Mercury'' mentions talk at a ball hosted by Queen Victoria on the same day about what people intend to wear:<blockquote>The Marchioness of Londonderry and her sister-in-law Lady Aline Beaumont, intend to wear Polish costumes at the Duchess of Devonshire’s fancy ball. The Duke has almost decided to wear a dress copied from a Titian painting of Charles the Fifth. Lady Gosford, his step-daughter, will personate a lady of his court. The Princess of Wales has not yet chosen her dress. This ball is making a great sensation in aristocratic circles.<ref>“Our London Letter. Up and Down.” ''Reading Mercury'' 12 June 1897, Saturday: 8 [of 12], Col. 7c. ''British Newspaper'' ''Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000369/18970612/100/0008.</ref></blockquote> === 19 June 1897, Saturday === On Monday, 21 June 1897, the ''Pall Mall Gazette'' reported a dinner party at the Savoy Hotel hosted by Madame Melba:<blockquote>Mdme. Melba entertained a large party at dinner on Saturday evening at the Savoy Hotel. The tables were most equisitely [sic] decorated with the rarest flowers, and the menus bore the combined flags of England and Australia. The guests included Mr. and Mrs. Kenyon Mason, M. Bemberg, [[Social Victorians/People/Craven|Mr. Caryl Craven]], Mr. Theodore Byard, Miss Ada Crossley, Mr. J. M. Bruce (Melbourne), Mr. Leo Stern, the Misses Donaldson (Australia), and Mr. Landon Ronald. Following the dinner was music, in which the distinguished hostess took part, to the delight of her guests.<ref>"Pall Mall Gazette Office." ''Pall Mall Gazette'' 21 June 1897 Monday: 8 [of 10], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970621/020/0008.</ref></blockquote>The ''British Australasian'' has more detail:<blockquote>Madame Melba entertained a large party of friends at dinner on Saturday evening at the Savoy Hotel. The distinguished singer received the guests in her own salon, and from there the party passed to the grand dining hall, where tables were specially reserved and decorated for the party, which included several of her compatriots. An exquisite array of flowers lent particular beauty to a scene which was much enhanced by a novel and exceedingly pretty scheme of illumination. The menu cards were embellished with the flags of England and Australia, most effectively embossed in gold and colours, and each guest took away a card, which during the evening was made doubly interesting by the addition of the diva's autograph. The dinner was as follows:— Fantaisies Muscovite.<br />Poule au Pat.<br />Veloute à la Reine.<br />Truite à la d'Orleans.<br />Volaille à la Diva.<br />Baron d'Agneau de lait à la Broche.<br />Petits pots à la Francaise [sic].<br />Pommes noissettes.<br />Mousse d'Ecrevisses Rossini.<br />Cailles Rôties aux Feuilles de Vigne.<br />Salade Rachel.<br />Aubergines au Gratin.<br />Peches Glacées Vanille.<br />Friandises. Fruits.<br />Vins.<br />Hochheimer. Bollinger, 1889.<br />Cantenac Brown 1884.<br />Café Turc. Liqueurs. The guests included M. Bemberg, the composer, Baronet Von Zedlitz, Miss Ada Crossley, Mr. Theodore Byard, Miss Dora Mitchell (sister of the hostess), Mr. Landon Ronald, Mr. and Mrs. Kenyon Mason — the latter being a niece of Dr. O'Hara, of Melbourne — Miss Agnes Murphy, [[Social Victorians/People/Craven|Mr. Caryl Craven]], Mr. J. M. Bruce of Melbourne, Misses Donaldson, Madame Guy d'Hardelot (the composer), Mr. Charles Ellis (the diva's American manager), Mr. Leo Stern, and other well-known people. A feature of the dinner was an ice-boat of beautiful design, with the word "Melba" in frozen letters graduating from stem to stern. After the dinner the guests returned to Madame Melba's roems [sic], where the pleasure of the evening was crowned by the hostess herself singing with all the purity of tone and perfection of phrasing which have won her the distinction of "prima donna of the world." Miss Ada Crossley's lovely voice was also heard, and Mr. Theodore Byard contributed two songs, and Mr. Stern a 'cello solo. The game of American Post, with the gifted hostess as Postal Director, closed an evening of unique enjoyment.<ref>["Madame Melba..."]. ''British Australasian'' 24 June 1897 Thursday: 35 [of 68], Col. 1b–c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003365/18970624/150/0035.</ref></blockquote>"American Post" — or, in the U.S., "Post Office" — is still a kissing game played at parties, often among teenagers.<ref>{{Cite journal|date=2022-03-01|title=Post office (game)|url=https://en.wikipedia.org/w/index.php?title=Post_office_(game)&oldid=1074645964|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Post_office_(game).</ref> ===20 June 1897, Sunday=== Accession Day: the official Jubilee Hymn, music by Arthur Sullivan and lyrics by William Waltham How, Bishop of Wakefield, was "used in all churches and chapels"; Sullivan's tune is called Bishopgarth and "was later offered in The Methodist Hymnal as an alternative for the Harvest hymn by William Chatterton Dix To Thee, O Lord, Our Hearts We Raise" (Richards 406). Queen Victoria "attended a Thanksgiving service at St George's Chapel, Windsor, at which ... Sullivan's ... hymn ... [was] performed."<ref>Richards 137.</ref> Later that day, perhaps, Alfred Austin (appointed Poet Laureate after [[Social Victorians/People/William Morris|William Morris]] had turned it down) presented his "Victoria," composed for the occasion, to Victoria. === 21 June 1897, Monday === The events of this day were in London. First the Queen, who was still at Windsor, took the train to Paddington Station. She "hosted a State Banquet at in the State Supper Room at Buckingham Palace"; Mr. J. Sommer, Bandmaster, conducted the Band of Royal Engineers.<ref name=":1">"The Queen's Diamond Jubilee — The Music in 1897." The Classical Reviewer 5 May 2012 <nowiki>http://theclassicalreviewer.blogspot.com/2012/05/queens-diamond-jubilee-music-in-1897.html</nowiki> (accessed August 2020).</ref> A reception followed in the Ballroom for guests who had been invited to the Diamond Jubilee celebration, most of them the usual crowd and heads of state from Europe and the Empire. ===22 June 1897, Tuesday=== Diamond Jubilee Day<ref>Mackenzie-Rogan, Lt. Colonel John. Fifty Years of Army Music. London: Methuen, 1926: 124.</ref>; a "thanksgiving service" was held in St. Paul's Cathedral in honor of Queen Victoria's Diamond Jubilee.<ref>Murphy, Sophia. ''The Duchess of Devonshire's Ball''. Sidgwick & Jackson, 1984: 12.</ref> Murphy describes the procession to St. Paul's:<blockquote>The procession which took place on 22 June was the culmination of the patriotic fervour that inspired the nation in that summer of 1897. The Queen, accompanied by 50,000 troops, was driven through the streets of London for the thanksgiving service outside St Paul's Cathedral, where she was greeted by her family, headed by the Prince and Princess of Wales. The crowds turned out in their thousands. Every window overlooking the six-mile route, every inch of space available on the streets, was filled with cheering, flag-waving subjects, the majority of whom had never known another sovereign.<ref>Murphy 15.</ref></blockquote>Mackenzie-Rogan describes the procession and service like this:<blockquote>The procession [from Buckingham Palace to St. Paul's] was led by the great imperial warrior Field Marshal Lord Roberts and included Canadian Mounties, Jamaica Artillery, Royal Nigerian Constabulary, the Cape Mounted Rifles, the New South Wales Lancers, Trinidad Light Horse, and New Zealand Mounted Troops, along with a variety of Indian troops. The service took place on the steps of St. Paul's, with the Queen remaining seated in her carriage. The choir contained many of the most famous musicians of the day joining in singing: [[Social Victorians/People/Arthur Sullivan|Sir Arthur Sullivan]], Sir Walter Parratt, Dr Hubert Parry, Dr Frederick Bridge, Alberto Randegger, Dr A. H. Mann, Barton McGuckin, John E. West, and Joseph Bennett. Sir / George Martin conducted his Jubilee Te Deum, and this was followed by the intoning of the Lord's Prayer, the singing of All People that On Earth Do Dwell to the familiar tune The Old Hundredth and then the first verse of the national anthem. Then the Archbishop of Canterbury on an impulse called for three cheers for the Queen. They could be heard in Trafalgar Square. The service was accompanied throughout by the military bands of the Royal Artillery and the Royal School of Military Music, Kneller Hall.<ref>Mackenzie-Rogan 137–138.</ref></blockquote>Sir George Martin was organist at St. Paul's and knighted in 1897.<ref name=":1" /> About 15,000 people were in the congregation, but Queen Victoria did not leave her carriage, so some ceremony took place on the steps, including a Te Deum written by Prince Albert before his death. The Queen's carriage then went to Mansion House for a ceremony with the Lord Mayor of the City of London. She was then driven around the city, taking her, essentially, to the people, so they could see her. She then was driven back to Windsor. According to the Classical Reviewer,<blockquote>In the evening, a torchlight procession of boys from Eton School sang for Queen Victoria in the Quadrangle of Windsor Castle and the boys created formations on the ground including the letters ‘V.R’. They were accompanied by the band and drums of the Coldstream Guards, performing a number of songs including ‘Auld Lang Syne” and “God Save The Queen”. Afterwards the boys gave Queen Victoria three cheers.<ref name=":1" /></blockquote>Later that evening was called "Jubilee Night."<ref>Murphy 14.</ref> ===26 June 1897, Saturday=== There was apparently a regular celebration of Arthur Collins' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902. === 28 June 1897, Monday === Queen Victoria hosted an enormous garden party at Buckingham Palace, with many royals and foreign dignitaries in attendance because they were in London for the Diamond Jubilee. Many of the dignitaries, especially from South Asia, seem likely to have been at the [[Social Victorians/1897 Fancy Dress Ball |Duchess of Devonshire's fancy-dress ball]] on 2 July 1897. The ''Morning Post'' covered the garden party with a great deal of detail about the guests who were invited and who attended.<ref>“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4–5 [of 12], Cols. 1a–1c. 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> ===End of June 1897=== A few days before 2 July 1897:<blockquote>A few days before the Devonshire House Ball, Joseph Chamberlain had given a party at which the crush had been so great that Princess Louise, the fourth daughter of Queen Victoria who was married to the Marquis of Lorne, had been overcome and had nearly fallen underfoot. So dense was the crowd at this party that it had been impossible to clear a path for the Prince and Princess of Wales. The Prince was so angry that he left the party without even being received, much to the shame and embarrassment of his hostess.<ref>Murphy 39.</ref></blockquote> ==July 1897== ===2 July 1897, Friday=== The [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]]. Earlier in the day, the [[Social Victorians/Derby Day at Epsom Downs|derby at Epsom Downs]]. === 6 July 1897, Tuesday === Just 4 days after the 2 July fancy-dress ball, Louisa, Duchess of Devonshire and Spencer, Duke of Devonshire hosted a garden party at Devonshire House. No royals are reported as having been present; the list of people who attended is specific and arranged like other kinds of similar parties — including those hosted by Albert Edward, Prince of Wales and Alexandra, Princess of Wales — with the list generally organized by rank, with immediate families listed together, especially parents and a daughter, though adult children with their own titles are listed by them and their general rank. The list of South-Asian dignitaries is specific and useful to know who was in London from South Asia.<blockquote>DEVONSHIRE HOUSE. The Duchess of Devonshire gave a garden party yesterday afternoon at Devonshire House, when a distinguished company assembled. The band of the Scots Guards, under the ''bâton'' of Mr. Dunkerton, performed a spirited selection of music throughout the afternoon. The company arrived in quick succession from four till nearly seven o'clock by the two side entrances to the grounds, as well as the principal entrance in Piccadilly, and the presence of our Indian and Colonial visitors in their picturesque and varied uniforms testified to the far-reaching popularity of the Duke of Devonshire and the hostess. The Maharajah of Kapurthala, the Thakur Sahib of Gondal and the Maharanee, the Maharajah Sir Pertab Singh, Thakur Hari Singh, Kumar Dhopal Singh, Rajah Khetri Singh, Rajah Agit Singh, Raj Kuman Umaid of Shapura, Bijey Singh, Sir Jamsetjee Jejeebhoy and Miss and the Messrs. Jejeebhoy, and the Maha Mudalayar of Kandy, besides the officers of the Imperial Service Troops and the Officers of the Native Cavalry Corps were present. Among those who attended were: Dona Solomon Dias Bandaranaike and Miss Amy Dias Bandaranaike, Senathi Rajah, Deir Senathi Rajah, Don and Donna F. de Zea Bermudez, Prince and Princess Edward of Saxe-Weimar, the Princess Sophia Dhuleep Singh and Countess of Selkirk, the Austrian Ambassador and Countess Deym and Countess Isabella Deym, the French Ambassador and Baroness de Courcel and Mesdlles. de Courcel, the United States Ambassador, the Spanish Ambassador, Countess Casa Valencia and Madlle. and Madlle. Consuelo de Alcala Galiano, the Turkish Ambassador and Madame Anthopoulos, the Belgian Minister, the Chinese Minister, the Japanese Minister and Madame Kato, the Brazilian Minister, the Portuguese Minister, the Netherlands Minister and Baroness de Goltstein, Madame de Staal, Count Hermann Hatzfeldt, Count Albert Mensdorff, Count Costa (Secretary Netherlands Legation), M. Boulatzeel, M. and Madame Geoffray, Count Alexander Munster, the Duke of Hamilton and the Ladies Douglas-Hamilton, the Duchess of Buccleuch, the Duchess of Newcastle, the Duchess of Montrose and Lady Helen Graham, the Duchess of Portland, the Duke of Fife, the Duchess of Roxburghe and Ladies Margaret and Victoria Innes-Ker, the Duchess of Cleveland, the Duchess of Buckingham and Chandos and Lord Egerton of Tatton, the Duke of Grafton, Cardinal Vaughan, the Marchioness of Tweeddale, the Marchioness of Headfort, the Dowager Marchioness of Londonderry, Georgiana Marchioness of Downshire, the Earl of Clarendon and Lady Edith Viiliers, the Earl and Countess of Mayo, the Countess of Carnarvon, the Earl and Countess of St. Germans, the Countess of Belmore and Ladies Corry, the Countess of Lichfield and Lady Bertha Anson and Miss Mills, the Countess of Caledon, the Earl and Countess of Dunraven aud Lady A. Wyndham-Quin, Evelyn Countess Bathurst and Lady Evelyn Bathurst, Countess Cadogan, the Countess of Derby and Lady Isabel Stanley, the Earl and Countess of Coventry and Lady Anne Coventry, the Countess of Gosford, the Earl and Countess of Cork, Earl and Countess Annesley, the Countess of Kintore and Ladies Keith-Falconer, the Earl of Kenmare, the Countess of Strafford and Misses Egerton, the Countess of Yarborough, the Earl and Countess of Listowel and Lady Beatrice Hare, the Countess of Lathom and Ladies Wilbraham, the Countess of Ancaster and Ladies Willoughby, the Earl and Countess of Ellesmere and Ladies Mabel and Catherine Egerton, the Countess d'Hautpoul and Mrs. Iznaga, the Countess of Galloway and Miss Stewart, Victoria Countess of Yarborough and Mr. Richardson, the Dowager Countess of Mayo and Lady Florence Bourke, the Dowager Countess of Harewood and Lady Mary Lascelles, Countess Manvers and Lady Mary Pierrepont, the Countess of Enniskillen and Lady Florence Cole, the Earl of Leven and Melville, Viscount Gort, Viscountess Raincliffe and Lady Mildred Denison, Viscount and Viscountess Halifax, Viscountess Boyne and the Hon. Maud and the Hon. Florence Hamilton Russell, Viscountess Portman and the Hon. Mary Portman, Viscountess Helmsley, Viscount and Viscountess Cross and the Hon. Miss Cross, Viscount and Viscountess Knutsford, the Master of the Rolls and Lady Esher, Lord and Lady Morris and Miss Morris, Lady Helen Grimston and Miss Mackintosh, Lady Rossmore, Lady Reay, Lady Moreton, Emily Lady Ampthill and the Hon. Romola Russell, Lady Florence Duncombe, Lady Templemore and the Hon. Hilda Chichester, Lady and the Misses Walrond, Lady Mabel Howard, Lady Evelyn Macdonald, Lord and Lady Thring, Lord and Lady Alington, Lord and Lady Blythswood, Lord Dynevor, Lady Margaret Graham, Lord and Lady Hopetoun, Lady Harris and the Misses Maxwell, Lord Stanmore and Miss Gordon, Lady Ardilaun, Helen Lady Forbes and Miss Forbes, Lord and Lady Rothschild, Lady Mary Trefusis and Miss Adela Trefusis, Lady Sykes, Lord and Lady Muncaster, Lord Morpeth, Lord Lawrence, Lord Hencage, Chief Justice Way (South Australia), Mary Lady Vivian, Lord and Lady Inchiquin and the Hon. Miss O'Brien, Lady Stratheden and the Hon. Miss Campbell, Lady Belper and the Hon. Norah and the Hon. Lilian Strutt, Lady Lawrence and the Hon. Anna Lawrence and Miss Deichman, Lady Lilian Yorke and Miss Pelly, Lady Fanny Lambart and Miss Lambart, Lady Arthur Hill and Miss Hill, Lord Barnard and Lady Louisa Cecil, Lord and Lady Pirbright, Lady Lurgan, Lady Elizabeth Williamson and Mr. Williamson, Lady Calthorpe and Hon. Misses Calthorpe, Lady Muriel Boyle, Lady Constance Leslie, Lord and Lady Wantage, Lord and Lady Connemara, Lord and Lady John Cecil, Lady Cynthia Graham, Lady Heneage and Hon. Margaret Heneage, Lady Halsbury and Hon. Evelyn Giffard, Lord and Lady Roberts, Lord Shand, Lord and Lady Ashbourne and the Hon. Violet Gibson, Lord and Lady Saltoun, Lady Herschell, Lady Lysons, Lord James and Miss James, Lady Tweedmouth, Lady Castletown, Lady Eustace Cecil and Miss Cecil, Lady Burton, Lord and Lady Hillingdon and Hon. Miss Mills, Lady Ventry and the Hon. Miss de Moleyns, Lady Prinsep, Lady Victoria Russell, the Hon. Mrs. Eliot and Miss Evelyn Eliot, the Hon. Charles and Mrs. Ramsay, the Hon. FitzRoy and Mrs. Stewart, the Hon. F. Gavan and Miss Duffy, the Hon. D. K. Congden and Mrs.Congden (Western Australia), the Hon. Mr. and Mrs. Arthur Elliot, the Hon. Mrs. Mallet, the Hon. Percy and Mrs. Wyndham and Miss Wyndham, the Hon. J. B. Whyte and Miss Whyte, the Hon. Mrs. R. Moreton and Miss E. Moreton, Captain the Hon. Arthur and Mrs. Somerset, the Hon. Mrs. Anstruther and the Hon. Miss Hanbury Tracy, the Hon. Mrs. and Miss Haig, Captain the Hon. A. Bagot, the Hon. Mrs. E. Talbot, the Hon. H. Littleton, the [[Social Victorians/People/Keppel|Hon. G. and Mrs. Keppel]], the Hon. Mrs. W. Farquhar, the Right Hon. R. W. Hanbury, M.P., and Mrs. Hanbury, the Right Hon. Sir William and Lady Marriott, the Right Hon. Ian Hamilton, the Right Hon. Sir John and Lady Lubbock, the Right Hon. Sir Mountstuart and Lady Grant-Duff and Miss Grant-Duff, the Right Hon. Sir Wilfrid and Lady Laurier, the Right Hon. Sir H. and Lady de Villiers, the Right Hon. the Lord Mayor and Lady Mayoress and Miss Faudel-Phillips, the Right Hon. Lord Justice Lopes and the Misses Lopes, the Attorney-General and the Misses Webster, the Right Hon. Sir Richard Paget and Lady and the Misses Paget, Baron and Baroness d'Erlanger, Sir John and Lady Barron and Miss Barron, Sir Henry and Lady Meysey-Thompson, Sir Waiter Peace, Sir Wilfrid Lawson and Miss Josephine Lawson, Sir Thomas and Lady Sutherland, Sir James and the Hon. Lady Miller, Sir Reginald and Lady Anson, Sir John and Lady Bramston, Sir Gordon and Lady Sprigg, Sir Charles and Lady Jessel, Sir George and Lady Petre, Sir James and Lady Mackenzie and Miss Mackenzie, Sir George and Lady Allen, Sir William and Lady Whiteway and Miss Whiteway, Mr. and Lady Moyra Cavendish, Mr. and Lady Aline Beaumont, Mr. and Lady Angela Forbes, Sir Cecil Clementi and Lady Smith, Sir E. M. Nelson, Sir Edward and Lady Hertslet, Sir John Donnelly and Lady and the Misses Donnelly, Sir Frederick and Lady Wigan, Sir Henry Doulton, Sir W. J. Farrer, Sir C. and Lady Douglas Fox, Sir M. W. Collett, Sir George Hayter and Lady Chubb, Sir Richard and Miss Temple, Sir George Arthur, Sir Edward Birkbeck and Miss Jolliffe, Sir Edward and Lady Carbutt, Sir Bartle Frere, Sir William and Lady Quayle Jones, Sir Courtenay and Lady Ilbert and Miss Ilbert, Sir George Young, Sir Arthur Sullivan, Sir Weetman and Lady Pearson, Sir W. and Lady Percival, Sir Frederick and Miss Sanders, Sir Augustus and Lady Adderley, Sir David Gamble, Sir Lewis and Lady M'lver, Sir Frederick Young and Miss Young, Admiral and Lady Edith Adeane and Lady Ida Dalzell, Sir Oswald and the Misses Moseley, Sir Henry and Lady Meysey-Thompson, Mr. Alfred de Rothschild, Mr. L. de Rothschild, Mrs. C. Wilson and Miss Wilson, Madame von André, Mr. W. Ganz, Mr. Frederick Fitch, Mr. and Lady Evelyn Mason, Mr. Algernon Peel, Colonel and the Hon. Mrs. Herbert, Mr. and Mrs. Dunbar Buller, Mr. and Mrs. George Cawston, Mr. Buckle, Mr. and Mrs. Herbert Daw, Mr. James Judd, Mr. Wootton Isaacson, M.P., and Mrs. Isaacson, General and Mrs. Pemberton, Mrs. Graham Murray, Mr. and Mrs. Arthur James, Mr. L. Harcourt, Mrs. Ashurst Morris, Mrs. John Delacour, Dr. and Mrs. Wrench, Mr. Sidney Buxton, M.P., Colonel and the Hon. Mrs. Neeld, M. and Madame Van Raalte, Mr. and Mrs. Albert Sandeman, Mr. Frederick Dutton, Mr. Blyth, Mrs. Buxton and Miss Buxton, Canon and Mrs. Wiiberforce and Miss Wiiberforce, Mr. Kimber, M.P., Mr. Rochfort Maguire, M.P., and Mrs. Maguire, Mr. F. Morgan Harvey, Mr. and Mrs. Scott Boys, Mr. and Mrs. Herbert Stoneham, Mr. H. S. Ashbee, Mr. C. A. Prescott, Mrs. Baker, Mr. Hugh Leonard, Mr. and Mrs. Henry Homewood Crawford, Mr. and Mrs. Paddon, Mr. and Mrs. Alfred Beebe, Mr. T. N. Christie, Mr. T. R. Dewar, Mr. H. M. Simons, Mr. J. Matthey, Mr. J. Mackrell, Mr. and Mrs. W. A. Briscoe, Mr. and Mrs. W. H. Adler, Mr. and Mrs. James R. Bluff, Mr. N. L. Cohen, Mr. William Paterson, Mr. D. M. Fox, Mr. J. Gordon Smith, Mr. Alfred Huth, Miss Harris, Mr. M. P. Grace, Mr. T. W. Aldwinckle, Mr. M. G. Hale. Mr. T. G. Jackson, Mr. Cornelius Hanbury, Mr. A. M'Ilwraith, Mr. and Mrs. Robert Russell, Mrs. Arthur Wilson and Miss [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], Mr. Y. Nakai, Mr. A. d'A. Seneviratne (Colombo), Surgeon-General Ryerson (Canada), Mrs. Gurdon, Mrs. Molesworth, Mr. Bagot Molesworth, Dr. and Mrs. Broadbent, Mr. Christopher Sykes, Captain Harrison, Mrs. Mackenzie and Miss Maud Mackenzie, Mr. Rankin, M.P., and Mrs. Rankin, Mr. Seton Karr, M.P., and Mrs. Seton Karr, Mr. and Mrs. W. Bridgeman, Miss Quain, Mrs. and Miss Walter, Mr. Herbert Praed, Colonel Harris, Mr. and Mrs. Frank Gossett, Mr. and Miss Lowther, Mrs. Macdonald, Mr. and Mrs. Beck, Mr. Alfred Duncombe, Colonel and Mrs. Bevington, M. L. Albu, Mrs. Freeman Murray and Miss Newton, Mr. William Murray, Colonel and Mrs. lnnes, [[Social Victorians/People/Fanny Ronalds|Mrs. Ronalds]], Mrs. and Miss Gully, Mr. Guy Pym, M.P., and Mrs. Pym, Mr. Hamilton Aïdé, Colonel and Mrs. Fludyer, Major Symon, Colonel Nisbet, Major Drummond, Mr. and Mrs. F. C. Dawson, Dr. and Mrs. Maclagan and Miss Maclagan, Mr. Moon, M.P., Mr. and Mrs. W. H. Grenfeli, Mr. and Mrs. North, Mr. and Mrs. W. W. Mitchell and Miss Mitchell (Ceylon), Mr. Cecil Anstruther, Colonel Swaine, Mr. and Mrs. Boulton, Mr. Fleetwood Wilson, Mr. A. F. Wallace, Lieutenant-Colonel Arthur, Captain Abney, Rev. Herbert Rowseil, Mr. Kenny, M.P., and Mrs. and Miss Kenny, Mr. and Mrs. A. H. P. Stoneham, Mr. Robert Duncan, Mr. T. S. Hull, M. and Madame Romero, Mr. J. T. Davis and Mrs. and Miss Davis, Dr. and Mrs. James, Mr. and Mrs. A. H. P. Stoneham [sic repetition], Mr. L. C. Russell Jones, M.P., Mr. G. Lawson Johnston, and many others.<ref>"Devonshire House." ''Morning Post'' 07 July 1897 Wednesday: 7 [of 12], Col. 6a–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970707/069/0007.</ref></blockquote> ===8 July 1897, Thursday=== From this day until the end of the run in December 1897, Sullivan's ballet ''Victoria and Merrie England'' at the Alhambra Theatre, Leicester Square, "included a cinematograph film of the Jubilee procession."<ref name=":0" /> === 11–16 July 1897, Week Of === The Ladies' Kennel Association show in the Royal Botanic Gardens in Regent's Park:<blockquote>The Ladies’ Kennel Association had a fine show last week in the Royal Botanic Gardens at Regent’s Park. The Princess of Wales, says the ''Daily News'', was represented in several classes. In basset-hounds she sent her rough Sandringham Vivian and smooth Zero. Among the black pugs, one of the most largely contested in all the classes, Her Royal Highness’s representative was Black Gin, while to the Dachshund section she sent her favourite Wanghee, which won three prizes at the recent Norwich show. The best of all the Princess’s dogs, however, was her beautiful Borzoi Alex, bred by Mr. Rouseau, and described by Ataman the Great — Outeheschka. Alex, who has already taken seven first and four second prizes, secured two more firsts — that in the challenge class for both sexes and that in the open class for dogs. He also took a special prize a "premiership dog.” Among the prize-takers were the Countess of Carnarvon, Lady Kathleen Pilkington, Lady Cathcart, the [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]], the Hon. Mrs. Morrison, Lady Granville Gordon, the Hon. Mrs. Baillie, Mrs. Panmure Gordon, Mrs. Rowland Ward, and Major Davis, of whose prizewinners and of the prize collie print Snap-shots.<ref>"Ladies' Kennel Association." ''Penny Illustrated Paper'' 17 July 1897, Saturday: 7 [of 16], Col. 2c [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000693/18970717/050/0007. Print p. 39.</ref></blockquote> === 13 July 1897, Tuesday === Mr. Schreiber attended a dance at Montagu House, hosted by the Duchess of Buccleuch. [[Social Victorians/People/Churchill|Mr. "Winstone" Churchill]] is listed immediately after Mr. Schreiber.<blockquote>THE DUCHESS OF BUCCLEUCH'S BALL. The Duchess of Buccleuch gave a dance last evening at Montagu House, Whitehall. The garden was festooned with fairy lamps and Chinese lanterns, and the approaches and paths were outlined with the former. Music was supplied by Herr Wurm's Viennese White Band, and dancing was kept up until an early hour. Among those present were: The Princess Mary Adelaide and the Duke of Teck and Prince Alexander of Teck, Prince and Princess Hohenlohe, Prince and Princess Pless, Princess Sophia Duleep Singh and the Countess of Selkirk, Mr. and Mrs. Whitelaw Reid, the United States Ambassador and Miss Hay, the Austrian Ambassador and Countess Deym and Countess Isabella Deym, the Spanish Ambassador and Countess Casa Valencia, the Portuguese Minister, Mr. Henry White, the Russian Ambassador, the Duchess of Montrose and Lady Helen Graham, the Duchess of Manchester, tbe Duchess of Roxburgh and Lady Victoria lnnes-Ker, the Duchess of Cleveland, the Marchioness of Tweeddale, the Marchioness of Zetland, the Marchioness of Lansdowne and Lady Beatrix Fitzmaurice, the Marchioness of Hastings and Miss Olive Chetwynd, the Dowager Marchioness of Downshire, the Countess of Lichfield and Lady Bertha Anson, Earl Beauchamp, the Countess of Dunmore and Lady Mildred Murray, the Countess of Powis, the Dowager Countess of Harewood, Lady Mary Lascelles and the Hon. Mary Portman, the Countess of Lathom and Lady Edith Wilbraham, the Countess of Jersey and Lady Mary Villiers, the Countess of Antrim and Miss Grenfell, the Countess of Enniskillen and Lady Florence Cole, the Countess of Ancaster and the Ladies Wiiloughby, the Countess of Erne and the Hon. Miss Crichton, the Earl of Granard, Earl and Countess Carrington and the Hon. Bridget Harbord, Countess Grey and Lady Victoria Grey, the Countess of Verulam and Miss Mackintosh, Earl Granville, Viscountess Newport and the Hon. Misses Bridgeman, Viscountess Milton, Viscount and Viscountess Emlyn and Miss Campbell, Viscount Doneraile, Viscountess Duncannon and the Hon. Irene Ponsonby, Viscount and Viscountess Halifax and the Hon. Miss Wood, Lady Mary Trefusis and Miss Adela Trefusis and Miss Carpenter Garnier, the Ladies Egerton, Lord Clinton and the Hon. Miss Trefusis, Lord and Lady George Hamilton, Lady Mabel Howard and Miss Howard, Lady Evelyn Macdonald, Mr. Victor and Lady Evelyn Cavendish, and Miss Egerton, Lord and Lady Stratheden and the Hon. Miss Campbell, Lady Evelyn Goschen and Miss Goschen, Lady Lilian Yorke and Miss Pelly, Lord and Lady Penrhyn and Miss Bromley-Davenport, Mr. and Lady Sybil Smith and Miss Baring, Lady Mary Lygon, Lady Lucy Hicks-Beach and Miss Hicks-Beach, Lady Leconfield and the Hon. Miss Wyndham, Lord Dunluce, Lord and Lady Inchiquin and the Hon. Miss O'Brien, Lord and Lady Burton, Lord Iveagh and the Hon. R. Guinness, Lord B. Blackwood, Lady Arthur Hill and Miss Hill, Lady Templemore and the Hon. Hilda Chichester and Lady Beatrice Meade, Lady Edward Cavendish, the Ladies Spencer Churchill, Lady Florence Astley, Lord Herbert Vane Tempest, Lady Cecil Scott Montagu and the Ladies Ker, Emily Lady Russell and the Misses Russell, Lord Ennismore, Lady E. Balfour, Lady Helen Stewart Murray, Lord George Stewart Murray, Lord and Lady Balfour of Burleigh, the Hon. Jean Bruce, the Lord Mayor and Lady Mayoress and Miss Faudel-Phillips, Lady Mary Fitzwilliam and Miss Elsie Fitzwilliam. Lord Aberdare, Lord Stanmore, Lord Frederick Hamilton, Count and Countess Gleichen, the Hon. Alec Yorke, the Hon. Kenneth Campbell, the Hon. Reginald Coventry and Lady Dorothy Coventry, the Hon. Bertha Lambert and Miss M. Cochrane, the Hon. J. Maxwell Scott and Miss Maxwell Scott, the Hon. Arthur Brodrick, the Hon. Benjamin Bathurst, the Hon. A. Anson, the Hon. N. Hill-Trevor, the Hon. G. Browne Guthrie, the Hon. William Maxwell, the Hon. Gertrude Walsh, the Hon. W. Walsh, the Hon. Mrs. Baillie of Dochfour, the Hon. Schomberg M'Donnell, Sir C. Dalrymple and Miss Dalrymple, Sir Edgar Sebright, Sir George and Miss Sterling, Sir Robert and Lady Frances Gresley, Mr. and Lady Mary Hope and Miss Hope, Captain Arthur Bagot, Mr. George Phipps, Mr. Francis Egerton, Mr. Wilfred Egerton, Mr. John Thynne and the Misses Thynne, Mrs. and Miss Stanley, Miss Mildmay, Miss Margaret and Miss Isabel Thynne, Mr., Mrs., and Miss Hope, Mrs. C. and Miss Ker, Mrs. Grahame Murray and Miss Murray, Captain and Mrs. Makin, Mr. Packe, Colonel and Mrs. Hegan Kennard, Miss Campbell, Miss Finch, Miss Cornwallis West, Mr. Scott Montagu, Mrs. Charles Wilson and Miss Enid Wilson, Mr. J. Cavendish. Mr. Lane-Fox, Mr. and Mrs. Cazalet, Mr. Guy Campbell, Mr. and Lady Moyra Cavendish, Mr. Arthur James, Mr. Donald Cameron, Mr. Milbanke, Mr. Ferguson, Mr. Erskine, Mr. Everard Doyle, Mr. Herbert Praed, Mr. Christopher Sykes, Mr. Leech, Mr. and Mrs. Cavendish Bentinck, Mr. Nigel Campbell, Mr. Vivian, Mr. and Mrs. Ogden Mills, Mr. Cecil Lowther, Mr. A. M. Southey, Mr. Walter Erskine, Mr. Victor Corkran, Mr. Gully, Mr., Mrs., and Miss Fletcher, Captain Doyle, Captain Holford, Mr. Ralli, Mr. Victor Russell, Mr. Frederick Russell, Captain Cook, [[Social Victorians/People/Schreiber|Mr. Schreiber]], [[Social Victorians/People/Churchill|Mr. Winstone Churchill]], Mr. Bailey, Mr. Kemp, and many others.<ref>"The Duchess of Buccleuch's Ball." ''Morning Post'' 14 July 1897, Wednesday: 7 [of 12], Col. 6a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970714/067/0007.</ref></blockquote> === 16 July 1897, Friday === Dinner and ball at Londonderry House, hosted by the Marquis and Marchioness of Londonderry:<blockquote>LONDONDERRY HOUSE. Their Royal Highnesses the Prince and Princess of Wales and Princess Victoria honoured the Marquis and Marchioness of Londonderry by their presence at dinner last evening at Londonderry House. Subsequently a large ball, for which over 1,000 invitations had been issued, was given. Among those who had the honour to join the dinner circle were: The Portuguese Minister, the Duke and Duchess of Devonshire, the Duke and Duchess of Abercorn, the Duke of Roxburghe and Lady Margaret Innes Ker, the Duke and Duchess of Marlborough and Lady Lilian Churchill, the Marchioness of Lansdowne and Lady Beatrix Fitzmaurice, the Earl and Countess of Enniskiilen and Lady Florence Cole, the Earl and Countess of Pembroke and Lady Beatrix Herbert, the Earl and Countess of Derby and Lady Isobel Stanley, the Earl and Countess of Ellesmere and Lady Katherine Egerton, the Earl of Crewe, Georgina Countess of Dudley, Viscount and Viscountess Coke, Viscount Crichton, Lady Moreton, Lord Annaly, Lord Charles Montagu, Lord and Lady Lurgan, Lady Randolph Churchill, Lord Henry Vane-Tempest, Lord Herbert Vane-Tempest. Lady Gerard, the Hon. Dudley Marjoribanks, the Hon. G. Hamilton Russell, the Hon. H. and Mrs. Bourke, Sir Francis and Lady Jeune, Sir Arthur Ellis, Sir Samuel and Lady Sophie Scott, Lady de Trafford, Captain and Lady Sarah Wilson, Major and Lady Guendolen Little, Mr. and Lady Aline Beaumont, Mr. and Mrs. Leopold de Rothschild, Captain Ricardo, Colonel and Mrs. Paget, Mr. and the Hon. Mrs. Beckett, Mr. K. Moncreiffe, Mr. and Mrs. Menzies, Mr. and Mrs. Oppenheim and Miss Oppenheim, Mr. Sykes, Miss Stanley, Miss Chaplin, and Lady Helen Stewart, Lord Castlereagh, and the [[Social Victorians/People/Ancaster#Mr. C. Willoughby|Hon. C. Willoughby]]. Their Royal Highnesses the Duke and Duchess of York arrived for the ball at eleven o'clock, attended by Lady Mary Lygon and the Hon. Derek Keppel, when dancing immediately commenced to Gottlieb's Viennese Orchestra. His Royal Highness Prince Christian of Schleswig-Holstein, the Duke of Teck, Prince Francis and Prince Alexander of Teck, and the Duke of Cambridge, attended by Colonel FitzGeorge were also present. Among others attending were: Princess Henry of Pless and Miss Cornwallis West, the Russian Ambassador, the Austrian Ambassador, Countess Deym and Countess Isabella Deym, the French Ambassador and Baroness de Courcel, Mr. and Mrs. Whitelaw Reid, the United States Ambassador and Mrs. and Miss Hay, the Danish Minister and Madame de Bille, the Belgian Minister, the Brazilian Minister, Count Kinsky, the Lord Chancellor, Lady Halsbury and the Hon. Evelyn Giffard, the Duchess of Buccleuch, the Countess of Dalkeith and Lady Constance Scott, the Duke and Duchess of Portland, the Duke and Duchess of Sutherland, the Duchess of Montrose and Lady Helen Graham, the Duke and Duchess of Somerset, the Duchess of Newcastle, the Duchess of Cleveland, the Duchess of Manchester and Lady Alice Montagu, the Duchess of Roxburghe and Lady Victoria Innes-Ker, the Marquis and Marchioness of Zetland, Viscountess Milton, the Marchioness of Hastings, Miss Olive Chetwynd, Miss Mary Dyke and Miss de Winton, Marquis Camden, the Marchioness of Headfort and Lady Beatrix Taylour, the Dowager Marchioness of Londonderry, the Marchioness of Blandford and Lady Norah Spencer Churchill, the Earl and Countess of Lathom, Lady Edith Wilbraham and Lady Florence Cecil, Countess Spencer, the Countess of Warwick, the Earl of Cork[,] the Countess of Aberdeen, the Earl and Countess of Yarborough, the Earl of llchester and Lady Muriel Fox-Strangways, the Earl of Lonsdale, the Earl of Arran, the Countess of Powis and Miss Cotterell, Theresa Countess of Shrewsbury, Countess Stanhope and Lady Katherine Stanhope, the Countess of Verulam and Mackintosh, the Earl of Denbigh, the Countess of Suffolk and Lady Eleanor Howard, Earl and Countess Annesley, the Countess of Eglinton and Lady Edith Montgomery, Countess Howe, the Countess of Eldon and Lord Kilmarnock, the Countess of Coventry and Lady Barbara Smith, the Earl and Countess of Mayo, the Earl and Countess of Erne and Lady M. Crichton, the Countess of Kilmorey, the Earl of Listowel and Lady Beatrice Hare, the Earl of Granard, Countess Cadogan, the Countess of Jersey and Lady Margaret Villiers, Isabella Countess of Wilton, the Earl of Shrewsbury, Countess Carrington, the Earl and Countess of Carnarvon, the Earl of Erroll, the Earl and Countess of Dunraven and Lady Eileen Wyndham Quin, the Earl of Chesterfield, the Countess of Huntingdon, the Earl of Durham, the Countess of Ancaster and Lady Alice Willoughby, Viscount and Viscountess Falmouth, Viscount and Viscountess Templetown, Viscount and Viscountess Cranborne, Viscountess Boyne and the Hon. Florence Hamilton Russell, Viscountess Duncannon and the Hon. Irene Ponsonby, Viscountess Hood and the Hon. Dorothy Hood, Viscountess Barrington, Viscountess Newport and the Hon. Miss Bridgeman, Viscount and Viscountess Raincliffe and Lady Mildred Denison, Viscount Valentia, Viscount St. Cyres, the Marchesa Santurce, Lord and Lady Ashbourne, Lady Cardross, Miss Erskine, Miss Chaplin and Miss Baird, Lord and Lady Castletown and Lady Florence Bourke, Lord and Lady Glenesk, Lord and Lady de Ros, Lady Cynthia Graham and Lady Ulrica Duncombe, Lord and Lady Tweedmouth, Lord and Lady St. Oswald, Lord aud Lady Algernon Gordon-Lennox, Lady Elcho, Lord and Lady Inchiquin and the Hon. Miss O'Brien, Lady O'Brien and Miss O'Brien, Lady Aiington, Lady Blythswood, Lady Violet Brassey, Lady Evelyn Cotterell, Lady O'Neill and the Hon. Henrietta O'Neill, Lord and Lady Clonbrock and the Hon. Miss Dillon, Lady Llangattock and the Hon. Miss Rolls, Lady Hillingdon and the Hon. Miss Mills, Lady de Ramsey, Lord Churchill, Lord Lovat, Lord and Lady Fitzgerald, Lord Hyde, Lord Alexander Thynne, Lady Hopetoun, Lord and Lady Edmund Talbot, Lord Villiers, Lady Alwyne Compton, Lady Penrhyn and the Hon. Violet Douglas-Pennant, Emily Lady Ampthill and the Hon. Romola Russell, Lady Lister Kaye, Lady Cicely Gathorne-Hardy and Miss Gathorne-Hardy, Lady Ardilaun, Lord and Lady Burton, Lady Constance Combe, Mr. and Lady Florence Astley, Lady Evelyn Ewart, Lord Harris, Lord Morris and Miss Morris, Lady Arthur Hill and Miss Hill and the Hon. Miss Hill Trevor, Lord Garioch, Lord Kenyon, Lady Hartopp, Lord Iveagh, Lord Rowton, Lord de L'Isle and Dudley, Lady Helen Munro-Ferguson and Lady Hermione Blackwood, Lady Brabourne, Lady Magheramorne, Lord and Lady Edward Cecil, Lady Audrey Buller and Miss Howard, Lady Emily Van de Weyer and Miss Van de Weyer, Lord and Lady Henry Bentinck, Lord and Lady Charles Beresford, Lord William Beresford, the Right Hon. Sir William Harcourt, M.P., and Lady Harcourt, the Right Hon. James and Mrs. Lowther, the Right Hon. Walter Long, M.P., and Lady Doreen Long, the Right Hon. Henry Chaplin, M.P., the Right Hon. Sir Michael Hicks-Beach, M.P., and Mr. and Miss Hicks-Beach, the Right Hon. G. J. Goschen M.P., and Mrs. and Miss Goschen, the Right Hon. Arnold Morley, M.P., the Right Hon. Sir Henry Drummond Wolff, the Hon. Henry Fitzwilliam and Miss Fitzwilliam, General the Hon. Charles and Miss Thesiger, the Hon. Schomberg M'Donnell, General and the Hon. Mrs. Hugh M'Calmont, the Hon. Mrs[.] Lowther, Colonel the Hon. Heneage Legge, the Hon. Mrs. Hill, the Hon. Mrs. Carpenter, Miss Talbot Carpenter, the Hon. Mrs. Oliphant, Mr. Francis and the Hon. Mrs. Fitzgerald, the Hon. Mrs. Benyon, the Hon. Benjamin Bathurst, the Hon. Mrs. Gerard, the Hon. Arthur Brodrick, the Hon. Mrs. Stirling, the Hon. Charles Harris, the Hon. Gerald Portman, the Hon. Claud Hay, the Hon. Mrs. Baillie of Dochfour, Major the Hon. C. Lambert, the Hon. S. Ormsby Gore, the Hon. A. Yorke, the Hon. Michael and Mrs. Herbert, the Hon. Walter Rice, Major the Hon. D. Lawless, Baron and Baroness Deichmann, Baron von Oppell, Baron Mirbach, Baron [[Social Victorians/People/Watson|Meyer Watson]], Count Hadik, Count Hermann Hatzfeldt, Sir Henry and Lady Meysey-Thompson, Sir Henry Edwardes, Sir Robert Hamilton, Sir Donald Wallace, Sir Robert and Lady Penrose.FitzGerald [sic dot], Sir George Stirling, Sir Archibald Edmonstone, Sir Charles Hartopp, Sir Edgar Sebright, Sir G. Arthur, Sir Condie Stephen, Sir Charles Hall, Sir George Wombwell, Mrs. Edward Barclay, Mrs. Hope, Captain Milner, Major Poulteney, Major Wickham, Mrs. Ronalds, Mr. and Mrs. Atkinson Clerk, Colonel and Mrs. Eminson, Mr. Moon, M.P., Mr. Beaumout, Mr. Edward Packe, Mr. and Miss Sanders, Mr. and the Hon. Mrs. Philips Roberts, Miss Thellusson, Mr. and Mrs. Jarvis, Mr. George Phipps, Captain Dundas, Captain and Mrs. Wilfrid Marshall, Captain Feilden [sic], Major Shuttleworth, Mr. Hugh Gaisford, Mr. and Mrs. Rupert Beckett, Mr. Monro Ferguson, Mr. Henry Petre, Mr. Somerset Onslow, Mr. W. Gillett, Mr. and Mrs. Hamar Bass, Mr. and Mrs. Rupert Beckett, Mr. Millbanke, Mrs. Hwfa Williams, Mr. Graham Vivian, Mr. Evelyn Cecil, Mr. Herbert Praed, Mr. Farquhar, Miss Cockrell, Major and Mrs. Coddrington, Mr. and Mrs. Fred Villiers and Miss Villiers, Mr. C. Webb, Major Jenkins, Captain and Mrs. Fitzgerald, Mr. Macgregor, Mr. Roberts, Mr. Winston Churchill, Mr. Colin Keppel[,] Mr. Lougley, Miss Herbert, Mrs. Loder, Mr. and Mrs. Arthur James, Mrs. Moberly Bell, Mr. Moberly Bell, Mr. Gregson, Mr. R. D. Norton, Captain Elsworthy, Captain Johnson, Colonel Paget Moseley, Mr. and Mrs. Ogden Mills, Mr. Vincent, Mr. Francis Whitmore, Mr. Ward Cook, Mr. Balmain, Dr., Mrs., and Miss Maclagan, Mrs. and Miss Chamberlain, and many others.<ref>"Court Circular." "Londonderry House." ''Morning Post'' 17 July 1897 Saturday: 7 [of 12], Col. 7a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970717/076/0007.</ref> </blockquote> ===17 July 1897, Saturday=== <blockquote>During his holiday, Mr. Wyndham has found a sub-tenant in Mr. Horniman, who produced a farce called "Four Little Girls," by Mr. Walter Stokes Craven, on July 17. It tells how two widowers of Wimbledon (Mr. Barnes and Mr. Blakeley) resolve to marry their housekeepers (Miss M. A. Victor and Miss Emily Miller) — four capital studies in old age — on condition that their sons marry the duaghters of the two ladies. The boys, with the connivance of their tutor, Mr. Nuggeridge (Mr. Welch), have already married (one of the brides being Miss Mabel Beardsley, the artist's sister), so that extravagant farce is the result. The piece was played in a lively key, Mr. Blakeley, Mr. Welch (masquerading as a Scot under such a very English name as Muggeridge), and Mr. Kenneth Douglas (as one of the young men) being specially amusing.<ref>"The Playhouses: 'Four Little Girls,' at the Criterion Theatre." ''Illustrated London News'' (London, England), Saturday, July 24, 1897; Issue 3040, Col. A.</ref></blockquote> === 23 July 1897, Friday === 23 July 1897 — or 30 July 1897 – Friday, Lady Burton's party at Chesterfield House:<blockquote>Chesterfield House is so beautiful, says the "Daily Mail," as be independent of floral decoration; there were, in fact, no more than the usual amount of flowers and palms on Friday night, so that nothing distracted the attention from the beautiful pictures. The lighting was wonderfully good, and refreshments were served in the library until the supper room was opened — at about half-past twelve. The rooms were never too crowded, and Lady Burton, who looked very well in grey embroidered in black, and who wore a big diamond tiara, had a word of welcome for one of her guests. Prince Francis of Teck represented royalty and took Lady Burton down to supper: while Lord Burton took Countess Deym, who looked very handsome in white. Lady Londonderry went down with Mr. Alfred Rothschilde; Lady Gerard, who looked very pretty in white with blue in and out of her hair, went in with Mr. Craven; Lady Howe went in with Lord Kenyon; and Lady Eva Dugdale went in with Mr. Arthur James. Far the prettiest women in the room were Lady Henry Bentinck (who looked perfectly lovely in pale yellow, with a Iong blue sash; and [[Social Victorians/People/Bourke|Mrs. Algernon Bourke]], who was as smart as possible in pink, with pink and white ruchings on her sleeves and a tall pink feather in her hair. Naturally, one great topic of conversation was the marriage of the Duchess of Hamilton, and people talked, too, of the religious meetings which have been held lately at Lady Henry Bentinck's house, when the Archbishop of Canterbury has delivered lectures to "young girls only." These have made a very great impression.<ref>"Lady Burton's Party at Chesterfield House." ''Belper & Alfreton Chronicle'' 30 July 1897, Friday: 7 [of 8], Col. 1c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004151/18970730/162/0007. Print title: ''Belper and Alfreton Chronicle''; n.p.</ref></blockquote> ===31 July 1897, Saturday=== The London ''Morning Post'': the wedding of Mabel Caroline Wombwell and Henry R. Hohler: <blockquote>Mr. Henry R. Hohler, eldest son of Mr. Henry Booth Hohler, of Fawkham Manor, Kent, was married to Mabel Caroline, second daughter of Sir George and Lady Julia Wombwell, in St. Paul's Church, Knightsbridge, on Saturday afternoon. His Royal Highness the Duke of Cambridge attended the ceremony, accompanied by Colonel FitzGeorge. The bride arrived shortly after half-past two o'clock, and was led to the chancel by her father, who gave her away. She was followed by two pages, Master Alastair Graham Menzies, son of Mr. and Mrs. Graham Menzies, and Master W. Rollo, son of the Hon. Eric and Mrs. Rollo, who were in Highland costume, with their respective tartans. Eight bridesmaids followed: Lady Mary Villiers and the Hon. Ella Peel, cousins of the bride; Miss Hohler, sister of the bridegroom; the Ladies Edith and Mary Dawson, nieces of the bride; Miss Amy Hohler and Miss Torfrida Rollo, nieces of the bridegroom; and the Hon. Theresa Fitzwilliam. The bride wore a white satin gown embroidered with pearls, diamonds, and silver sequins, and trimmed with accordion-pleated chiffon, trails of orange blossoms, and a chiffon sash. Her veil was of lovely old Brussels lace, and her jewels included a large diamond star in her hair, the gift of the Earl and Countess of Dartrey; a diamond heart locket, her mother's gift, and a diamond bracelet, Mr. Holder's present. The bridesmaids were attired in white French muslin over white satin, trimmed with Valenciennes lace insertion and runners of white baby ribbon; they wore fichus of white point d'esprit, and white straw hats trimmed with bows of point d'esprit and pink roses. Gold bangles with light blue enamel and diamond heart lockets and shower bouquets of pink carnations were the bridegroom's gifts. The Service was choral. The Bishop of Sodor and Man, the Rev. H. Montagu Villiers, vicar of St. Paul's, and the Rev. T. Blackall, rector of Fawkham, Kent, were the officiating clergy. Mr. Gerald Hohler acted as best man to his brother. After the ceremony Sir George and Lady Julia Wombwell welcomed the relations and friends of both families at the residence of the Earl and Countess of Dartrey, sister of the bride, in Eaton-square, lent for the occasion. Among those present were the Duke of Cambridge, the Marchioness of Blandford, Theresa Countess of Shrewsbury, the Earl and Countess of Jersey and the Ladies Villiers, the Earl and Countess of Dartrey, the Countess of St. Germans, the Countess of Carnarvon, Viscount Peel, Lady Penrhyn and the Hon. Miss Douglas-Pennant, Lady Constance Gore, Lord Villiers, the Bishop of Sodor and Man and Mrs. Straton, Lord Abinger, Lady Cynthia Graham, Captain the Hon. E. Dawson, the Hon. Mrs. Baillie of Dochfour and Miss Bruce, Lady Hartopp and Miss Enid Wilson, the Hon. Mrs. Wood, the Hon. Mabel Murray, the Hon. Helen Henniker, the Hon. Mrs. Maguire and the Hon. Ella Peel, the Hon. Reginald Villiers, the Hon. Eric and Mrs. Rollo, the Hon. Agnes Peel, Mrs. Hohler, Mrs. Frederick Hohler, Mrs. Wombwell, Mr. F. Wombwell, Captain and Mrs. Wombwell, Colonel Wombwell, Mr. and Mrs. C. Hohler, Mr. Hohler, Mr. Tremayne, Madame and Miss de Bunsen, Mrs. Harry Goschen, Mrs. Charles Van Raalte, Mrs. Gunston, Mrs. Wilfrid Marshall, Mr. Francis Gregson, [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Wilson]] and [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Mr. Alfred de Rothschild, Mis. Ernest Villiers and Miss Villiers, Mr. Ward Cook, Mr. Frank Green, Mr. and Mrs. Fane, Mr. Robinson, Mr. and Mrs. Deacon, Miss Willoughby and Miss Gertrude Willoughby, Colonel and Mrs. Livesey Wardle, Mr. and Mrs. Graham Menzies, Miss Gordon, Mr. and Mrs. Du Plat Taylor, Miss Muriel Blundell, Mr. Edward Hare, Major Victor Farquharson, Mr. and Mrs. William Aston, Miss Julia Ponsonby, Mr. Launcelot Smith, Mrs. Richard Martin, Major Lawes, Rev. T. Blackall, Colonel and Mrs. Alan Gardner, Mr. and Mrs. Harris Saunders, &c. Mr. and Mrs. Henry R. Hohler subsequently left for Clumber, Notts, lent by the Duke and Duchess of Newcastle, for the honeymoon. Among the numerous presents to the bride were: From his Royal Highness the Duke of Cambridge, a gold antique tortoise inkstand, set with various stones; the Bridegroom, diamond tiara, turquoise and diamond bracelet, emerald and diamond clover-leaf bracelet, diamond and turquoise heart, diamond combs, several gem rings; Sir George Wombwell, large diamond cross set with a black pearl, diamond, a hack and hunter; Lady Julia Wombwell, large diamond heart and old lace; the Earl and Countess of Dartrey, large diamond star; Mr. Stephen Wombwell, diamond and sapphire chain bracelet; Mr. Hohler, diamond bracelet, horse and harness; Mrs. Hohler, diamond ring; Prince Frederick Dhuleep Singh, diamond and ruby brooch; Mr. and Mrs. Frederick Hohler, diamond sword; Miss Hohler, a very handsomely-fitted dressing bag and enamel pencil bracelet; Mrs. F. Wombwell, diamond crescent; Mr. and Mrs. Graham Menzies of Hallyburton, diamond tiara; the Earl of Jersey, hair ornament in emeralds and diamonds; the Countess of Jersey, necklet of amethysts and diamonds; the Earl and Countess of Ellesmere, gold and jewelled heart looking-glass; the Marquis and Marchioness of Londonderry, large diamond and sapphire crescent; Mr. Alfred de Rothschild, sapphire and diamond brooch; the Earl and Countess of Carnarvon, sapphire and diamond bracelet; Lady Alice Egerton and the Ladies Ada and Alexandra G. Osborne, amethyst and pearl locket; Lord Balvaird, Tay pearl bracelet; the Duke and Duchess of Marlborough, old silver pen tray and tea caddy; the Marchioness of Blandford, large frame; Lady Helen Vincent, gold-handled umbrella; Mrs. Ernest Villiers, old silver wheelbarrow; the Ladies Edith and Mary Dawson, silver frame; the Countess of Lathom, brooch; the Ladies Maud, Bertha, and Edith Wil_____m, scarf pin; Mr. and the Hon. Mrs. Maguire, diamond and sapphire brooch; Mr. and Mrs. Arthur Wilson and Miss Muriel Wilson, turquoise and diamond hair ornament; the Marquis of Abergavenny, pair of silver candlesticks; Viscount Peel, large silver dish; Lord and Lady Dorchester, large silver scent bottles; Viscountess Milton, parasol with jewelled handle; Lady Cynthia Graham, enamel and diamond fox head pencil; the [[Social Victorians/People/Keppel|Hon. George and Mrs. Keppel]], white enamel and turquoise sleeve links; Lord and Lady Deramore, tortoiseshell fan; Mr. and Mrs. Vyner, gold pencil studded with emeralds and diamonds; the [[Social Victorians/People/Feversham|Earl and Countess of Feversham]], silver-gilt inkstand and candlesticks; Viscountess Helmsley, gold-handled umbrella; Lord and Lady Burton, agate and gold-mounted paper knife; the Dowager Marchioness of Londonderry, standard lamp; Isabella Countess of Wilton, pair of silver candlesticks; Sir Henry Edwardes, old étui case; the Earl and Countess of Wharncliffe, a fan; the Countess of Ancaster, marqueterie table; the Dowager Countess of Craven, illuminated clock; the Hon. Cecil and Mrs. Bingham, silver dishes; the Hon. Reginald and Mrs. Parker, silver inkstand; the Earl and Countess of St. Germans, silver box; the Hon. Hugh W. Fitzwilliam, walking-stick with gold handle encrusted with jewels; Mr. and Mrs. Leopold de Rothschild, diamond and pearl clover-leaf brooch; the Earl and Countess of Yarborough, silver salver; the Earl and Countess of Harewood, silver and enamel smelling bottle; Miss E. Wombwell, silver hand glass and pair of silver-backed brushes; the Earl and Countess of Coventry, large silver tea pot; the Earl and Countess of Ilchester, a fan; Elizabeth Countess of Wilton and Mr. Pryor, large silver inkstand; Lady Rothschild, umbrella with gold handle set with jewels; Captain H. Wombwell, gold-mounted claret bottle; Lord Abinger, Louis XIV. clock; Mr. F. Wombwell, four silver bonbonniere dishes; the Hon. Hubert Duncombe, gold and onyx bangle; the Hon. R. Villiers, set of silver brushes, looking-glass, comb, and tray; the Hon. Mabel and Theresa W. Fitzwilliam, clock with electric light; Lady Skelmersdale, silver-mounted purse; Lady Mildred Denison, writing case; Mr. A. C. Wombwell, breakfast service; Mrs. Graham Menzies, large silver inkstand; General Wombwell, large silver scent bottle; the Countess of Selkirk, four silver bonbonniere dishes and spoons; Sir H. and Lady Evelyn Ewart, silver sugar basin; Sir A. and Lady Edmonstone. silver tea caddy; the Tenantry on the Newburgh and Wass Estates, large silver tea tray; the Tenantry on the Old Byland Estate, alabaster clock; the Indoor and Outdoor Servants of Newburgh, large silver coffee pot; the Indoor and Outdoor Servants of Hallyburton, marble clock and address; School Children of Old Byland, prayer and hymn books. Among the gifts received by the bridegroom were: From the Dowager Duchess of Newcastle, service of silver table plate, 308 pieces; the Duke of Newcastle, silver-mounted spirit table; the Duchess of Newcastle, gold, diamond, and enamel sleeve links; the Duke of Wellington, pair of silver candlesticks; Sir George Wombwell, two Queen Anne silver salvers; Mr. Gerald F. Hohler, cheque; Mr. Stephen Wombwell, old Sheffield and cut glass cruet; Mr. and Mrs. E. T. Hohler, Dresden dessert service and antique cut glass decanters; Mr. Thomas B. Hohler, Persian carpet; Sir Robert and Lady Affleck, pair of jewelled gold links; Mr. and Mrs. C. Hohler, silver soup tureen; the Hon. Mr. and Mrs. Rolls, pair of silver baskets; Mr. and Mrs. F. S. Hohler, silver flower basket; Mr. H. B. Hohler. silver George 111. centre piece; the Hon. Algernon Mills, silver hot-water jug; Miss Hohler, pair of old silver sauce boats; Tenants on the Fawkham Estate, barometer; Servants at Fawkham Manor, large silver salver; Men on the Home Farm, Fawkham, silver-mounted walking stick; Major Lawes, dessert service; Mrs. Owen Williams, silver cigarette lighter; Mr. E. P. Hare, gold matchbox; Mrs. Hohler, writing table; Miss C. Gordon, gold-mounted amber cigarette-holder, in silver case; Captain A. Hicks-Beach, pair of silver photograph frames; Colonel and Mrs. Antrobus, gold and amethyst seal; Mr. and Mrs. Fletcher of Saltoun. pair of silver candlesticks; Mr. Arthur Capel Cure, large silver cigarette box; Mr. E. A. Franklin, tortoiseshell and gold cigar case; Mr. T. L. Hare, M.P., double reading lamp; Sir Basil Hall, crystal decanter in silver stand; Captain and Mrs. Torrens, double silver inkstand; Captain Hon. E. Hanbury, silver-mounted letter case; Mr. Frank Deacon, large silver bowl; Colonel and Mrs. Brownrigg, silver salver; Major the Hon. C. Lambton, standard lamp; Mr. and the Hon. Mrs. Tremayne, antique silver candlesticks; Colonel and Mrs. Barrington [?] Campbell, settee; besides other presents.<ref>"Marriage of Mr. H. R. Hohler and Miss Wombwell." ''Morning Post'' 2 August 1897, Monday: 6 [of 8], Col. 3a–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970802/067/0006 (accessed June 2019).</ref></blockquote> ==August 1897== === 18 August 1897, Wednesday === The celebration of the birthday of Austro-Hungarian Emperor-King Francis Joseph:<blockquote>The Austro-Hungarian Colony. — Austro-Hungarians in London celebrated the birthday of the Emperor-King Francis Joseph by a dinner at the Trocadero Restaurant last night. Count Albert Mensdorff, Chargé d'Affaires for Austro Hungary, presided, and amongst the leading members present were [[Social Victorians/People/Hadik|Count Hadik]], Secretary of the Embassy, Captain Sztrany Asky, the Naval Attaché; Chevalier Princig De Harwaldt, Acting Consul-General; Mr. Leopold Pam, Chairman of the Austro-Hungarian Aid Society; Mr. Pillischer, Vice-Chairman of the Hungarian Association; Mr. Louis Felbermann, hon. secretary of the Hungarian Association; Mr. S. Bodascher, hon. secretary of the Austro-Hungarian Aid Society; Mr. Politzer, Almoner of the Austro-Hungarian Aid Society; Mr. M. Weiss, and Mr. J. Kaufmann, Almoner of the Hungarian Association; Colonel Hain, and many others. A telegram expressing the loyalty of the Austro-Hungarian Colony was despatched to his Majesty the Emperor-King.<ref>"The Austro-Hungarian Colony." ''Morning Post'' 19 August 1897 Thursday: 3 [of 8], Col. 4c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970819/017/0003.</ref></blockquote> ===30 August 1897, Monday=== Summer Bank Holiday ==September 1897== Sometime in September 1897, the Inner Order of the Golden Dawn moved its headquarters from 62 Oakley Square, where they had been from March 1896, to 36 Blythe Road.<ref>Howe 126.</ref> Also during this month a subgroup in the Golden Dawn, including Annie Horniman and Frederick Gardner were meeting in Talgarth Road, West Kensington, London.<ref>Howe 197.</ref> === 16 September 1897 === [[Social Victorians/1897 Fancy Dress Ball/anthology#Production of The White Heather at the Drury Lane Theatre in London|''The White Heather'' opened at the Drury Lane Theatre]]. ''The White Heather'' is a melodrama by Cecil Raleigh and Henry Hamilton that opened at the Drury Lane Theatre on 16 September 1897, had another run in spring 1898 and was revived at the Princess in 1899; it ran on Broadway in New York beginning in November 1897.<ref>{{Cite journal|date=2023-08-10|title=The White Heather (play)|url=https://en.wikipedia.org/w/index.php?title=The_White_Heather_(play)&oldid=1169721008|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/The_White_Heather_(play).</ref> The V&A has a collection of 31 photographs of some of the sets for the London production (https://collections.vam.ac.uk/search/?page=1&page_size=15&q=%22White+Heather%22). One scene was a fancy-dress ball that used originals and copies of what was worn at the Duchess of Devonshire's ball on 2 July 1897; the newspaper reports and review articles assume that readers would recognize the reference to the July ball and be interested on that account. The play was produced by Arthur Collins (the theatre producer and director, not the courtier). A story in The ''Hartlepool Northern Daily Mail'' about reports that ''The White Heather'', by Cecil Raleigh and Henry Hamilton, performed at the Drury Lane theatre, included a scene that reproduced the Duchess of Devonshire’s Ball:<blockquote>"The White Heather" is strikingly realistic, the scenes including a reproduction of the recent famous Duchess of Devonshire's ball, a diving expedition, Battersea Bark (that beautiful resort), &c. With respect to the ball scene, the critic quoted above [describing the actress Mrs. John Wood] says: — "The mass of gorgeous colour and dazzling brilliancy in the ball scene was simply overpowering." The drama is of sufficient importance to warrant a little extra space being devoted to it. The following, taken from a contemporary, will be of interest to my lady readers particularly: — The ball scene offers almost a surfeit of brilliant colour to the spectators. The actual costumes worn by some two hundred of the guests at the Duchess of Devonshire's ball have been secured by the management, and are worn by the actors and actresses. Conspicuous among these is the purple velvet of the Earl of Leicester. An exact reproduction of the costume of Grand Master of the Knights Hospitallers of Malta, as worn by the Prince of Wales, is another striking dress. Mrs John Wood, as Queen Elizabeth, wears a costume consisting of a dress with long pointed bodice and farthingale in rich cream broché, the entire petticoat and sleeves in cream duchesse satin, handsomely embroidered in gold and jewels, stomacher and sides of overdress elaborately trimmed with the same ornaments, and the double wing collar and large cuffs in fine lace, thickly studded with jewels. Her mantle is of Venetian red and gold brocade, lined throughout with red and gold-shot gauze, while the headdress and crown are in fine paste emeralds set in gold. Miss Kate Rorke has a Marie Stuart costume in black silk velvet and duchesse satin, handsomely trimmed with gold passementerie and jewels, with large pearl girdle. Her large headdress and collar, combined with long mantle, are in fine white silk gauze-trimmed lace. The principal costumiers in the Metropolis, it should be added, have been employed in making gowns for the play.<ref>"The Stage." The ''Hartlepool Northern Daily Mail'' 25 September 1897: 6 [of 8], Col. 4A–B. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000378/18970925/118/0006.</ref></blockquote>The Queen also ran a review:<blockquote>DRURY LANE THEATRE. REDECORATED, re-upholstered, under new management, and with a brand new melodrama, the "National Theatre," as its late lessee loved to have it called, has once more opened its doors to an eager public. Judging by the warmth of the reception given to Mr Arthur Collins, who has assumed the mantle of the late Sir Augustus Harris, and by the enthusiasm which prevailed throughout the presentation of "The White Heather," there is no manner of doubt as to the success of the new management and the new melodrama. The former has been inaugurated under the happiest possible circumstances, for Mr Collins has had the invaluable assistance of Messrs Cecil Raleigh and Henry Hamilton, than whom none better understand the tastes and requirements of a Drury Lane audience, while never, it can be safely said, has an autumn melodrama, even here, been more strongly cast or better acted than the present one. If there is a fault to be found with the new play, it is, perhaps, that the story is somewhat less powerful than those to which the audiences at this theatre have previously been accustomed. It certainly furnishes incidents thrilling and enthralling enough to satisfy the most captious critic, while it possesses the merit of providing that delightful actress, Mrs John Wood, with as good a part as she has had for years. But the plot is not precisely a strong one, and there is in truth a vast amount of magnificent spectacle and scenery and dresses and effects to a very small pennyworth of story. If, however, Mr Raleigh and Mr Hamilton Lave not much of a tale to tell, they tell it very well indeed, and the dialogue of the new play is wonderfully bright and amusing. It is on the old theme of a Scotch marriage that the authors have built the new drama. Lord Angus Cameron, a stock Exchange "plunger," and heir to a Dukedom, has, during a voyage north on his yacht "The White Heather," married pretty Marion Hume, daughter of a Stock Exchange jobber, according to Scottish law. In course of time he sees someone whom it would be much more convenient for him to make Lady Cameron in proper form, and as the yacht has conveniently sunk off the English coast with the evidences of the marriage with Marion, and only one of the two necessary witnesses survives, he promptly repudiates the poor girl, and brands their small boy as illegitimate. Her father, on learning her story, plunges heavily in South Africans, in order, as he hopes, to obtain money enough to fight Lord Angus at law, but the shock brought about by a disastrous decline in the South African market, and the worry in connection with his daughter, cause his sudden death, and Marion then finds her warmest friend in the kindly Lady Janet McLintock, [53, Col. 1c / 2a] Lord Angus's sister. All manner of troubles beset the heroine, of course, and her cause seems indeed lost when Lord Angus actually descends below the sea to the sunken White Heather, in order to destroy the evidences of his marriage with her. But the inventive genius of the authors does not stop here. They take the audience with them down, down among the fishes and the seaweed in company with a lowly champion of poor Marion, one Dick Beach. The two men fight for the precious document beneath the rolling wave. The villain fatally stabs poor Beach, but he contrives to cut the air apparatus attached to Lord Angus's diving dress, speedily causing his death, of course, and to live long enough to be hauled to the surface and leave the "marriage lines" in safe hands. After this it is easy enough to make matters smooth for the heroine, but this is not accomplished until the great ball scene, which everybody has been anticipating for weeks, and which all London will be anxious to see. A richer or more interesting spectacle than this has never been presented even on the Drury Lane stage. Many of the dresses were actually worn at the now famous ball given during the late season at Devonshire House, and the effect produced by the admirably managed crowds of guests in their fancy costumes is one of exceeding brilliance. The gowns worn in this scene are really magnificent, but the whole play is marvellously well dressed, and on this account is sure to find favour with lady playgoers, who will see here the very newest and prettiest conceits in walking, evening, boating, cycling, and shooting costumes. The same lavishness that marks the dressing of the play is shown in the scenic effects. There is something to interest everybody and excite their enthusiasm in "The White Heather," for, besides the original submarine scene already referred to, there is a most realistic presentment of "Boulter's Lock" on a Sunday afternoon; by a mechanical contrivance we see the lock emptied and the boats, filled with their gaily clad occupants, descend, a scene which provokes the wildest cheers from the audience. But the spectators are transported, too, to a realistic Scotch moor, where real heather is blooming, and where real dogs and gillies and guns and sportsmen are duly to the fore. Nor does this exhaust the wonderful scenic attractions of the new piece, for there is, besides, a lovely picture of Battersea Park with its crowds of fair cyclists in every kind of natty dress, from the tweed skirt and dainty blouse to unmistakable "rationals," and a thrilling Stock Exchange scene, which, in their way, will appeal no less forcibly to many playgoers. Little space is left to speak in detail of the splendid work done by Mrs John Wood, whose vivacity and resource seem to be limitless; by Mr Henry Neville, who astonishes his old admirers by appearing for once as an uncompromising villain; and by Miss Kate Rorke, who invests the heroine with all possible womanly charm, and wins for her the fullest sympathy of the audience. Conspicuous success is achieved by Mr R. Loraine as the faithful Dick Beach, Mr F. B. Gordon as the old Stockjobber, Hume, and by Mr Dawson Millward, the ''jeune premier'' of the piece, which, indeed, is thoroughly well acted all round, Miss Beatrice Lamb, Miss Pattie Browne, Mr H. de Lange, and Mr C. M. Lowne all contributing very well-finished character studies. "The White Heather" promises to bring abundant good luck to Drury Lane.<ref>"Drury Lane Theatre." "The Drama." The ''Queen'' 25 September 1897, Saturday: 53 [of 80], Col. 1c–2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18970925/293/0053.</ref> </blockquote>"Mrs. Aria" has an article in the ''Queen'' about the women's dresses, gowns and cloaks in the play. Here is what she says about the fancy-dress ball scene:<blockquote>The fancy dresses in the ball scene — those which did not previously do their duty at the Duchess of Devonshire's ball, and there are many of these amongst the number — were designed by that clever artist, Mr Cumelli [sic, s/b ''Comelli'']; and very gorgeous they are, all velvet and satin and gold embroidery and lace, a decorative note being struck on Miss Kate Rorke's dress, as Marie Antoinette, by long gauze draperies falling from her hair to the hem of the black gown.<ref>Mrs. Aria. "A Vista of Fashion." The ''Queen'' 2 October 1897, Saturday: 58 [of 134; print page # is 628], Col. 1c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971002/171/0068.</ref></blockquote>This article by Mrs. Aria says sketches to accompany this story appear on p. 618 (this article is on p. 628 of the print newspaper). ==October 1897== ===3 October 1897, Sunday=== William Morris died, at Kelmscott House. At Morris's request Arnold Dolmetsch came to play The Earle of Salisbury's Pavin on the virginals.<ref>Campbell 103.</ref> ===5 October 1897, Tuesday=== The Princess Mary, Duchess of Teck, the Duke of Teck, and Alexander of Teck and retinue visited Henry James Tufton, 1st Baron Hothfield of Hothfield in Appleby. Muriel Wilson was in the houseparty to which the Tecks were travelling. Also in the party were "the Hon. John and Lady Ierne Tufton, Mr. and Mrs. William Portal, [[Social Victorians/People/Arthur Stanley Wilson|Mr. Arthur Wilson]] and [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Sir George Arthur, Bart., Mr. and Mrs. Charles Van Raalte; the Hon. Rosamond Tufton, the Hon. Sackville Tufton, the Hon. Charles Tufton, Captain George Tufton, Lady Clementine Walsh, and Mr. Leo. Trevor." Hothfield hosted a garden party the next day but otherwise the visit was “of a strictly private character.”<ref>"Arrival of the Duke and Duchess of Teck. The House Party.” ''Penrith Observer'' 5 October 1897, Tuesday: 5 [of 8], Col. 5a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002296/18971005/088/0005 (accessed July 2019).</ref> ===20 October 1897, Wednesday=== 1897 October 16?: the wedding of Lord Waterford and Lady Beatrix Fitzmaurice. [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Wilson]]'s and [[Social Victorians/People/Muriel Wilson|Muriel Wilson]]'s gifts to her were a “red leather writing pad”<ref>"Marriage of Lord Waterford and Lady Beatrix Fitzmaurice." ''The Waterford Standard'' 20 October 1897, Wednesday: 2 [of 3], Cols. 5c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001678/18971020/024/0002 accessed June 2019).</ref>; Muriel Wilson and Kenneth Wilson also are listed as having given a gift to him, “brocaded stationery case and blotting book.”<ref>"Marriage of Lord Waterford and Lady Beatrix Fitzmaurice." ''The Waterford Standard'' 20 October 1897, Wednesday: 2 [of 3], Cols. 6c [of 7]. ''British Newspaper Archive''https://www.britishnewspaperarchive.co.uk/viewer/bl/0001678/18971020/024/0002 accessed June 2019).</ref> The [[Social Victorians/People/Albert Edward, Prince of Wales|Prince]] and [[Social Victorians/People/Alexandra, Princess of Wales|Princess of Wales]] as well as many celebrities and familiar names attended and sent gifts. ===31 October 1897, Sunday=== Halloween. ==November 1897== ===1 November 1897, Monday=== The Holderness Hunt:<blockquote>The season of the Holderness pack commenced yesterday, when there was a large gathering at Rise. Before commencing operations the company were hospitably entertained by Mr and the Hon. Mrs Bethell. Amongst those present were the Master ([[Social Victorians/People/Arthur Stanley Wilson|Mr Arthur Wilson]]), Mr and Mrs Stanley Wilson, Mr and Mrs Kenneth Wilson, Mr Clive Wilson, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Mr Wellesley Wilson, J. Simons Harrison, Mr R. D. Richardson, Mr Robert Voase, Mr and Mrs Robinson, Miss Bethell, Commander Bethell, M.P., Mr and Mrs Hutchinson, Mr H. Richardson, Captain Samman, Mr W. England. Mr G. England, Mr T. Jackson. Mr Harry E. Bainton, Mr J. J. Ridley, Mr T. Dixon, Mr R. Dixon. Mr Heslop, Riby Wright, Mr W. Todd, Mr Wilfred Harrison, Mr William Robinson, Captain Short, Mr Fisher. Mr E. Harland, Mr J. Nutchey, and others. The staff from the kennels sported their new scarlet, and the whole turn out was excellent. Ash, who had the bitch pack out, first tried the woods, where plenty of foxes were to be found and after a bit of brushing about, two were killed in cover. At Farnton the hounds unkennelled a useful fox, which for nearly half an hour afforded good sport, taking a wide ring by way of Sigglesthorne and back to cover, where he was lost. Catwick Thorns were then successfully drawn. Reynard, on making for the open, shot away in the direction of Brandesburton. Scent being good, the hounds hunted him in grand style. Leaving Brandesburton to the left, he took the direction of Lord Mayor's Whin. Then he made for Newsome and round by Nunkeeling, almost as far Seaton. Swinging back to the left, he ran round Star Carr Hill and Brandesburton village. He then turned and sought his old retreat at Catwick Thorns, where saved his brush, after giving the followers a rattling gallop of nearly an hour. Several were out for the first time in pink yesterday.<ref>"Tally-Ho! The Hunting Season Begins." ''Hull Daily Mail'' 2 November 1897, Tuesday: 5 [of 6], Col. 3a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000324/18971102/083/0005 (accessed July 2019).</ref></blockquote> ===2 November 1897, Tuesday=== <quote>Dolmetsch, writing to Horne on 2 November, mentions another visit to Florence which seems to be 'more certain than ever'. The performance of The Tempest at the Mansion House has been postponed until 5 November because of the death of the Duchess of Teck and in turn Dolmetsch's own concert date has been changed. 'But that will do no harm, rather some good, as I have obtained invitations to Mansion House for all my subscribers, and that has been paragraphed in the press (not The Times!). I have got 30 subscribers now. So that is not so bad."<quote><ref>Campbell, Margaret. ''Dolmetsch: The Man and His Work''. U of Washington Press, 1975: 119.</ref> [Was AEFH one of those subscribers? Who was?] ===5 November 1897, Friday=== Guy Fawkes Day ===17 November 1897, Wednesday=== Muriel Wilson took part in the meet of the Duke of Rutland’s hounds:<blockquote>The opening meet of these hounds, which should have taken place at Leadenham a fortnight ago, but was postponed owing to the hard condition of the ground, took place Wednesday at Croxton Park, where a large and fashionable gathering assembled, including several followers of the Quorn and Cottesmore packs. Amongst those present were Sir Gilbert Greenall, Miss Greenall, Mr. Cyril Greenall, Lord Robert Manners, Colonel Theobald, Major Longstaffe (Little Ponton), Colonel Hutchinson, Baron and Baroness Max de Tuyll, Mr. Algernon and Lady Henrietta Turner, the Hon. Lancelot and Mrs. Lowther, the Hon. Gavin Hamilton, Major Bradford Atkinson, Captain and Mrs. Lawson, Captain Timson, Sir Henry Rawlinson, Mr. and Miss Hodgson, the Hon. H. R. Scott, Colonel Ashton, Captain Boyce, Mr. and Mrs. Long, Mrs. Ellison, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Mr. Otho Paget, the Misses Markham, Mr. Maxwell Angus, the Rev. J. P. Seabrooke (Waltham), Mr. Gerald Hansom, the Rev. R. Mirehouse (Colsterworth), and others. In Freeby Wood there was a good show of foxes, and the pack got on the line of one which made for Waltham Village, and then back through the Ashes. From this point Reynard crossed the Grantham road and entered covert known the “Brooms,” but he was lost shortly afterwards in the neighbourhood of Thorpe lngold. Another short spin was had from Brentingby Wood, but the fox, like his predecessor, made good his escape, as did also a third, which went away from Brentingby Spinney. Subsequently the pack was taken to a hen roost at Freeby, where a fox in search of poultry had been incarcerated. On quitting these quarters Reynard was quickly caught and killed. It was a poor scenting day, but in other respects the conditions were nearly perfect. The opening of the active season on the Lincolnshire side will commence to-day, the meet being at Syston Park — the seat of Sir Joint Thorold.<ref>"Hunting. The Duke of Rutland’s Hounds." ''Yorkshire Post'' 19 November 1897, Friday: 10 [of 10], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000687/18971119/183/0010.</ref></blockquote> ===20 November 1897, Saturday=== 1897 November 20, parties for the Derby (reported on Wednesday, 24 November 1897):<blockquote>Ten years ago (writes a correspondent to the Daily Mail) how few of the fashionable people went to Derby races: now all the smartest people go, and it is one of the most important meetings, rivalling Doncaster in popularity. The big houses which are in the neighbourbood, no doubt, have considerable influence in this matter, and since the Duke and Duchess of Devonshire had house-parties at Chatsworth, there has been a marked increase in tha attendance. Lord and Lady Burton, Mr. and Mrs. Miller Mundy, Mr. and Mrs. Hamar Bass, and others, are also bent on hospitalities, and consequently the latter end of last week London was quite empty. The Duchess of Devonshire's party included Lord and Lady Londonderry, Lady Wolverton, Lord and Lady Essex, Mrs. Chaine, Lord Marcus Beresford, and Lord Crewe, as well as Lord and Lady Cadogan, Lord and Lady Howe, Lord and Lady Lurgan, and Mrs. Leo Rothschild, whose names I have already mentioned as staying at the house. Then, at Rangemore there is a very gay party indeed, including Lady Angela and Mr. James Forbes, Lady Sarah Wilson, Lady de Traflord, [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur]] and [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Mr. and Mrs. Hwfa Williams, Lady Norreys, Mr. Cecil Howard, Mr. Greenfell, Mr. Wilson, and Mr. Ludwig Neumann. At Shipley are Sir Charles and Lady Hartopp, [[Social Victorians/People/Keppel|Mr. and Mrs. George Keppel]], Mrs. de Winton, Lord Athlumney, and Mr. Sturt, among others. At Foston Lord and Lady Magheromorne are staying, as well as Mrs. Farquharson; while Mrs. Hamar Bass seemed to have a large party, including the Maurice FitzGeralds. Very few people looked better than Lady Essex, in brown, with soft white ermine fur; she is a very pretty woman, with delicate features and complexion, and big, soft, dark eyes. Lady Sarah Wilson, in brown with black braid, and a hat of brown chenille, looked remarkably smart; as also did Mrs. Jack Cumming, who came with Mrs. Bass, and her dress was a tobacco-coloured cloth, with a delightful Russian coat of velvet to match, with a sable collar and small sable toque. Lady Angela Forbes, who dresses almost as well as Lady Algernon Lennox, wore a sort of greeny-blue homespun; and Miss Muriel also looked so well, in simplest tweeds. Lady Norreys, in spite of a cold, looked very pretty, and was warmly clad in an astrachan coat, and sable hat; Mrs. Hamar Bass looked very nice in dull green and chinchilla; Mrs. Farquharson was also at her best; Lady de Trafford was very quietly dressed, while Lady Hartopp wore a very quaint coat of white sheepskin. Among the very many other men were Mr. Montagu Guest, Mr. Hungerford, and Mr. Combe.<ref>"Hints for Ladies. Fashion at Derby Races." ''Derby Mercury'' 24 November 1897, Wednesday: 6 [of 8], Col. 5a [of 7]. British Newspaper Archive <nowiki>https://www.britishnewspaperarchive.co.uk/viewer/bl/0000052/18971124/050/0006</nowiki>.</ref></blockquote> ==December 1897== === 4 December 1897, Saturday === The wedding of [[Social Victorians/People/Mount Stephen|Miss Gian Tufnell and George, Baron Mount-Stephen]], which took place shortly after the death of Princess Mary Adelaide of Cambridge, Duchess of Teck, for whom Gian Tufness was a lady in waiting and attendant:<blockquote>MARRIAGES. LORD MOUNT-STEPHEN AND MISS TUFNELL. The marriage of Lord Mount-Stephen with Gian, daughter of the late Robert George Tufnell, Commander R.N., formerly of Cheney Court, Box, Wilts, and afterwards of Kensington, Bath, was celebrated on Saturday week in the Church of St. Margaret, Westminster. The ceremony was performed by the Bishop of Peterborough, assisted by the Rev. F. E. Coggin, vicar of Lemsford, Herts. The bride, who was given away by her uncle, Mr. Tufnell, was attended by four bridesmaids, all children, viz.: Miss Williams, niece of the bride; Miss Berkeley, Miss Corbet, and Miss Kirby. Major-General Sir John Carstairs M'Neil, K.C.B., V.C., Equerry to the Queen, accompanied Lord Mount-Stephen as best man. The wedding was attended by only the relations and most intimate friends of the bride and bridegroom. The bride wore a simply-made, slightly trained gown of ivory satin, a wreath of orange blossoms, and Brussels lace veil. Her ornaments were pearls. The little girls were in blue silk, veiled with lace, wore lace hats, trimmed with blue bows, and carried baskets of pink roses. Lord and Lady Mount-Stephen left London in the afternoon for Dover ''en route'' for Paris. Among the wedding presents were: From the Prince of Wales, trefoil moonstone and diamond brooch. The Duke and Duchess of Connanght, silver gilt cup. The Duchess of Albany, casket in wood, with ormolu mounts. The Duke and Duchess of York, diamond brooch. Prince and Princess Edward of Saxe-Weimar, silver-gilt salts bottle with medallion in top. Princess Victor of Hobenlohe, silver cigar lighter. Duke of Teck, diamond and sapphire brooch. Field-Marshal Sir Donald and Lady Stewart, silver cigar box. Sir Stafford and Lady Northcote, gold and enamel whist box and markers. Colonel and Hon. Mrs. Egerton, silver gilt jewel box. Mr. L. Iveson, silver matchbox. Miss Willmott, cigarette case and matchbox in gun metal studded with gems. Countess of Selkirk, silver inkstand. Countess Somers, tortoiseshell and silver photograph case. Lord de Mauley, ivory paper knife. Colonel Rowland Egerton, tortoiseshell cigar box. Sir Stanley and Lady Clarke, Empire box. Mrs. Arkwright, a "Where is it?" silver mounted. Hon. Harriett Phipps, Diamond Jubilee coin. Colonel Bruce Fellows, tortoiseshell and gold mounted walking stick. Lord and Lady Clifford, similar gift. Sir John McNeill, gold mounted umbrella. Mr. Beverley McJones, silver framed almanack. Mr. Vander, tortoiseshell and silver photograph frame. Household and employés at Brocket Hall, silver bowl. Colonel and Mrs. R. B. Lane, travelling clock. Lord and Lady Eustace Cecil, Turkish inlaid table. Miss Flora MacLeod, silver whist markers. Lady Roberts, "Lord Roberts: Forty-one years in India," bound in green morocco. Mr. and Mrs. Thomas Skinner, opera glasses in gun metal. Mr. Robert Berkeley, book for notes, ormolu mounted. Mr. Douglas, silver antique cigar ashtray. Hon. Alex. York, silver box. Lord and Lady William Seymour, tortoiseshell and silver paper knife. Mr. Frank Farrer, umbrella with cork handle. Mrs. Vyner, long chain, blue and red enamel. Hon. Mrs. Sartoris, silver pincushion. Hon. Mrs. Halford, silver gilt seal. Mrs. Jago, tortoiseshell and ormolu casket. Lady Mary Lloyd crystal seal. Mr. Alfred Sartoris, silver pencil. Lady Eva Dugdale, buckle. Viscountess Doneraile, china for plant. Miss G. Curtis, gold thimble set with stones. Mr. and Mrs. Charles Hopwood, gold, ruby and diamond links. Mr. and Hon. Mrs. West, silver box. Miss Still, silver paperknife. Sir Frederick and Lady Wigan, tortoiseshell and silver clock. Earl Beauchamp and Lady Mary Lygon, enamel and pearl clasp. Mrs. George Peacocke, parasol with violets. Dowager Lady Aylesford, china basket for flowers. Mary Lady Raglan, china bell. Miss Halford, enamel chain, four-leaved shamrock. Miss Willmott, gold hunter repeater watch. Hon. D. Keppel and Captain R. Peel, antique silver box. Mr. Thomas Baring, antique box with miniature. Hon Caroline Roche, silver taper holder. Hon. Mrs. Mitford, photograph frame. Hon. Helen Henniker, cherry silver mounted walking stick. Major Keppel and Mrs. Stephenson, writing pad in heliotrope morocco. Dowager Marchioness Conyngham, tooled leather box with print on lid. Mr. W. Ward Cook, old Sheffield canister. Hon. Mary Thesiger and Miss Wauchope, green morocco writing case. Hon. Mrs. Corbet, enamel charm in case. Mrs. Maxwell Williams, silver and red leather Prayer-book. Mrs. Charles Inge, china ornament. Mr. and Mrs. Reginald Cookson, gold pencil case with diamond monogram. Mr. and Mrs. Robert Berkeley, antique diamond heart bangle. Miss Eleanor Berkeley, mother-o'-pearl fan. Hon. Charles Ellis, sapphire and diamond ring. Mr. and Mrs. Cotton Curtis, silver box. Hon. Mrs. Dalrymple Hamilton, silver heart shaped frame. Miss M. Greathead, heart-shaped photo frame. Mr. Labalmondiere and Mrs. Hext, water colour by Rheam. Lady Maud Warrender, antique silver tray. Viscount and Viscountess Wolseley, stamped leather cabinet box with looking-glass top. Mr. and Mrs. V. F. Tufnell, tortoiseshell tray inlaid gold monogram. Mr. Maxwell Williams, large silver topped salts bottle. Mrs. H. C. Gunston, white leather cardcase with coronet in diamonds. Hon. Osbert Molyneux, tortoiseshell inlaid black lace fan with name. Mr. Gaspard Farrer, enamel and diamond topaz necklace. Mrs. Pratt Barlow, Hymn-book in silver case. Miss Hunter, silver hammered tray. Mrs. Basil Ellis, white silk cushion embroidered. Lady Harcourt, book — "Richard Conway," by G. Williamson. Mr. Laurence Currie, antique tortoiseshell and gold piqué box. Lady Katharine Coke, portrait of H.R.H. Princess Mary Adelaide Duchess of Teck in enamel and silver frame. Miss Addie Paget, piece of Turkish embroidery. Mrs. Gannon, heart-shaped silver box. From household 16, James's-street, S. Scun, L. Oldcum, L. Morgan, and A. Butler, silver panel photo frame with monogram. Mrs. Crutchley, white leather purse and cardcase. Mrs. Charles Harbord, silver and tortoiseshell tea canister. Hon. Mr. A. N. Hood, green enamel heartshaped sleeve links. Mr. and Mrs. Wyndham Portal, embossed leather book slide. Hon. Mr. and Mrs. Sidney Glyn, pair of antique Sheffield candlesticks. Mr. A. O. Kirby, enamel and gold perforated casket for potpourri. Countess of Sefton, Vienna inlaid leather despatch box, with name in gold. Mrs. [Col. 2c/3a] Adair Bruce, ivory carved cardcase. Miss Sybil Corbet, book bound in brown morocco, "The Child of the House." Mrs. Willmott, "Queen Victoria," by Holmes, bound in green crushed morocco, lined with watered silk. Sir Stafford and Lady Northcote, travelling bag with silver gilt fittings. Lady Margaret Levett, miniature frame in case. Arthur St. Leger Glyn, antique Empire ring stand. Miss Sand, tortoiseshell comb with pearl and diamond top. Lord and Lady Strathcona and Mount Royal, diamond brooch. Blanche Lady Rosslyn, ''Christian Year'' in white vellum binding. Miss Nimmo, red glass jar. Miss J. R. Nimmo, glass box with tray. Dowager Countess of Iddesleigh and Mr. Oliver Northcote, silver box with Turkish d'oyleys. Countess Helena Gleichen, antique ivory and gold fan. Rev. F. and Mrs. Colman, silver notebook. Sir Guy and Lady Campbell, white feather fan with tortoiseshell stick. Mr. William Gillett. silver repoussé hand mirror; Countess Vaida Gleichen, book, "Rubiat of Omar Khayyam," bound in Persian red morocco. Colonel Jago, dark blue despatch box with name. Mr. and Mrs. Philip Walker, silver photograph frame. Sir Whittaker and Lady Ellis, silver gilt sugar sifter and salver. Colonel and Mrs. Tufnell, antique silver box. Countess of Chesterfield, pair of cut glass and silver topped scent bottles. Mrs. Tighe, silver menu holder.<ref>"Marriages. Lord Mount-Stephen and Miss Tufnell." ''Clifton Society'' 09 December ''1897 Thursday: 14 [of 16], Cols. 2a–3a [of 3]. British'' Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0002164/18971209/071/0014.</ref></blockquote> ===25 December 1897, Saturday=== Christmas Day 25 December 1897 or so, Sullivan's ballet Victoria and Merrie England closed at the Alhambra Theatre, Leicester Square (Richards 31). According to Richards, "members of the royal family attended on nineteen occasions.<ref name=":0">Richards, Jeffrey. ''Imperialism and Music: Britain, 1876–1953''. Manchester University Press, 2001: 31.</ref> Since 8 July 1897, the program "included a cinematograph film of the Jubilee procession."<ref name=":0" /> ===26 December 1897, Sunday=== Boxing Day === 30 December 1897, Thursday === An entertainment at a party at Blenheim Palace in which some of the costumes from the [[Social Victorians/1897 Fancy Dress Ball|July 1897 Duchess of Devonshire's fancy-dress ball]] were worn:<blockquote>Dramatic entertainment in aid of the restoration fund of Woodstock parish church were given yesterday afternoon and evening in the long library at Blenheim Palace. The first portion of the entertainment consisted of a series of tableaux in which those who look part included the Duchess of Marlborough, Lady Sarah Wilson, Lord Chesterfield, Lord Churchill, Lord and Lady Curzon, Lady Blandford, Ladies Lilian and Norah Spencer Churchill, the Hon. Mrs. A. Bourke, [[Social Victorians/People/Henry White|Mr. and Mrs. Henry White]], and Mr. H. Milner. Except in two cases the tableaux were of an historical character, and they were picturesquely portrayed. Many of the costumes were those worn at the Devonshire House fancy ball last June.<br /><br />The second part of the entertainment consisted of a new musical burlesque in two acts, entitled "œAn Idle Hour," written by Mr. lan Malcolm, M.P. The Countess of Clondyke was taken by the Duchess of Marlborough, and the Duke represented Septimus Sand in a typical get-up of Cousin Jonathan. Lady Randolph Churchill, in sprightly and amusing fashion, delineated an up-to-date lady journalist in the guise of Mrs. Jubilee Junius. Another character deserving of notice was that of Mrs. Oshant, which was entrusted to Lady Churchill. This was an obvious burlesque on Mrs. Ormiston Chant. The principal portion of the incidental music was furnished by Mr. C. W. Perkins, organist to the Birmingham Corporation.<ref>"Politics and Persons." ''St James's Gazette'' 31 December 1897 Friday: 13 [of 16], Col. 1a [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0001485/18971231/069/0013?browse=true.</ref></blockquote> ==Footnotes== {{reflist}} khgfygo6j1r2w57rhpbdbljitqq14i3 0 285384 2693584 2693250 2024-12-27T05:53:46Z Young1lim 21186 /* Linking Libraries */ 2693584 wikitext text/x-wiki === Workings of the GNU Compiler for IA-32 === ==== Overview ==== * Overview ([[Media:Overview.20200211.pdf |pdf]]) ==== Data Processing ==== * Access ([[Media:Access.20200409.pdf |pdf]]) * Operators ([[Media:Operator.20200427.pdf |pdf]]) ==== Control ==== * Conditions ([[Media:Condition.20230630.pdf |pdf]]) * Control ([[Media:Control.20220616.pdf |pdf]]) ==== Function calls ==== * Procedure ([[Media:Procedure.20220412.pdf |pdf]]) * Recursion ([[Media:Recursion.20210824-2.pdf |pdf]]) ==== Pointer and Aggregate Types ==== * Arrays ([[Media:Array.20211018.pdf |pdf]]) * Structures ([[Media:Structure.20220101.pdf |pdf]]) * Alignment ([[Media:Alignment.20201117.pdf |pdf]]) * Pointers ([[Media:Pointer.20201106.pdf |pdf]]) ==== Integer Arithmetic ==== * Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]]) * Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]]) * Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]]) * Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]]) * Borrow ([[Media:Borrow.20241223.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241227.pdf |pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241226.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241226.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241226.pdf |pdf]]) ==== Library Search Path ==== * Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]]) * Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]]) ==== Linking Process ==== * Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]]) * Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]]) * Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]]) * Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]]) * Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]]) * Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]]) * Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]]) ==== Example I ==== * Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]]) * Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]]) * Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]]) ==== Examples II ==== * analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]]) * analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]]) * analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming language]] 1o6e6ko0kxkrtzodb1u5pu3spna1vet 2693591 2693584 2024-12-27T10:45:36Z Young1lim 21186 /* Integer Arithmetic */ 2693591 wikitext text/x-wiki === Workings of the GNU Compiler for IA-32 === ==== Overview ==== * Overview ([[Media:Overview.20200211.pdf |pdf]]) ==== Data Processing ==== * Access ([[Media:Access.20200409.pdf |pdf]]) * Operators ([[Media:Operator.20200427.pdf |pdf]]) ==== Control ==== * Conditions ([[Media:Condition.20230630.pdf |pdf]]) * Control ([[Media:Control.20220616.pdf |pdf]]) ==== Function calls ==== * Procedure ([[Media:Procedure.20220412.pdf |pdf]]) * Recursion ([[Media:Recursion.20210824-2.pdf |pdf]]) ==== Pointer and Aggregate Types ==== * Arrays ([[Media:Array.20211018.pdf |pdf]]) * Structures ([[Media:Structure.20220101.pdf |pdf]]) * Alignment ([[Media:Alignment.20201117.pdf |pdf]]) * Pointers ([[Media:Pointer.20201106.pdf |pdf]]) ==== Integer Arithmetic ==== * Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]]) * Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]]) * Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]]) * Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]]) * Borrow ([[Media:Borrow.20241224.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241227.pdf |pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241226.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241226.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241226.pdf |pdf]]) ==== Library Search Path ==== * Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]]) * Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]]) ==== Linking Process ==== * Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]]) * Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]]) * Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]]) * Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]]) * Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]]) * Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]]) * Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]]) ==== Example I ==== * Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]]) * Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]]) * Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]]) ==== Examples II ==== * analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]]) * analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]]) * analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming language]] gw8ujx42k5ka2zu7s28lbq0sn6qzthu 2693593 2693591 2024-12-27T10:46:30Z Young1lim 21186 /* Integer Arithmetic */ 2693593 wikitext text/x-wiki === Workings of the GNU Compiler for IA-32 === ==== Overview ==== * Overview ([[Media:Overview.20200211.pdf |pdf]]) ==== Data Processing ==== * Access ([[Media:Access.20200409.pdf |pdf]]) * Operators ([[Media:Operator.20200427.pdf |pdf]]) ==== Control ==== * Conditions ([[Media:Condition.20230630.pdf |pdf]]) * Control ([[Media:Control.20220616.pdf |pdf]]) ==== Function calls ==== * Procedure ([[Media:Procedure.20220412.pdf |pdf]]) * Recursion ([[Media:Recursion.20210824-2.pdf |pdf]]) ==== Pointer and Aggregate Types ==== * Arrays ([[Media:Array.20211018.pdf |pdf]]) * Structures ([[Media:Structure.20220101.pdf |pdf]]) * Alignment ([[Media:Alignment.20201117.pdf |pdf]]) * Pointers ([[Media:Pointer.20201106.pdf |pdf]]) ==== Integer Arithmetic ==== * Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]]) * Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]]) * Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]]) * Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]]) * Borrow ([[Media:Borrow.20241225.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241227.pdf |pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241226.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241226.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241226.pdf |pdf]]) ==== Library Search Path ==== * Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]]) * Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]]) ==== Linking Process ==== * Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]]) * Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]]) * Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]]) * Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]]) * Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]]) * Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]]) * Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]]) ==== Example I ==== * Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]]) * Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]]) * Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]]) ==== Examples II ==== * analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]]) * analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]]) * analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming language]] qwtl4oi0u3x89v7fjvhrnmw6hiddmtr 2693596 2693593 2024-12-27T10:47:34Z Young1lim 21186 /* Integer Arithmetic */ 2693596 wikitext text/x-wiki === Workings of the GNU Compiler for IA-32 === ==== Overview ==== * Overview ([[Media:Overview.20200211.pdf |pdf]]) ==== Data Processing ==== * Access ([[Media:Access.20200409.pdf |pdf]]) * Operators ([[Media:Operator.20200427.pdf |pdf]]) ==== Control ==== * Conditions ([[Media:Condition.20230630.pdf |pdf]]) * Control ([[Media:Control.20220616.pdf |pdf]]) ==== Function calls ==== * Procedure ([[Media:Procedure.20220412.pdf |pdf]]) * Recursion ([[Media:Recursion.20210824-2.pdf |pdf]]) ==== Pointer and Aggregate Types ==== * Arrays ([[Media:Array.20211018.pdf |pdf]]) * Structures ([[Media:Structure.20220101.pdf |pdf]]) * Alignment ([[Media:Alignment.20201117.pdf |pdf]]) * Pointers ([[Media:Pointer.20201106.pdf |pdf]]) ==== Integer Arithmetic ==== * Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]]) * Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]]) * Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]]) * Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]]) * Borrow ([[Media:Borrow.20241226.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241227.pdf |pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241226.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241226.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241226.pdf |pdf]]) ==== Library Search Path ==== * Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]]) * Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]]) ==== Linking Process ==== * Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]]) * Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]]) * Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]]) * Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]]) * Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]]) * Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]]) * Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]]) ==== Example I ==== * Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]]) * Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]]) * Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]]) ==== Examples II ==== * analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]]) * analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]]) * analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming language]] 8nn4hfzn1skyyql2b56qa9lx01ifiap 2693598 2693596 2024-12-27T10:48:42Z Young1lim 21186 /* Integer Arithmetic */ 2693598 wikitext text/x-wiki === Workings of the GNU Compiler for IA-32 === ==== Overview ==== * Overview ([[Media:Overview.20200211.pdf |pdf]]) ==== Data Processing ==== * Access ([[Media:Access.20200409.pdf |pdf]]) * Operators ([[Media:Operator.20200427.pdf |pdf]]) ==== Control ==== * Conditions ([[Media:Condition.20230630.pdf |pdf]]) * Control ([[Media:Control.20220616.pdf |pdf]]) ==== Function calls ==== * Procedure ([[Media:Procedure.20220412.pdf |pdf]]) * Recursion ([[Media:Recursion.20210824-2.pdf |pdf]]) ==== Pointer and Aggregate Types ==== * Arrays ([[Media:Array.20211018.pdf |pdf]]) * Structures ([[Media:Structure.20220101.pdf |pdf]]) * Alignment ([[Media:Alignment.20201117.pdf |pdf]]) * Pointers ([[Media:Pointer.20201106.pdf |pdf]]) ==== Integer Arithmetic ==== * Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]]) * Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]]) * Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]]) * Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]]) * Borrow ([[Media:Borrow.20241227.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241227.pdf |pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241226.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241226.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241226.pdf |pdf]]) ==== Library Search Path ==== * Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]]) * Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]]) ==== Linking Process ==== * Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]]) * Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]]) * Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]]) * Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]]) * Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]]) * Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]]) * Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]]) ==== Example I ==== * Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]]) * Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]]) * Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]]) ==== Examples II ==== * analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]]) * analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]]) * analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming language]] ha3dkfdslqfkunykw7rvtk463qi2k6y 2 289273 2693349 2693264 2024-12-26T19:19:09Z Dc.samizdat 2856930 revised abstract 2693349 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with these objects distributed in its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. Each galactic 3-sphere is expanding radially from its center and origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4</math> is <math>c < c_4 < 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and the quantum mechanical atomic theory. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} keaiscwn1bt1j21hybld5dl1s330ybq 2693361 2693349 2024-12-26T19:49:45Z Dc.samizdat 2856930 2693361 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with these objects distributed in its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. Each galactic 3-sphere is expanding radially from its center and origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4</math> is <math>c < c_4 < 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and the quantum mechanics atomic theory. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} omcprt30vbsirmlixef341o5w7w7m6y 2693364 2693361 2024-12-26T19:57:35Z Dc.samizdat 2856930 2693364 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with these objects distributed in its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. Each galactic 3-sphere is expanding radially from its center and origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4</math> is <math>c \leq c_4 \leq 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and the quantum mechanics atomic theory. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} 0w5hc0prxbaulw6vxpu1804xfvyfw0j 2693369 2693364 2024-12-26T20:07:53Z Dc.samizdat 2856930 2693369 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with those objects distributed in its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. Each galactic 3-sphere is expanding radially from its center and origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4</math> is <math>c \leq c_4 \leq 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and the quantum mechanics atomic theory. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} 138wta37k8j8iznxejn619fj0k97u8j 2693370 2693369 2024-12-26T20:09:31Z Dc.samizdat 2856930 2693370 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with those objects distributed in its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. Each galactic 3-sphere is expanding radially from its center and origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4</math> is <math>c \leq c_4 \leq 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and the atomic theory of quantum mechanics. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} bkbbbtb8k588aza72dq1bood88nq4i8 2693372 2693370 2024-12-26T20:17:31Z Dc.samizdat 2856930 2693372 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with those objects distributed in its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. Each galactic 3-sphere is expanding radially from its center and origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4 = 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and the atomic theory of quantum mechanics. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} hkjt5o3qdrmgyojqg813iqp89qw4eav 2693376 2693372 2024-12-26T20:20:16Z Dc.samizdat 2856930 2693376 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with these objects distributed in its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. Each galactic 3-sphere is expanding radially from its center and origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4 = 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and the atomic theory of quantum mechanics. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} cnwhtual42lmllf93wuwciam46rpdan 2693378 2693376 2024-12-26T20:21:32Z Dc.samizdat 2856930 2693378 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with these objects distributed in its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. Each galactic 3-sphere is expanding radially from its center and origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4 = 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} 6c3ow9weusy0rhrhpzblua8vcjl4ajo 2693571 2693378 2024-12-27T03:01:31Z Dc.samizdat 2856930 2693571 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with these objects distributed in its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. The observable universe may be properly visualized as a 4-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4 = 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} a7cj5ak5wm0tz4bz02qq8sonn6z27z5 2693572 2693571 2024-12-27T03:02:26Z Dc.samizdat 2856930 2693572 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with these objects distributed in its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. The observable universe may be visualized as a 4-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4 = 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} a4hm5jjbjjdk0i0lfljv4td5xsmx74x 2693573 2693572 2024-12-27T03:05:48Z Dc.samizdat 2856930 2693573 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with these objects distributed on its 3-dimensional surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. The observable universe may be visualized as a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4 = 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} o63a3a4p0782esnp15qmdrnle9fh3wf 2693574 2693573 2024-12-27T03:06:13Z Dc.samizdat 2856930 2693574 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|June 2023 - December 2024}} <blockquote>'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere, with these objects distributed on its surface. The black hole at a galaxy's center is the 4-ball of empty space they surround. The observable universe may be visualized as a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects though 4-space, also the speed of light through 3-space. The propagation speed of light through 4-space <math>c_4 = 2c</math>. This model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories as expressions of intrinsic symmetries.</blockquote> == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]]. === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|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}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position. .... == Galilean relativity in a space of four orthogonal dimensions == Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions. General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space <math>R^4</math>, spherical 4-space <math>S^4</math>, or any orthogonal 4-manifold. Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics. Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light. === Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions === Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries. === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... === Physics === .... === Thoreau's spherical relativity === Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}]}} .... == Conclusions== === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== ... ==== Galaxies ==== The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side). == Revolutions == The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe. When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity <math>c</math>. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since. == Origins of the theory == Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.'' The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}} == Notes == {{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 (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. 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. 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}} {{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}} {{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=}} {{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 vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the 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.}}|name=24-cell vertex figure}} {{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=}} {{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. [[24-cell#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 [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#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 [[24-cell#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. 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 [[24-cell#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° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#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'' completely orthogonal directions 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 as in a simple rotation.{{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 by rotating toward it; it can only reach the 16-cell ''beyond'' it. 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 sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{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 the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{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'''. 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 (after ''two'' revolutions). 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 a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{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 [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[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]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[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|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, 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}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#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]] equal to the square root of four times the square of that distance. 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}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} 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}} {{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}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{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}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{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}} {{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}} {{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 [[W:hyperplane|hyperplane]] 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 (completely orthogonal){{Efn|name=completely orthogonal planes}} 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 are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{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 diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{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. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but 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. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Notelist|40em}} == Citations == {{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.}} {{Reflist|40em}} == 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|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston}} * {{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 journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1-3 | pp=59-71 }} * {{Cite journal|last=Stillwell|first=John|author-link=W:John Colin Stillwell|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} * {{Cite book | last1=Conway | first1=John H. | author-link1=W:John Horton Conway | last2=Burgiel | first2=Heidi | last3=Goodman-Strauss | first3=Chaim | author-link3=W:Chaim Goodman-Strauss | year=2008 | title=The Symmetries of Things | publisher=A K Peters | place=Wellesley, MA | title-link=W:The Symmetries of Things }} * {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}} * {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}} * {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}} * {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. 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 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 | pages=1423-1449 | doi=10.3390/sym2031423 }} * {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} *{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}} {{Refend}} poaktwo1pjtibygpq5a8xbabc84jspy 3 290866 2693453 2682784 2024-12-26T23:41:00Z Tule-hog 2984180 /* BookCat on main page */ new section 2693453 wikitext text/x-wiki {{User talk-page header}} {{#babel:custodian|curator|global rollbacker|en-N|ja-N|Commons|Wiktionary}} {{Userboxtop}} {{User Wikipedia admin|simple}} {{User_admin Wiktionary|Simple English Wiktionary|lang_code=simple}} {{User Meta-Wiki}} {{User Wikidata}} {{User Wikiquote}} {{User contrib|13,000}} {{User contrib SUL|440,000}} {{Userboxbottom}} == Archives == *[[/2023]] *[[/2024]] *[https://en.wikiversity.org/wiki/Special:Log?type=delete&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Deletion log] *[https://en.wikiversity.org/wiki/Special:Log?type=protect&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Protection log] *[https://en.wikiversity.org/wiki/Special:Log?type=import&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Import log] *[https://en.wikiversity.org/wiki/Special:Log?type=move&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Move log] *[https://en.wikiversity.org/wiki/Special:Log?type=block&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Block log] == Welcome == {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] MathXplore!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:49, 1 December 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} == BookCat on main page == Hello! I noticed [[Introduction to graph theory]] is not in [[:Category:Introduction to graph theory]]. Should it be, or is that part of the categorization scheme? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:41, 26 December 2024 (UTC) kembtjj4l73rbs3jc17n47nfjkngnqp 2693456 2693453 2024-12-26T23:43:41Z MathXplore 2888076 /* BookCat on main page */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2693456 wikitext text/x-wiki {{User talk-page header}} {{#babel:custodian|curator|global rollbacker|en-N|ja-N|Commons|Wiktionary}} {{Userboxtop}} {{User Wikipedia admin|simple}} {{User_admin Wiktionary|Simple English Wiktionary|lang_code=simple}} {{User Meta-Wiki}} {{User Wikidata}} {{User Wikiquote}} {{User contrib|13,000}} {{User contrib SUL|440,000}} {{Userboxbottom}} == Archives == *[[/2023]] *[[/2024]] *[https://en.wikiversity.org/wiki/Special:Log?type=delete&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Deletion log] *[https://en.wikiversity.org/wiki/Special:Log?type=protect&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Protection log] *[https://en.wikiversity.org/wiki/Special:Log?type=import&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Import log] *[https://en.wikiversity.org/wiki/Special:Log?type=move&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Move log] *[https://en.wikiversity.org/wiki/Special:Log?type=block&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Block log] == Welcome == {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] MathXplore!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:49, 1 December 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} == BookCat on main page == Hello! I noticed [[Introduction to graph theory]] is not in [[:Category:Introduction to graph theory]]. Should it be, or is that part of the categorization scheme? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:41, 26 December 2024 (UTC) : The author may have forgotten or didn't know about the template. I have added the template accordingly. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 23:43, 26 December 2024 (UTC) 7vlycjwc4rhkqeanu6kk65hzy4lf64s 2693459 2693456 2024-12-26T23:46:53Z Tule-hog 2984180 /* BookCat on main page */ Reply 2693459 wikitext text/x-wiki {{User talk-page header}} {{#babel:custodian|curator|global rollbacker|en-N|ja-N|Commons|Wiktionary}} {{Userboxtop}} {{User Wikipedia admin|simple}} {{User_admin Wiktionary|Simple English Wiktionary|lang_code=simple}} {{User Meta-Wiki}} {{User Wikidata}} {{User Wikiquote}} {{User contrib|13,000}} {{User contrib SUL|440,000}} {{Userboxbottom}} == Archives == *[[/2023]] *[[/2024]] *[https://en.wikiversity.org/wiki/Special:Log?type=delete&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Deletion log] *[https://en.wikiversity.org/wiki/Special:Log?type=protect&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Protection log] *[https://en.wikiversity.org/wiki/Special:Log?type=import&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Import log] *[https://en.wikiversity.org/wiki/Special:Log?type=move&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Move log] *[https://en.wikiversity.org/wiki/Special:Log?type=block&user=MathXplore&page=&wpdate=&tagfilter=&subtype=&wpFormIdentifier=logeventslist Block log] == Welcome == {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] MathXplore!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:49, 1 December 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} == BookCat on main page == Hello! I noticed [[Introduction to graph theory]] is not in [[:Category:Introduction to graph theory]]. Should it be, or is that part of the categorization scheme? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:41, 26 December 2024 (UTC) : The author may have forgotten or didn't know about the template. I have added the template accordingly. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 23:43, 26 December 2024 (UTC) ::Thanks. Tangential follow-up, do you think there would be any value in creating {{tlx|Resource category}}/[[:Category:Resource categories]] to track categories dedicated to a single resource and its subpages? It might make for easier maintenance, but I don't have much experience with {{tlx|BookCat}} or other solutions. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:46, 26 December 2024 (UTC) 3hpl82g7iz3ybd5x00ffayymp9fieax 0 292742 2693319 2691303 2024-12-26T18:20:32Z TMorata 860721 added link, corrected spacing 2693319 wikitext text/x-wiki {{:Global Audiology/Header}} You can contribute to the online information about audiology services and practices worldwide by creating content on Global Audiology at Wikiversity. Share your knowledge and experiences to help others learn more about audiology. Your contributions to Wikiversity will help ensure that everyone has access to reliable information about audiology services and practices. Here is a guide for adding or editing content on audiology practices on Wikiversity. Whether you're a Wikimedia user or not, we have prepared steps for you. We also provide resources and tips to help you create and edit content (located in the Resources section). Tutorials specific to editing Wikipedia are at the bottom of this page. We hope this guide can help you get started on your contribution. We welcome all contributions, no matter how small. We are always open to feedback, so please let us know if you have any questions or suggestions. Thank you for your interest in contributing to Global Audiology at Wikiversity! {{HRow}} ===Suggested article structure=== The following is the suggested article structure for a country-specific page. Working on a small group preferably working closely with the local audiology society or a professional body (e.g., [https://isa-audiology.org/affiliates/overview ISA's Affiliated Societies]) is suggested to ensure the country specific page will include comprehensive information. Also, providing links to relevant websites as well as adding media (images, videos) is likely to enhance reading experience of the content. *Brief Country Information *History of Audiology and Aural Care *Incidence and Prevalence of Hearing Loss *Hearing Care Services **Professionals providing hearing care services **Audiological services **Services offered by Otolaryngologists, Otologists, and Otoneurologists **Role of primary health care providers and community health workers in hearing care **Laws related to hearing care services *Education and Professional Practice **Education of professionals working in hearing care services **Professional and Regulatory Bodies **Scope of Practice and Licensing *Audiology Research *Audiology Charities *Challenges and Opportunities *Acknowledgments *References *Author Information {{HRow}} ===Step-by-step=== ====For Wikimedia Users==== ===== Step 1: Create an account ===== Before creating content on Wikiversity, you need to create a Wikimedia account. If you already have an account, you can skip this step and move on to step 2. Go to the Wikimedia homepage click "Create account" in the upper right-hand corner, and follow the prompts to create your account (or directly to [https://meta.wikimedia.org/w/index.php?title=Special:CreateAccount&returnto=Main+Page Create Account]. You must provide your username and email address and choose a password. ===== Step 2: Familiarize yourself with the community guidelines ===== Before contributing to Global Audiology at Wikiversity, take the time to review existing entries and read the community guidelines. This will help you understand what content is acceptable and what isn't and help you avoid mistakes that could result in your contributions being edited or deleted. For instance, the language is neutral, the content you develop is politically neutral, does not celebrate one individual, company, or institution, and instead presents a more balanced view of the structure and status of audiology in the country or region. ===== Step 3: Visit the Global Audiology Homepage ===== The Global Audiology Homepage provides a wealth of information about the project's mission and purpose, as well as resources and other materials related to hearing health and audiology. Check out the [https://en.wikiversity.org/wiki/Global_Audiology homepage] and explore to learn more about our initiatives and how you can get involved. ===== Step 4: Conduct research ===== Find the Global Audiology content you want to add to or modify. Prepare your content by doing some research on your chosen topic. Use reliable sources, such as peer-reviewed academic journals or bona fide websites and/or news services. We suggest you include references to key statements. It is easy, Wikimedia formats it for you). However, it may be difficult to find references to all the content. ===== Step 5: Create your content ===== As soon as you have gathered your research, start creating content. Click on the "Edit" button at the page's top, and using the pencil icon, select “Visual editor”. The toolbar lets you format your text and add headings, links, and other formatting elements. You can edit or add new text, images, or multimedia and edit existing content. The article structure should be followed and your content should be neutral, well-organized, easy to read, accurate, relevant, and properly cited (more training resources are in the Resources section). ===== Step 6: Preview and save ===== Before you publish your changes, preview them to ensure they look as you intended. Click the "Apply changes" button and choose the "Show preview" option. You will be able to see how your changes will look before they are actually published. If you need to make further changes to the content, click on the "Edit" button again. To discuss your changes with other contributors or get feedback, click the "Discussion" tab. ===== Step 7: Publish ===== Once you are satisfied with your changes, click "Publish". Please describe succinctly what you have done (created content, added citation, added category, added hyperlink, etc.). Well done! You have successfully added or modified content for the Global Audiology Wikiversity! Continue to update and improve the content over time based on feedback and changes in the field of audiology. Also, consider joining the Discussion page to connect with other Global Audiology contributors. Volunteer subject matter editors and Global Audiology representatives will be alerted of your edit and review it. They might suggest edits or contact you if they have questions. ===== Step 8. More publishing ===== With style modifications, your contribution could become an article for a peer-reviewed journal, such as [https://www.tandfonline.com/doi/full/10.1080/14992020701770843 Audiology in Brazil], [https://www.tandfonline.com/doi/abs/10.3109/00206097409071699 Audiology in Greenland], [https://www.tandfonline.com/doi/abs/10.3109/00381796809075452 Audiology in India], [https://www.tandfonline.com/doi/full/10.1080/14992020500485650 Audiology in South Africa], [https://www.tandfonline.com/doi/full/10.1080/14992020802203322 Audiology education and practice from an international perspective], and others. ====For Non-Wikimedia Users==== ===== Step 1: Visit the Global Audiology homepage ===== The Global Audiology Homepage provides a wealth of information about the project's mission and purpose, as well as resources and other materials related to hearing health and audiology. Check out the [https://en.wikiversity.org/wiki/Global_Audiology homepage] and explore to learn more about our initiatives and how you can get involved. ===== Step 2: Familiarize yourself with the community guidelines ===== Before contributing to Global Audiology at Wikiversity, take the time to read the community guidelines. This will help you understand what content is acceptable and what isn't, and help you avoid mistakes that could result in your contributions being edited or deleted. For instance, the language is neutral, the content you develop is politically neutral, does not celebrate one individual, company, or institution, and instead presents a more balanced view of the status of audiology in the country or region. ===== Step 3: Conduct research ===== Find the Global Audiology content you want to add or modify. Prepare your content by doing some research on your chosen topic. Use reliable sources, such as peer-reviewed academic journals or reputable websites and/or news services. We suggest you include references to key statements and facts. It is easy, Wikimedia formats it for you. However, it may be difficult to find references to all the content. ===== Step 4: Create your content ===== As soon as you have gathered your research, start creating content. Your article can be formatted in Word or PDF. Article structure should be followed, and your content should be neutral, well-organized, easy to read, accurate, relevant, and properly cited. ===== Step 5: Submit your article ===== Please send your article to the Global Audiology team or ISA administrators, so we can publish it on Wikiversity for you. {{HRow}} ===Notes=== * For other Wikiversity training resources, check-out the [https://dashboard.wikiedu.org/training dashboard]. * If the country you wish to contribute to or write an article about is not yet available on Global Audiology at Wikiversity, please reach out to our Global Audiology team or ISA administrators. Let us know what country you would like to be added and we will create it for you. * If you teach audiology, you can ask your students to work on this project, as many others have done for audiology content. Students usually find this activity very motivating. Some examples specific to audiology include: ;Various universities outside the US and Canada: [https://outreachdashboard.wmflabs.org/campaigns/hearing_health__20222024/programs Hearing health campaign 2022-2024] ;University of the Witwatersrand, South Africa: [https://outreachdashboard.wmflabs.org/courses/University_of_the_Witwatersrand/Pathology_of_the_ear_(2nd_Semester)/home Pathology of the ear] ;University of Montreal, Canada: [https://dashboard.wikiedu.org/courses/University_of_Montreal/Promotion_and_prevention_in_audiology-Promotion_et_pr%C3%A9vention_en_audiologie_(Winter-Spring) Promotion and prevention in audiology] ;University of Northern Colorado, USA: [https://dashboard.wikiedu.org/courses/University%20of%20Northern%20Colorado/Hearing%20Loss%20Prevention%20(Fall%20Semester) Hearing loss prevention] {{HRow}} ===Tutorials=== |[[File:GLAM logo transparent.png|left|100 px]] |This step-by-step guide brings together some of the best resources to help you get started in Wikipedia. It is based on a [[w:WP:GLAM/TCMI/MAP|guide]] originally created by [[w:User:LoriLee|LoriLee]] for middle and high school students to edit Wikipedia. If they can do it, you can! If you would like more general information on why you should contribute to Wikipedia, please see [https://outreachdashboard.wmflabs.org/training training library] ===Training video=== [[File:Editing Wikipedia.webm|Editing Wikipedia]] {{:Global Audiology/footer}} {{HRow}} ===Suggested themes=== Did not find the topic you are looking for? Please let us know: {{Clickable button 2|CONTACT US|url=mailto:contact@globalaudiology.org}} d8qfyr8yki4nem9jfnhmcxp8y5330jv 2693320 2693319 2024-12-26T18:22:35Z TMorata 860721 /* Tutorials */ moved link 2693320 wikitext text/x-wiki {{:Global Audiology/Header}} You can contribute to the online information about audiology services and practices worldwide by creating content on Global Audiology at Wikiversity. Share your knowledge and experiences to help others learn more about audiology. Your contributions to Wikiversity will help ensure that everyone has access to reliable information about audiology services and practices. Here is a guide for adding or editing content on audiology practices on Wikiversity. Whether you're a Wikimedia user or not, we have prepared steps for you. We also provide resources and tips to help you create and edit content (located in the Resources section). Tutorials specific to editing Wikipedia are at the bottom of this page. We hope this guide can help you get started on your contribution. We welcome all contributions, no matter how small. We are always open to feedback, so please let us know if you have any questions or suggestions. Thank you for your interest in contributing to Global Audiology at Wikiversity! {{HRow}} ===Suggested article structure=== The following is the suggested article structure for a country-specific page. Working on a small group preferably working closely with the local audiology society or a professional body (e.g., [https://isa-audiology.org/affiliates/overview ISA's Affiliated Societies]) is suggested to ensure the country specific page will include comprehensive information. Also, providing links to relevant websites as well as adding media (images, videos) is likely to enhance reading experience of the content. *Brief Country Information *History of Audiology and Aural Care *Incidence and Prevalence of Hearing Loss *Hearing Care Services **Professionals providing hearing care services **Audiological services **Services offered by Otolaryngologists, Otologists, and Otoneurologists **Role of primary health care providers and community health workers in hearing care **Laws related to hearing care services *Education and Professional Practice **Education of professionals working in hearing care services **Professional and Regulatory Bodies **Scope of Practice and Licensing *Audiology Research *Audiology Charities *Challenges and Opportunities *Acknowledgments *References *Author Information {{HRow}} ===Step-by-step=== ====For Wikimedia Users==== ===== Step 1: Create an account ===== Before creating content on Wikiversity, you need to create a Wikimedia account. If you already have an account, you can skip this step and move on to step 2. Go to the Wikimedia homepage click "Create account" in the upper right-hand corner, and follow the prompts to create your account (or directly to [https://meta.wikimedia.org/w/index.php?title=Special:CreateAccount&returnto=Main+Page Create Account]. You must provide your username and email address and choose a password. ===== Step 2: Familiarize yourself with the community guidelines ===== Before contributing to Global Audiology at Wikiversity, take the time to review existing entries and read the community guidelines. This will help you understand what content is acceptable and what isn't and help you avoid mistakes that could result in your contributions being edited or deleted. For instance, the language is neutral, the content you develop is politically neutral, does not celebrate one individual, company, or institution, and instead presents a more balanced view of the structure and status of audiology in the country or region. ===== Step 3: Visit the Global Audiology Homepage ===== The Global Audiology Homepage provides a wealth of information about the project's mission and purpose, as well as resources and other materials related to hearing health and audiology. Check out the [https://en.wikiversity.org/wiki/Global_Audiology homepage] and explore to learn more about our initiatives and how you can get involved. ===== Step 4: Conduct research ===== Find the Global Audiology content you want to add to or modify. Prepare your content by doing some research on your chosen topic. Use reliable sources, such as peer-reviewed academic journals or bona fide websites and/or news services. We suggest you include references to key statements. It is easy, Wikimedia formats it for you). However, it may be difficult to find references to all the content. ===== Step 5: Create your content ===== As soon as you have gathered your research, start creating content. Click on the "Edit" button at the page's top, and using the pencil icon, select “Visual editor”. The toolbar lets you format your text and add headings, links, and other formatting elements. You can edit or add new text, images, or multimedia and edit existing content. The article structure should be followed and your content should be neutral, well-organized, easy to read, accurate, relevant, and properly cited (more training resources are in the Resources section). ===== Step 6: Preview and save ===== Before you publish your changes, preview them to ensure they look as you intended. Click the "Apply changes" button and choose the "Show preview" option. You will be able to see how your changes will look before they are actually published. If you need to make further changes to the content, click on the "Edit" button again. To discuss your changes with other contributors or get feedback, click the "Discussion" tab. ===== Step 7: Publish ===== Once you are satisfied with your changes, click "Publish". Please describe succinctly what you have done (created content, added citation, added category, added hyperlink, etc.). Well done! You have successfully added or modified content for the Global Audiology Wikiversity! Continue to update and improve the content over time based on feedback and changes in the field of audiology. Also, consider joining the Discussion page to connect with other Global Audiology contributors. Volunteer subject matter editors and Global Audiology representatives will be alerted of your edit and review it. They might suggest edits or contact you if they have questions. ===== Step 8. More publishing ===== With style modifications, your contribution could become an article for a peer-reviewed journal, such as [https://www.tandfonline.com/doi/full/10.1080/14992020701770843 Audiology in Brazil], [https://www.tandfonline.com/doi/abs/10.3109/00206097409071699 Audiology in Greenland], [https://www.tandfonline.com/doi/abs/10.3109/00381796809075452 Audiology in India], [https://www.tandfonline.com/doi/full/10.1080/14992020500485650 Audiology in South Africa], [https://www.tandfonline.com/doi/full/10.1080/14992020802203322 Audiology education and practice from an international perspective], and others. ====For Non-Wikimedia Users==== ===== Step 1: Visit the Global Audiology homepage ===== The Global Audiology Homepage provides a wealth of information about the project's mission and purpose, as well as resources and other materials related to hearing health and audiology. Check out the [https://en.wikiversity.org/wiki/Global_Audiology homepage] and explore to learn more about our initiatives and how you can get involved. ===== Step 2: Familiarize yourself with the community guidelines ===== Before contributing to Global Audiology at Wikiversity, take the time to read the community guidelines. This will help you understand what content is acceptable and what isn't, and help you avoid mistakes that could result in your contributions being edited or deleted. For instance, the language is neutral, the content you develop is politically neutral, does not celebrate one individual, company, or institution, and instead presents a more balanced view of the status of audiology in the country or region. ===== Step 3: Conduct research ===== Find the Global Audiology content you want to add or modify. Prepare your content by doing some research on your chosen topic. Use reliable sources, such as peer-reviewed academic journals or reputable websites and/or news services. We suggest you include references to key statements and facts. It is easy, Wikimedia formats it for you. However, it may be difficult to find references to all the content. ===== Step 4: Create your content ===== As soon as you have gathered your research, start creating content. Your article can be formatted in Word or PDF. Article structure should be followed, and your content should be neutral, well-organized, easy to read, accurate, relevant, and properly cited. ===== Step 5: Submit your article ===== Please send your article to the Global Audiology team or ISA administrators, so we can publish it on Wikiversity for you. {{HRow}} ===Notes=== * For other Wikiversity training resources, check-out the [https://dashboard.wikiedu.org/training dashboard]. * If the country you wish to contribute to or write an article about is not yet available on Global Audiology at Wikiversity, please reach out to our Global Audiology team or ISA administrators. Let us know what country you would like to be added and we will create it for you. * If you teach audiology, you can ask your students to work on this project, as many others have done for audiology content. Students usually find this activity very motivating. Some examples specific to audiology include: ;Various universities outside the US and Canada: [https://outreachdashboard.wmflabs.org/campaigns/hearing_health__20222024/programs Hearing health campaign 2022-2024] ;University of the Witwatersrand, South Africa: [https://outreachdashboard.wmflabs.org/courses/University_of_the_Witwatersrand/Pathology_of_the_ear_(2nd_Semester)/home Pathology of the ear] ;University of Montreal, Canada: [https://dashboard.wikiedu.org/courses/University_of_Montreal/Promotion_and_prevention_in_audiology-Promotion_et_pr%C3%A9vention_en_audiologie_(Winter-Spring) Promotion and prevention in audiology] ;University of Northern Colorado, USA: [https://dashboard.wikiedu.org/courses/University%20of%20Northern%20Colorado/Hearing%20Loss%20Prevention%20(Fall%20Semester) Hearing loss prevention] {{HRow}} ===Tutorials=== |[[File:GLAM logo transparent.png|left|100 px]] |This step-by-step [[w:WP:GLAM/TCMI/MAP|guide]] brings together some of the best resources to help you get started in Wikipedia. It is based on a guide originally created by [[w:User:LoriLee|LoriLee]] for middle and high school students to edit Wikipedia. If they can do it, you can! If you would like more general information on why you should contribute to Wikipedia, please see [https://outreachdashboard.wmflabs.org/training training library] ===Training video=== [[File:Editing Wikipedia.webm|Editing Wikipedia]] {{:Global Audiology/footer}} {{HRow}} ===Suggested themes=== Did not find the topic you are looking for? Please let us know: {{Clickable button 2|CONTACT US|url=mailto:contact@globalaudiology.org}} 7vllaze8txq6lzcjeiq8r2j9swqdwbt 0 297051 2693469 2543737 2024-12-27T00:08:26Z Tule-hog 2984180 Bot: Replacing category Networks with [[:Category:Networking|Networking]] 2693469 wikitext text/x-wiki ==Learning Content Summary== The Library Network learning resource provides librarians and library students with an understanding of library networking. Learners will grasp how multiple libraries collaborate, share resources, and improve services through interconnected information systems. By studying library networking, participants will learn how it enhances access to diverse information resources, complements traditional services with digital resources, and opens new roles for librarians. Additionally, learners will explore the benefits of Electronic Document Delivery (EDD) as a cost-effective solution for obtaining periodicals. This resource equips participants with the knowledge to support collaboration and adapt to evolving information technology trends for efficient library services. ==Goals== {{TOC right|limit|limit=2}} At the end of this study, learners should be able to: * Understand the concept of library networking and its significance in facilitating collaboration and resource sharing among multiple libraries. * Appreciate the benefits of library networking, including increased access to diverse information resources, complementing traditional services with digital resources, and improving information delivery. * Recognize the role of library networking in enabling new opportunities and roles for librarians, such as managing and reorganizing information, developing new applications, and customizing software to support teaching, research, and learning activities. * Gain insights into Electronic Document Delivery (EDD) as a viable alternative to expensive journal subscriptions, allowing libraries to efficiently meet the information needs of their patrons. * Acquire knowledge about adapting to evolving information technology trends for efficient and effective library services through collaborative efforts among libraries. ==Definition of Library Network== [[File:Social Network Diagram (segment).svg|alt=Network Diagram|thumb|Network Diagram]] A library network is widely defined as a group of libraries coming together with an understanding to help each other to meet the information needs of their patrons. It is a collection of interconnected information systems and communication facilities that work together through a more or less formal agreement to perform information handling operations to provide better services to users<ref>https://www.researchgate.net/publication/310673806_Use_and_Importance_of_Barcode_System_in_Libraries</ref>. As a result, it is crucial to emphasize that the critical aspect of library networking is the sharing of library materials among libraries through the use of information and communication technologies. ==Benefits of Networking in Nigeria Library== === Increased Access to Local and International Libraries and Information Resources === One of the most important benefits of networking is that it makes available the products of a variety of information suppliers and facilitates interaction with library and information resources. Due to budget shortages and growing library material prices, many libraries have turned to networks to access a variety of services. The realization of the global digital library is built on collaboration, sharing, and open technology, which is, of course, central to the delivery of the global digital library. Academic libraries can use the Joint Academic Network Service (JANET) for a variety of purposes, including access to textual and numerical databases, access to periodical agents and book suppliers, access to and transfer of bibliographic records, access to networked bibliographic and other databases, and so on. This ability to access national networks (such as JANET) enabled universities to use network technology more easily than by running their system. Once connected, the interface is the same and operates as if the resource was accessed locally; such users are likely to be reflected over the internet.<ref>https://nou.edu.ng/coursewarecontent/LIS%20320%20.pdf</ref> === Complement to other Resources/Services === The internet, as an international complement to traditional library reference books, has the potential to provide up-to-date information when more traditional publishing kinds may be locked. While most traditional reference resources, led to journal articles or books, www-based resources led to a variety of information sources, including unpublished documents, project proposals, online sites, and so on. A notable example of a library that incorporated the internet into its operations via a public access internet gateway system known as (Bodeleian Access to Remote Databases) is the Bodeleian Library. Its experience also demonstrates that it is possible to integrate electronic and printed content and present the two as complementary. === Improved Traditional Information Services === The integration of new electronic resources and services with traditional activities is a progression. Among the many and various chances that IT provides libraries to complement and improve current traditional services, the main factor underlying networks is that it allows libraries to give multimedia-based material in ways that they have not been able to do previously. Several academic libraries use the internet to showcase and explain the services they provide, either in-house or on the Internet, allowing end users to interact with them via the internet. General information about the library's location, rules, and registration procedures; information about reader services (e.g., loans, reservations, and available facilities); information about collection and subject access; and information about people are among the types of information libraries post on the web (e.g. staff profiles). Other libraries' OPACs, newsgroups, bibliographic databases, electronic journals, and other library-related websites, to name a few, are among the external services to which links are provided by a library's home page.<ref>Khalid, H.M. (2000). "Co-operation and networking in library and information systems of advanced countries: a framework for countries with less developed systems<nowiki>''</nowiki>, Library Review, Vol. 49 Nos. 1 and 2, pp. 57-63.</ref> === New Roles for Libraries and Librarians === Library networking improves access to information and gives libraries and librarians a plethora of new opportunities and roles, such as; creating, managing, filtering, locating, and reorganizing information, customizing software, developing new applications, and translating information into different formats to support teaching, research, and learning activities. The evolution of information technology necessitates new abilities for library employees to steer the evolution and avoid becoming obsolete in the corporate world of the twenty-first century. === Electronic Document Delivery (EDD) === Libraries and information centres in Nigeria are struggling to meet the demands of an ever-increasing number of clients while also dealing with steep increases in the cost of books and journals. The increasing difficulty in subscribing to core journals due to price increases and the necessity to make additional library space play a big part in the acceptability of the EDD service, which is a feasible alternative to pricey journal subscriptions. Instead of paying for periodical subscriptions, money may be spent on acquiring and delivering periodicals. The British Library Document Supply Center (BLDSC), the world's largest institution dedicated to the supply of documents on loan or as surrogate copies to remote users, is a notable example of Electronic Document Delivery. (Each year, it receives roughly 3.7 million queries.) However, as technology advances and a growing number of full-text journals become available on the internet, silver platter's goal to create a global library is "search by search." It is an internet-based service effort that allows users (libraries) to search the bibliographic records of Silver Platter for (free) and unlimited searches. The linker on Silver Platter connects users from the bibliographic record to the whole record. Users begin paying when they wish to see the search result or the actual (full text) record. Users must pay, just as they must for document delivery services; however, instead of receiving the document by fax or mail, the user will receive it automatically on the screen. Some databases, which a library is likely to access only a few times per year, do not require a full subscription. This internet-based new service adds resources to a library's collection without the need for additional annual memberships or online connections now that libraries can have free access to silver platter's search-by-search collection on a pay-as-you-go basis (paying only for what libraries use), Furthermore, because the balance of a library account is presented on the screen together with details of the databases searched, usage statistics can assist libraries in making the most use of their money.<ref>Williams, F. (1997). Electronic Document Delivery: A trial in an academic library. Retrieved from <nowiki>http://www.ariadne.ac.uk/issue10/</nowiki>. </ref> == Question for Practice == # Define library networking and explain its significance in supporting collaboration and resource sharing among multiple libraries. # What are the benefits of library networking, and how does it contribute to increased access to diverse information resources for library users? # How does library networking complement traditional library services with digital resources, and what are the advantages of integrating electronic resources with traditional activities? # In what ways does library networking create new opportunities and roles for librarians, and how can they leverage technology to support teaching, research, and learning activities? # Explain the concept of Electronic Document Delivery (EDD) in library networking, and discuss its role as a cost-effective solution for obtaining periodicals and meeting the information needs of library patrons. == References == [[Category:Library and Information Science]] [[Category:Networking]] k52pf1bzpd4askmnfy1un4v9a6b3pn8 0 301480 2693317 2693168 2024-12-26T18:16:26Z Private lecturer (celestial) 2975755 /* Image of God */ [[w:Trompenaars%27s_model_of_national_culture_differences|cultural dimension]] 2693317 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] {{-}} == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a [[w:Trompenaars%27s_model_of_national_culture_differences|cultural dimension]]. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attentional bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as '[[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]]'. === Getting a giraffe through an eye of a needle === The general recommendation, of course, is to be careful against the unknown requirements of heaven, which may be culturally unexpected, but logically sophisticated and therefore to prefer to err in favor of ethics rather than the opposite. The solution to the problem of getting a giraffe through an eye of a needle is an "animal trainer" (upbringing, education, mentoring, moral culture and ethics). In a capitalist society, when competitors (or even coworkers) may be seen as enemies on a regular basis, love of enemies could obviously also be seen to include granting natural rights to those “enemies” and neither choice-supportive bias nor attentional bias are helpful to do so. [[de:Ethik/Leben nach dem Tod]] hw56by6k2pe1vduklsebvvle58xei4p 2693318 2693317 2024-12-26T18:17:49Z Private lecturer (celestial) 2975755 /* Image of God */ [[File:7 Dimensions of culture.svg|thumbnail|7 Dimensions of Culture]] 2693318 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] {{-}} == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === [[File:7 Dimensions of culture.svg|thumbnail|7 Dimensions of Culture]] The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a [[w:Trompenaars%27s_model_of_national_culture_differences|cultural dimension]]. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attentional bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as '[[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]]'. === Getting a giraffe through an eye of a needle === The general recommendation, of course, is to be careful against the unknown requirements of heaven, which may be culturally unexpected, but logically sophisticated and therefore to prefer to err in favor of ethics rather than the opposite. The solution to the problem of getting a giraffe through an eye of a needle is an "animal trainer" (upbringing, education, mentoring, moral culture and ethics). In a capitalist society, when competitors (or even coworkers) may be seen as enemies on a regular basis, love of enemies could obviously also be seen to include granting natural rights to those “enemies” and neither choice-supportive bias nor attentional bias are helpful to do so. [[de:Ethik/Leben nach dem Tod]] klpyz96ip2w30el9jymg4pgkxuzpcx1 2693324 2693318 2024-12-26T18:53:56Z Private lecturer (celestial) 2975755 /* Love of enemies */ wording: particularly worthy 2693324 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] {{-}} == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === [[File:7 Dimensions of culture.svg|thumbnail|7 Dimensions of Culture]] The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a [[w:Trompenaars%27s_model_of_national_culture_differences|cultural dimension]]. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as particularly worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attentional bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as '[[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]]'. === Getting a giraffe through an eye of a needle === The general recommendation, of course, is to be careful against the unknown requirements of heaven, which may be culturally unexpected, but logically sophisticated and therefore to prefer to err in favor of ethics rather than the opposite. The solution to the problem of getting a giraffe through an eye of a needle is an "animal trainer" (upbringing, education, mentoring, moral culture and ethics). In a capitalist society, when competitors (or even coworkers) may be seen as enemies on a regular basis, love of enemies could obviously also be seen to include granting natural rights to those “enemies” and neither choice-supportive bias nor attentional bias are helpful to do so. [[de:Ethik/Leben nach dem Tod]] 3eax6hiwfxpiubchjfa2sws2aqe6zo2 2693344 2693324 2024-12-26T19:14:46Z Private lecturer (celestial) 2975755 /* The Twelve Apostles */ [[w:papal conclave|]] 2693344 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] {{-}} == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === [[File:7 Dimensions of culture.svg|thumbnail|7 Dimensions of Culture]] The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a [[w:Trompenaars%27s_model_of_national_culture_differences|cultural dimension]]. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as particularly worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a [[w:papal conclave|papal conclave]], which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attentional bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as '[[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]]'. === Getting a giraffe through an eye of a needle === The general recommendation, of course, is to be careful against the unknown requirements of heaven, which may be culturally unexpected, but logically sophisticated and therefore to prefer to err in favor of ethics rather than the opposite. The solution to the problem of getting a giraffe through an eye of a needle is an "animal trainer" (upbringing, education, mentoring, moral culture and ethics). In a capitalist society, when competitors (or even coworkers) may be seen as enemies on a regular basis, love of enemies could obviously also be seen to include granting natural rights to those “enemies” and neither choice-supportive bias nor attentional bias are helpful to do so. [[de:Ethik/Leben nach dem Tod]] fofkesrayofj31oktvnqetydruiu96a 2693350 2693344 2024-12-26T19:23:32Z Private lecturer (celestial) 2975755 /* Lex naturalis */ [[w:natural law|]] 2693350 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] {{-}} == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === [[File:7 Dimensions of culture.svg|thumbnail|7 Dimensions of Culture]] The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a [[w:Trompenaars%27s_model_of_national_culture_differences|cultural dimension]]. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as particularly worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a [[w:papal conclave|papal conclave]], which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis ([[w:natural law|natural law]]) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attentional bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as '[[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]]'. === Getting a giraffe through an eye of a needle === The general recommendation, of course, is to be careful against the unknown requirements of heaven, which may be culturally unexpected, but logically sophisticated and therefore to prefer to err in favor of ethics rather than the opposite. The solution to the problem of getting a giraffe through an eye of a needle is an "animal trainer" (upbringing, education, mentoring, moral culture and ethics). In a capitalist society, when competitors (or even coworkers) may be seen as enemies on a regular basis, love of enemies could obviously also be seen to include granting natural rights to those “enemies” and neither choice-supportive bias nor attentional bias are helpful to do so. [[de:Ethik/Leben nach dem Tod]] 637n7n0ex93yh2yijs2u8cxofw89gcn 2693354 2693350 2024-12-26T19:30:18Z Private lecturer (celestial) 2975755 /* Lex naturalis */ [[w:Contractualism|cultural social contract]] 2693354 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] {{-}} == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === [[File:7 Dimensions of culture.svg|thumbnail|7 Dimensions of Culture]] The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a [[w:Trompenaars%27s_model_of_national_culture_differences|cultural dimension]]. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as particularly worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a [[w:papal conclave|papal conclave]], which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis ([[w:natural law|natural law]]) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the [[w:Contractualism|cultural social contract]] and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attentional bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as '[[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]]'. === Getting a giraffe through an eye of a needle === The general recommendation, of course, is to be careful against the unknown requirements of heaven, which may be culturally unexpected, but logically sophisticated and therefore to prefer to err in favor of ethics rather than the opposite. The solution to the problem of getting a giraffe through an eye of a needle is an "animal trainer" (upbringing, education, mentoring, moral culture and ethics). In a capitalist society, when competitors (or even coworkers) may be seen as enemies on a regular basis, love of enemies could obviously also be seen to include granting natural rights to those “enemies” and neither choice-supportive bias nor attentional bias are helpful to do so. [[de:Ethik/Leben nach dem Tod]] 4bj4nmlqan6f9m2a6xhbl52v15eusrk 2693576 2693354 2024-12-27T03:42:04Z Private lecturer (celestial) 2975755 /* Image of God */ 2693576 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] {{-}} == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a [[w:Trompenaars%27s_model_of_national_culture_differences|cultural dimension]]. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as particularly worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a [[w:papal conclave|papal conclave]], which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis ([[w:natural law|natural law]]) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the [[w:Contractualism|cultural social contract]] and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attentional bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as '[[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]]'. === Getting a giraffe through an eye of a needle === The general recommendation, of course, is to be careful against the unknown requirements of heaven, which may be culturally unexpected, but logically sophisticated and therefore to prefer to err in favor of ethics rather than the opposite. The solution to the problem of getting a giraffe through an eye of a needle is an "animal trainer" (upbringing, education, mentoring, moral culture and ethics). In a capitalist society, when competitors (or even coworkers) may be seen as enemies on a regular basis, love of enemies could obviously also be seen to include granting natural rights to those “enemies” and neither choice-supportive bias nor attentional bias are helpful to do so. [[de:Ethik/Leben nach dem Tod]] b2pwhit5ty2tt4kq2cx8a9ztpnxucrl 0 302265 2693575 2682347 2024-12-27T03:27:08Z 193.116.193.217 2693575 wikitext text/x-wiki {{DISPLAYTITLE:WikiJournal Preprints/An Explanation for Dark Energy from Whittaker Potential Theory}} {{DISPLAYTITLE:WikiJournal Preprints/An Explanation for Dark Energy from Whittaker Potential Theory}} {{DISPLAYTITLE:WikiJournal Preprints/An Explanation for Dark Energy from Whittaker Potential Theory}} {{Article info | journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> | last1 = Titleman | orcid1 = | first1 = Mark | et_al = <!-- if there are >9 authors, hyperlink to the list here --> | affiliation1 = | correspondence1 = | correspondence = email@address.com | keywords = <!-- up to 6 keywords --> | license = <!-- default is CC-BY --> | abstract = A recent article found that black holes with posited vacuum energy interior solutions alongside cosmological boundaries have a cosmological coupling constant of k=3, meaning that black holes gain mass proportional to a3 in a parameterization equation within a Robertson-Walker cosmology – thus making black holes a cosmological dark energy species (Farrah et al. 2023). The mechanism for this is unknown. Two papers by E. T. Whittaker in 1903 and 1904 showed that all force potential could be understood as resulting from standing waves (static non-local solution) and propagating waves (local solution changing in time). This unification of gravitational and electromagnetic potential has been neglected even though it opens up new mathematical avenues and physical features. The mass-proportionality and preferred direction of the longitudinal waves within the two underlying Whittaker potentials can explain many features of General Relativity (Titleman 2022). They also offer a simple Newtonian explanation for expansion of the universe stemming from Whittaker potential theory – it is produced as longitudinal motion within the Whittaker potentials only when dynamic electromagnetism is separate from time-static gravity in intergalactic space. }} ==Introduction== The classic papers of E. T. Whittaker in 1903 and 1904 provided a general harmonic solution to the wave equation and Laplace equation in three dimensions, showing that both potentials could be analyzed into simple plane waves. Even though no action could be set up, it was important work that foresaw the Aharonov-Bohm effect and could be used to replace Dirac spinors in the Dirac equation (Ruse 1937). This “undulatory theory” could also explain several features of General Relativity. For example, gravitational lensing can be understood as resulting from the preferred direction of the potentials and their mass-proportionality (Titleman 2022). More specifically, a more massive observer would experience more longitudinal waves than only the two experienced by an observer as an electromagnetic wave, yet when ''observed'' at the speed of light the number of longitudinal waves would collapse into the orthogonal axis (non-local y-axis). This results from the simple fact that Whittaker’s analysis of the Laplace equation and wave equation is physically less arbitrary than the standard approach. The reduction of six degrees of freedom to two degrees of freedom provides a ''purely physical reason'' for the preferred directionality of the wave; an electromagnetic wave ''must'' have a preferred direction. The unity of gravity and electromagnetism by this analysis shows that they are mutually orthogonal. == A New Explanation for Expansion of the Universe == The new nature of the x, y, z axes permits each to be assigned a free parameter: longitudinal motion in the z is charge-proportional from the perspective of the observer (compressible potentials), number of longitudinal waves is mass-proportional from the perspective of the observer AND folds into the y-axis (static) when observed at high speed, and the x-axis or plane wave axis is related to amplitude, intensity, and soliton radius. Due to the dynamic longitudinal motion in the z-axis being additive, Whittaker’s potential theory provides a Newtonian explanation for expansion of the universe - it is merely dynamic light decoupled from static gravity and can only be produced intergalactically. If this is the case, there would be an inverse relation between the changing background intensity of the universe and expansion of the universe. Since the intensity of the universe is double that of all predicted stars (Lauer et al., 2022), the relation would be on the order of 3/2. <math>(1) \frac{3}{2} \Delta I_{v} = Expansion\ (plane)</math> The cosmological constant in the context of spacetime can potentially be found by implicating luminosity in (1). The 3 is the result of the new interpretation of three dimensions or three axes afforded by Whittaker’s undulatory theory. Longitudinal waves are additive in two directions – phase and antiphase z-directions. It is conjectured that black holes produce these longitudinal waves as scalar potentials, providing cosmological coupling, a third additive “direction” (y), another dynamic component, an important center for the scalar potentials, and a new understanding of waves as vorticity as the interface of plane and spherical rotations. This understanding could replace black hole singularities with vacuum energy interior solutions within a Robertson-Walker cosmology. ==A Relation to MOND?== Additionally, this can perhaps explain the relation of the MOND fitting parameter to dark energy understood as an energy density. <math>(2) a_0=\surd(\Lambda/3)</math> According to the new understanding of the three axes, the mass-proportional, static gravitational y-axis is related to the charge-proportional, dynamic electromagnetic z-axis by squaring. Two directions of dynamism are in the z, but one source of dynamism (black hole growth) is in all directions locally. Time-static gravity is only in the mass-proportional and thus limited-range observed y-axis. The potentials are non-local in most senses. As such, the dynamism at the interface of the cosmologically coupled z-axis and observed y-axis are related by squaring only within the limited range of nearby matter. Outside of this limited range there is simply expansion of the universe. Squaring must also be used for the cosmological constant in the context of spacetime – where the interface between dynamic z-axis and static y-axis is ''constantly'' ''implied.'' The MOND fitting parameter can thus be determined by an interaction between gravity purely in the Whittaker sense (limited by the presence of mass) and the cosmological constant in the context of the static-dynamic interactions implied by spacetime. The external field effect is the result of these interactions in conjunction with black hole cosmological coupling. == Conclusion == This understanding of Whittaker’s analytical papers in classical physics can provide a new understanding of expansion as simply purely dynamic longitudinal motion decoupled from static gravity. There would be a relation between dark energy in some sense and either intensity or luminosity. This may also explain the relation between the cosmological constant in the context of spacetime and the MOND fitting parameter. Ultimately, the gauge used by Whittaker in his 1904 paper to reduce the standard electromagnetic potentials to only two scalar potentials was oversimplified. It can be expanded through advances in computation and the Wick rotation which already links statistical mechanics to quantum mechanics and 4d Euclidean space to spacetime. Reactive probability, statistical mechanics and information (ternary) are of central importance. A language of Clifford algebra or the geometry of a Clifford torus (with luminosity and a black hole network phase space) can be developed. ==References== Farrah, Duncan, Kevin S. Croker, Michael Zevin, Gregory Tarlé, Valerio Faraoni, Sara Petty, Jose Afonso et al. "Observational evidence for cosmological coupling of black holes and its implications for an astrophysical source of dark energy." The Astrophysical Journal Letters 944, no. 2 (2023): L31. Lauer, Tod R., Marc Postman, John R. Spencer, Harold A. Weaver, S. Alan Stern, G. Randall Gladstone, Richard P. Binzel et al. "Anomalous flux in the cosmic optical background detected with new horizons observations." The Astrophysical Journal Letters 927, no. 1 (2022): L8. Ruse, H. S. "ON WHITTAKER'S ELECTROMAGNETIC ‘SCALAR POTENTIALS’." The Quarterly Journal of Mathematics 1 (1937): 148-160. Titleman, Mark. "Representations and Implications of Papers Written by ET Whittaker in 1903 and 1904." arXiv preprint arXiv:2205.08309 (2022). {{DEFAULTSORT:WikiJournal Preprints/An Explanation for Dark Energy from Whittaker Potential Theory}} [[Category:Dark energy]] __INDEX__ __NEWSECTIONLINK__ q5dl3putcmi34tod8mzjzox483bfz1e 0 307970 2693316 2652225 2024-12-26T18:01:24Z Tule-hog 2984180 c/e 2693316 wikitext text/x-wiki {| class="wikitable" |+ Acronyms that may be found on the exam. |- ! Acronym !! Full name |- | AAA || Authentication, Authorization, and Accounting |- | ACL || Access Control List |- | AES || Advanced Encryption Standard |- | AES-256 || Advanced Encryption Standards 256-bit |- | AH || Authentication Header |- | AI || Artificial Intelligence |- | AIS || Automated Indicator Sharing |- | ALE || Annualized Loss Expectancy |- | AP || Access Point |- | API || Application Programming Interface |- | APT || Advanced Persistent Threat |- | ARO || Annualized Rate of Occurrence |- | ARP || Address Resolution Protocol |- | ASLR || Address Space Layout Randomization |- | ATT&CK || Adversarial Tactics, Techniques, and Common Knowledge |- | AUP || Acceptable Use Policy |- | AV || Antivirus |- | BASH || Bourne Again Shell |- | BCP || Business Continuity Planning |- | BGP || Border Gateway Protocol |- | BIA || Business Impact Analysis |- | BIOS || Basic Input/Output System |- | BPA || Business Partners Agreement |- | BPDU || Bridge Protocol Data Unit |- | BYOD || Bring Your Own Device |- | CA || Certificate Authority |- | CAPTCHA || Completely Automated Public Turing Test to Tell Computers and Humans Apart |- | CAR || Corrective Action Report |- | CASB || Cloud Access Security Broker |- | CBC || Cipher Block Chaining |- | CCMP || Counter Mode/CBC-MAC Protocol |- | CCTV || Closed-circuit Television |- | CERT || Computer Emergency Response Team |- | CFB || Cipher Feedback |- | CHAP || Challenge Handshake Authentication Protocol |- | CIA || Confidentiality, Integrity, Availability |- | CIO || Chief Information Officer |- | CIRT || Computer Incident Response Team |- | CMS || Content Management System |- | COOP || Continuity of Operation Planning |- | COPE || Corporate Owned, Personally Enabled |- | CP || Contingency Planning |- | CRC || Cyclical Redundancy Check |- | CRL || Certificate Revocation List |- | CSO || Chief Security Officer |- | CSP || Cloud Service Provider |- | CSR || Certificate Signing Request |- | CSRF || Cross-site Request Forgery |- | CSU || Channel Service Unit |- | CTM || Counter Mode |- | CTO || Chief Technology Officer |- | CVE || Common Vulnerability Enumeration |- | CVSS || Common Vulnerability Scoring System |- | CYOD || Choose Your Own Device |- | DAC || Discretionary Access Control |- | DBA || Database Administrator |- | DDoS || Distributed Denial of Service |- | DEP || Data Execution Prevention |- | DES || Digital Encryption Standard |- | DHCP || Dynamic Host Configuration Protocol |- | DHE || Diffie-Hellman Ephemeral |- | DKIM || DomainKeys Identified Mail |- | DLL || Dynamic Link Library |- | DLP || Data Loss Prevention |- | DMARC || Domain Message Authentication Reporting and Conformance |- | DNAT || Destination Network Address Translation |- | DNS || Domain Name System |- | DoS || Denial of Service |- | DPO || Data Privacy Officer |- | DRP || Disaster Recovery Plan |- | DSA || Digital Signature Algorithm |- | DSL || Digital Subscriber Line |- | EAP || Extensible Authentication Protocol |- | ECB || Electronic Code Book |- | ECC || Elliptic Curve Cryptography |- | ECDHE || Elliptic Curve Diffie-Hellman Ephemeral |- | ECDSA || Elliptic Curve Digital Signature Algorithm |- | EDR || Endpoint Detection and Response |- | EFS || Encrypted File System |- | ERP || Enterprise Resource Planning |- | ESN || Electronic Serial Number |- | ESP || Encapsulated Security Payload |- | FACL || File System Access Control List |- | FDE || Full Disk Encryption |- | FIM || File Integrity Management |- | FPGA || Field Programmable Gate Array |- | FRR || False Rejection Rate |- | FTP || File Transfer Protocol |- | FTPS || Secured File Transfer Protocol |- | GCM || Galois Counter Mode |- | GDPR || General Data Protection Regulation |- | GPR || GNU Privacy Guard |- | GPO || Group Policy Object |- | GPS || Global Positioning System |- | GPU || Graphics Processing Unit |- | GRE || Generic Routing Encapsulation |- | HA || High Availability |- | HDD || Hard Disk Drive |- | HIDS || Host-based Intrusion Detection System |- | HIPS || Host-based Intrusion Prevention System |- | HMAC || Hashed Message Authentication Code |- | HOTP || HMAC-based One-time Password |- | HSM || Hardware Security Module |- | HTML || Hypertext Markup Language |- | HTTP || Hypertext Transfer Protocol |- | HTTPS || Hypertext Transfer Protocol Secure |- | HVAC || Heating, Ventilation Air Conditioning |- | IaaS || Infrastructure as a Service |- | IaC || Infrastructure as Code |- | IAM || Identity and Access Management |- | ICMP || Internet Control Message Protocol |- | ICS || Industrial Control Systems |- | IDEA || International Data Encryption Algorithm |- | IDF || Intermediate Distribution Frame |- | IdP || Identity Provider |- | IDS || Intrusion Detection System |- | IEEE || Institute of Electrical and Electronics Engineers |- | IKE || Internet Key Exchange |- | IM || Instant Messaging |- | IMAP || Internet Message Access Protocol |- | IoC || Indicators of Compromise |- | IoT || Internet of Things |- | IP || Internet Protocol |- | IPS || Intrusion Prevention System |- | IPSec || Internet Protocol Security |- | IR || Incident Response |- | IRC || Internet Relay Chat |- | IRP || Incident Response Plan |- | ISO || International Standards Organization |- | ISP || Internet Service Provider |- | ISSO || Information Systems Security Officer |- | IV || Initialization Vector |- | KDC || Key Distribution Center |- | KEK || Key Encryption Key |- | L2TP || Layer 2 Tunneling Protocol |- | LAN || Local Area Network |- | LDAP || Lightweight Directory Access Protocol |- | LEAP || Lightweight Extensible Authentication Protocol |- | MaaS || Monitoring as a Service |- | MAC || Mandatory Access Control |- | MAC || Media Access Control |- | MAC || Message Authentication Code |- | MAN || Metropolitan Area Network |- | MBR || Master Boot Record |- | MD5 || Message Digest 5 |- | MDF || Main Distribution Frame |- | MDM || Mobile Device Management |- | MFA || Multifactor Authentication |- | MFD || Multifunction Device |- | MFP || Multifunction Printer |- | ML || Machine Learning |- | MMS || Multimedia Message Service |- | MOA || Memorandum of Agreement |- | MOU || Memorandum of Understanding |- | MPLS || Multi-protocol Label Switching |- | MSA || Master Service Agreement |- | MSCHAP || Microsoft Challenge Handshake Authentication Protocol |- | MSP || Managed Service Provider |- | MSSP || Managed Security Service Provider |- | MTBF || Mean Time Between Failures |- | MTTF || Mean Time to Failure |- | MTTR || Mean Time to Recover |- | MTU || Maximum Transmission Unit |- | NAC || Network Access Control |- | NAT || Network Address Translation |- | NDA || Non-disclosure Agreement |- | NFC || Near Field Communication |- | NGFW || Next-generation Firewall |- | NIDS || Network-based Intrusion Detection System |- | NIPS || Network-based Intrusion Prevention System |- | NIST || National Institute of Standards & Technology |- | NTFS || New Technology File System |- | NTLM || New Technology LAN Manager |- | NTP || Network Time Protocol |- | OAUTH || Open Authorization |- | OCSP || Online Certificate Status Protocol |- | OID || Object Identifier |- | OS || Operating System |- | OSINT || Open-source Intelligence |- | OSPF || Open Shortest Path First |- | OT || Operational Technology |- | OTA || Over the Air |- | OVAL || Open Vulnerability Assessment Language |- | P12 || PKCS #12 |- | P2P || Peer to Peer |- | PaaS || Platform as a Service |- | PAC || Proxy Auto Configuration |- | PAM || Privileged Access Management |- | PAM || Pluggable Authentication Modules |- | PAP || Password Authentication Protocol |- | PAT || Port Address Translation |- | PBKDF2 || Password-based Key Derivation Function 2 |- | PBX || Private Branch Exchange |- | PCAP || Packet Capture |- | PCI DSS || Payment Card Industry Data Security Standard |- | PDU || Power Distribution Unit |- | PEAP || Protected Extensible Authentication Protocol |- | PED || Personal Electronic Device |- | PEM || Privacy Enhanced Mail |- | PFS || Perfect Forward Secrecy |- | PGP || Pretty Good Privacy |- | PHI || Personal Health Information |- | PII || Personally Identifiable Information |- | PIV || Personal Identity Verification |- | PKCS || Public Key Cryptography Standards |- | PKI || Public Key Infrastructure |- | POP || Post Office Protocol |- | POTS || Plain Old Telephone Service |- | PPP || Point-to-Point Protocol |- | PPTP || Point-to-Point Tunneling Protocol |- | PSK || Pre-shared Key |- | PTZ || Pan-tilt-zoom |- | PUP || Potentially Unwanted Program |- | RA || Recovery Agent |- | RA || Registration Authority |- | RACE || Research and Development in Advanced Communications Technologies in Europe |- | RAD || Rapid Application Development |- | RADIUS || Remote Authentication Dial-in User Service |- | RAID || Redundant Array of Inexpensive Disks |- | RAS || Remote Access Server |- | RAT || Remote Access Trojan |- | RBAC || Role-based Access Control |- | RBAC || Rule-based Access Control |- | RC4 || Rivest Cipher version 4 |- | RDP || Remote Desktop Protocol |- | RFID || Radio Frequency Identifier |- | RIPEMD || RACE Integrity Primitives Evaluation Message Digest |- | ROI || Return on Investment |- | RPO || Recovery Point Objective |- | RSA || Rivest, Shamir, & Adleman |- | RTBH || Remotely Triggered Black Hole |- | RTO || Recovery Time Objective |- | RTOS || Real-time Operating System |- | RTP || Real-time Transport Protocol |- | S/MIME || Secure/Multipurpose Internet Mail Extensions |- | SaaS || Software as a Service |- | SAE || Simultaneous Authentication of Equals |- | SAML || Security Assertions Markup Language |- | SAN || Storage Area Network |- | SAN || Subject Alternative Name |- | SASE || Secure Access Service Edge |- | SCADA || Supervisory Control and Data Acquisition |- | SCAP || Security Content Automation Protocol |- | SCEP || Simple Certificate Enrollment Protocol |- | SD-WAN || Software-defined Wide Area Network |- | SDK || Software Development Kit |- | SDLC || Software Development Lifecycle |- | SDLM || Software Development Lifecycle Methodology |- | SDN || Software-defined Networking |- | SE Linux || Security-enhanced Linux |- | SED || Self-encrypting Drives |- | SEH || Structured Exception Handler |- | SFTP || Secured File Transfer Protocol |- | SHA || Secure Hashing Algorithm |- | SHTTP || Secure Hypertext Transfer Protocol |- | SIEM || Security Information and Event Management |- | SIM || Subscriber Identity Module |- | SLA || Service-level Agreement |- | SLE || Single Loss Expectancy |- | SMS || Short Message Service |- | SMTP || Simple Mail Transfer Protocol |- | SMTPS || Simple Mail Transfer Protocol Secure |- | SNMP || Simple Network Management Protocol |- | SOAP || Simple Object Access Protocol |- | SOAR || Security Orchestration, Automation, Response |- | SoC || System on Chip |- | SOC || Security Operations Center |- | SOW || Statement of Work |- | SPF || Sender Policy Framework |- | SPIM || Spam over Internet Messaging |- | SQL || Structured Query Language |- | SQLi || SQL Injection |- | SRTP || Secure Real-Time Protocol |- | SSD || Solid State Drive |- | SSH || Secure Shell |- | SSL || Secure Sockets Layer |- | SSO || Single Sign-on |- | STIX || Structured Threat Information eXchange |- | SWG || Secure Web Gateway |- | TACACS+ || Terminal Access Controller Access Control System |- | TAXII || Trusted Automated eXchange of Indicator Information |- | TCP/IP || Transmission Control Protocol/Internet Protocol |- | TGT || Ticket Granting Ticket |- | TKIP || Temporal Key Integrity Protocol |- | TLS || Transport Layer Security |- | TOC || Time-of-check |- | TOTP || Time-based One-time Password |- | TOU || Time-of-use |- | TPM || Trusted Platform Module |- | TTP || Tactics, Techniques, and Procedures |- | TSIG || Transaction Signature |- | UAT || User Acceptance Testing |- | UAV || Unmanned Aerial Vehicle |- | UDP || User Datagram Protocol |- | UEFI || Unified Extensible Firmware Interface |- | UEM || Unified Endpoint Management |- | UPS || Uninterruptable Power Supply |- | URI || Uniform Resource Identifier |- | URL || Universal Resource Locator |- | USB || Universal Serial Bus |- | USB OTG || USB On the Go |- | UTM || Unified Threat Management |- | UTP || Unshielded Twisted Pair |- | VBA || Visual Basic |- | VDE || Virtual Desktop Environment |- | VDI || Virtual Desktop Infrastructure |- | VLAN || Virtual Local Area Network |- | VLSM || Variable Length Subnet Masking |- | VM || Virtual Machine |- | VoIP || Voice over IP |- | VPC || Virtual Private Cloud |- | VPN || Virtual Private Network |- | VTC || Video Teleconferencing |- | WAF || Web Application Firewall |- | WAP || Wireless Access Point |- | WEP || Wired Equivalent Privacy |- | WIDS || Wireless Intrusion Detection System |- | WIPS || Wireless Intrusion Prevention System |- | WO || Work Order |- | WPA || Wi-Fi Protected Access |- | WPS || Wi-Fi Protected Setup |- | WTLS || Wireless TLS |- | XDR || Extended Detection and Response |- | XML || Extensible Markup Language |- | XOR || Exclusive Or |- | XSRF || Cross-site Request Forgery |- | XSS || Cross-site Scripting |} {{BookCat}} ln96jb2tt5zzcghgs2zg3atvmwwez9p 0 307997 2693321 2652325 2024-12-26T18:35:29Z Tule-hog 2984180 finish pg 2693321 wikitext text/x-wiki {| class="wikitable" |+ Acronyms that may be found on the exam. |- ! Acronym !! Full name |- | AAA || Authentication, Authorization, and Accounting |- | AC || Alternating Current |- | ACL || Access Control List |- | ADF || Automatic Document Feeder |- | AES || Advanced Encryption Standard |- | AP || Access Point |- | APFS || Apple File System |- | APIPA || Automatic Private Internet Protocol Addressing |- | APK || Android Package |- | ARM || Advanced RISC [Reduced Instruction Set Computer] Machine |- | ARP || Address Resolution Protocol |- | ATA || Advanced Technology Attachment |- | ATM || Asynchronous Transfer Mode |- | ATX || Advanced Technology Extended |- | AUP || Acceptable Use Policy |- | BIOS || Basic Input/Output System |- | BSOD || Blue Screen of Death |- | BYOD || Bring Your Own Device |- | CAD || Computer-aided Design |- | CAPTCHA || Completely Automated Public Turing Test to Tell Computers and Humans Apart |- | CD || Compact Disc |- | CDFS || Compact Disc File System |- | CDMA || Code-Division Multiple Access |- | CERT || Computer Emergency Response Team |- | CIFS || Common Internet File System |- | CMD || Command Prompt |- | CMOS || Complementary Metal-Oxide Semiconductor |- | CPU || Central Processing Unit |- | CRL || Certificate Revocation List |- | DC || Direct Current |- | DDoS || Distributed Denial of Service |- | DDR || Double Data Rate |- | DHCP || Dynamic Host Configuration Protocol |- | DIMM || Dual Inline Memory Module |- | DKIM || DomainKeys Identified Mail |- | DMA || Direct Memory Access |- | DMARC || Domain-based Message Authentication, Reporting, and Conformance |- | DNS || Domain Name System |- | DoS || Denial of Service |- | DRAM || Dynamic Random-Access Memory |- | DRM || Digital Rights Management |- | DSL || Digital Subscriber Line |- | DVI || Digital Visual Interface |- | DVI-D || Digital Visual Interface-Digital |- | ECC || Error Correcting Code |- | EFS || Encrypting File System |- | EMI || Electromagnetic Interference |- | EOL || End-of-Life |- | eSATA || External Serial Advanced Technology Attachment |- | ESD || Electrostatic Discharge |- | EULA || End-User License Agreement |- | exFAT || Extensible File Allocation Table |- | ext || Extended File System |- | FAT || File Allocation Table |- | FAT12 || 12-bit File Allocation Table |- | FAT16 || 16-bit File Allocation Table |- | FAT32 || 32-bit File Allocation Table |- | FSB || Front-Side Bus |- | FTP || File Transfer Protocol |- | GFS || Grandfather-Father-Son |- | GPS || Global Positioning System |- | GPT || GUID [Globally Unique Identifier] Partition Table |- | GPU || Graphics Processing Unit |- | GSM || Global System for Mobile Communications |- | GUI || Graphical User Interface |- | GUID || Globally Unique Identifier |- | HAL || Hardware Abstraction Layer |- | HAV || Hardware-assisted Virtualization |- | HCL || Hardware Compatibility List |- | HDCP || High-bandwidth Digital Content Protection |- | HDD || Hard Disk Drive |- | HDMI || High-Definition Multimedia Interface |- | HSM || Hardware Security Module |- | HTML || Hypertext Markup Language |- | HTTP || Hypertext Transfer Protocol |- | HTTPS || Hypertext Transfer Protocol Secure |- | I/O || Input/Output |- | IaaS || Infrastructure as a Service |- | ICR || Intelligent Character Recognition |- | IDE || Integrated Drive Electronics |- | IDS || Intrusion Detection System |- | IEEE || Institute of Electrical and Electronics Engineers |- | IMAP || Internet Mail Access Protocol |- | IOPS || Input/Output Operations Per Second |- | IoT || Internet of Things |- | IP || Internet Protocol |- | IPS || Intrusion Prevention System |- | IPS || In-plane Switching |- | IPSec || Internet Protocol Security |- | IR || Infrared |- | IrDA || Infrared Data Association |- | IRP || Incident Response Plan |- | ISO || International Organization for Standardization |- | ISP || Internet Service Provider |- | ITX || Information Technology eXtended |- | KB || Knowledge Base |- | KVM || Keyboard-Video-Mouse |- | LAN || Local Area Network |- | LC || Lucent Connector |- | LCD || Liquid Crystal Display |- | LDAP || Lightweight Directory Access Protocol |- | LED || Light-emitting Diode |- | MAC || Media Access Control/Mandatory Access Control |- | MAM || Mobile Application Management |- | MAN || Metropolitan Area Network |- | MBR || Master Boot Record |- | MDM || Mobile Device Management |- | MFA || Multifactor Authentication |- | MFD || Multifunction Device |- | MFP || Multifunction Printer |- | MMC || Microsoft Management Console |- | MOU || Memorandum of Understanding |- | MSDS || Material Safety Data Sheet |- | MSRA || Microsoft Remote Assistance |- | MX || Mail Exchange |- | NAC || Network Access Control |- | NAT || Network Address Translation |- | NDA || Non-disclosure Agreement |- | NetBIOS || Networked Basic Input/Output System |- | NetBT || NetBIOS over TCP/IP [Transmission Control Protocol/Internet Protocol] |- | NFC || Near-field Communication |- | NFS || Network File System |- | NIC || Network Interface Card |- | NTFS || New Technology File System |- | NVMe || Non-volatile Memory Express |- | OCR || Optical Character Recognition |- | OLED || Organic Light-emitting Diode |- | ONT || Optical Network Terminal |- | OS || Operating System |- | PaaS || Platform as a Service |- | PAN || Personal Area Network |- | PC || Personal Computer |- | PCIe || Peripheral Component Interconnect Express |- | PCL || Printer Command Language |- | PE || Preinstallation Environment |- | PII || Personally Identifiable Information |- | PIN || Personal Identification Number |- | PKI || Public Key Infrastructure |- | PoE || Power over Ethernet |- | POP3 || Post Office Protocol 3 |- | POST || Power-on Self-Test |- | PPP || Point-to-Point Protocol |- | PRL || Preferred Roaming List |- | PSU || Power Supply Unit |- | PXE || Preboot Execution Environment |- | RADIUS || Remote Authentication Dial-in User Service |- | RAID || Redundant Array of Independent (or Inexpensive) Disks |- | RAM || Random-access Memory |- | RDP || Remote Desktop Protocol |- | RF || Radio Frequency |- | RFI || Radio-Frequency Interference |- | RFID || Radio-Frequency Identification |- | RJ11 || Registered Jack Function 11 |- | RJ45 || Registered Jack Function 45 |- | RMM || Remote Monitoring and Management |- | RTO || Recovery Time Objective |- | SaaS || Software as a Service |- | SAN || Storage Area Network |- | SAS || Serial Attached SCSI [Small Computer System Interface] |- | SATA || Serial Advanced Technology Attachment |- | SC || Subscriber Connector |- | SCADA || Supervisory Control and Data Acquisition |- | SCP || Secure Copy Protection |- | SCSI || Small Computer System Interface |- | SDN || Software-defined Networking |- | SFTP || Secure File Transfer Protocol |- | SIM || Subscriber Identity Module |- | SIMM || Single Inline Memory Module |- | S.M.A.R.T. || Self-monitoring Analysis and Reporting Technology |- | SMB || Server Message Block |- | SMS || Short Message Service |- | SMTP || Simple Mail Transfer Protocol |- | SNMP || Simple Network Management Protocol |- | SNTP || Simple Network Time Protocol |- | SODIMM || Small Outline Dual Inline Memory Module |- | SOHO || Small Office/Home Office |- | SPF || Sender Policy Framework |- | SQL || Structured Query Language |- | SRAM || Static Random-access Memory |- | SSD || Solid-State Drive |- | SSH || Secure Shell |- | SSID || Service Set Identifier |- | SSL || Secure Sockets Layer |- | SSO || Single Sign-on |- | ST || Straight Tip |- | STP || Shielded Twisted Pair |- | TACACS || Terminal Access Controller Access-Control System |- | TCP || Transmission Control Protocol |- | TCP/IP || Transmission Control Protocol/Internet Protocol |- | TFTP || Trivial File Transfer Protocol |- | TKIP || Temporal Key Integrity Protocol |- | TLS || Transport Layer Security |- | TN || Twisted Nematic |- | TPM || Trusted Platform Module |- | UAC || User Account Control |- | UDP || User Datagram Protocol |- | UEFI || Unified Extensible Firmware Interface |- | UNC || Universal Naming Convention |- | UPnP || Universal Plug and Play |- | UPS || Uninterruptible Power Supply |- | USB || Universal Serial Bus |- | UTM || Unified Threat Management |- | UTP || Unshielded Twisted Pair |- | VA || Vertical Alignment |- | VDI || Virtual Desktop Infrastructure |- | VGA || Video Graphics Array |- | VLAN || Virtual LAN [Local Area Network] |- | VM || Virtual Machine |- | VNC || Virtual Network Computer |- | VoIP || Voice over Internet Protocol |- | VPN || Virtual Private Network |- | VRAM || Video Random-access Memory |- | WAN || Wide Area Network |- | WEP || Wired Equivalent Privacy |- | WISP || Wireless Internet Service Provider |- | WLAN || Wireless LAN [Local Area Network] |- | WMN || Wireless Mesh Network |- | WPA || WiFi Protected Access |- | WWAN || Wireless Wide Area Network |- | XSS || Cross-site Scripting |} pzg6ou3h6kl2nwqlqlkiu2tlvdtg9dj 0 308102 2693365 2674993 2024-12-26T19:57:52Z Tule-hog 2984180 /* 1.6 Compare and contrast network topologies, architectures, and types. */ add further reading bx 2693365 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Multitier architecture|Multitier architecture]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} <noinclude> {{BookCat}} </noinclude> 3ilhy81n2k8pwktdendprhybewskcxg 2693366 2693365 2024-12-26T19:59:17Z Tule-hog 2984180 /* 1.7 Given a scenario, use appropriate IPv4 network addressing. */ add further reading 2693366 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Multitier architecture|Multitier architecture]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} <noinclude> {{BookCat}} </noinclude> a3h2vrxtoli34bv40rudmw17slfg8mp 2693368 2693366 2024-12-26T20:04:59Z Tule-hog 2984180 /* 1.8 Summarize evolving use cases for modern network environments */ add further reading bx 2693368 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Multitier architecture|Multitier architecture]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] }} <noinclude> {{BookCat}} </noinclude> j9pd0ef06109qmoyj0tq2p2a7buy1m9 2693371 2693368 2024-12-26T20:10:46Z Tule-hog 2984180 /* 1.8 Summarize evolving use cases for modern network environments */ add further item 2693371 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Multitier architecture|Multitier architecture]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Private network|Private network]] }} <noinclude> {{BookCat}} </noinclude> s8f4qdm8u49k1em97m4mbyc5nqvdx6k 2693373 2693371 2024-12-26T20:18:46Z Tule-hog 2984180 /* 1.8 Summarize evolving use cases for modern network environments */ add further items 2693373 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Multitier architecture|Multitier architecture]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Template:Area networks|Area networks]] * [[:w:Private network|Private network]] * [[:w:Local area network|Local area network]] (LAN) * [[:w:Wide area network|Wide area network]] (WAN) * [[:w:Wireless LAN|Wireless LAN]] (WLAN) * [[:w:Personal area network|Personal area network]] (PAN) * [[:w:Metropolitan area network|Metropolitan area network]] (MAN) * }} <noinclude> {{BookCat}} </noinclude> 0qpqlib8r58qkyn6oroshr230b8u97v 2693374 2693373 2024-12-26T20:19:35Z Tule-hog 2984180 /* 1.8 Summarize evolving use cases for modern network environments */ alter order 2693374 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Multitier architecture|Multitier architecture]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] * [[:w:Local area network|Local area network]] (LAN) * [[:w:Wide area network|Wide area network]] (WAN) * [[:w:Wireless LAN|Wireless LAN]] (WLAN) * [[:w:Personal area network|Personal area network]] (PAN) * [[:w:Metropolitan area network|Metropolitan area network]] (MAN) * }} <noinclude> {{BookCat}} </noinclude> q3gj0eu0uon2z8ztlw4qibtf3zmddea 2693377 2693374 2024-12-26T20:21:00Z Tule-hog 2984180 /* 1.8 Summarize evolving use cases for modern network environments */ c/e 2693377 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Multitier architecture|Multitier architecture]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] ** [[:w:Local area network|Local area network]] (LAN) ** [[:w:Wide area network|Wide area network]] (WAN) ** [[:w:Wireless LAN|Wireless LAN]] (WLAN) ** [[:w:Personal area network|Personal area network]] (PAN) ** [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> bqkhppyiglsk9h2nzpsjj6c19sfxwdc 2693379 2693377 2024-12-26T20:26:28Z Tule-hog 2984180 /* 1.8 Summarize evolving use cases for modern network environments */ c/e 2693379 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Multitier architecture|Multitier architecture]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> of0nwcofrm1j7nv252fkhl64vgfoc3j 2693381 2693379 2024-12-26T20:41:06Z Tule-hog 2984180 /* 1.6 Compare and contrast network topologies, architectures, and types. */ add further item 2693381 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Network architecture|Network architecture]] * [[:w:Multitier architecture|Multitier architecture]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> 6i4pi7onojkqhhwrqzku1u4ctotlhgh 2693382 2693381 2024-12-26T20:44:06Z Tule-hog 2984180 /* 1.6 Compare and contrast network topologies, architectures, and types. */ add further items 2693382 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Network architecture|Network architecture]] * [[:w:Multitier architecture|Multitier architecture]] * [[:w:Client–server model|Client–server model]] * [[:w:Peer-to-peer|Peer-to-peer model]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> azoqmzdn3d1upmlotqzfrup8vipohnj 2693389 2693382 2024-12-26T22:02:43Z Tule-hog 2984180 /* 1.5 Compare and contrast transmission media and transceivers. */ add further items 2693389 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{wikibox|<u>Further reading</u> * [[:w:Optical fiber connector|Optical fiber connector]] * [[:w:Fiber-optic cable|Fiber-optic cable]] * [[:w:Fiber media converter|Fiber media converter]] * [[:w:D-subminiature|D-sub]] * [[:w:Networking cable|Networking cable]] * [[:w:Broadband over power lines|Broadband over power lines]] * [[:w:Crimp connection|Crimp connection]] * [[:w:Punch down tool|Punch down tool]] * [[:w:Wire stripper|Wire stripper]] * [[:w:Cable tester|Cable tester]] }} {{col-end}} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Network architecture|Network architecture]] * [[:w:Multitier architecture|Multitier architecture]] * [[:w:Client–server model|Client–server model]] * [[:w:Peer-to-peer|Peer-to-peer model]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> fg68cmy9u85h4p6z9zek3ufnq956pbr 2693390 2693389 2024-12-26T22:03:59Z Tule-hog 2984180 /* 1.5 Compare and contrast transmission media and transceivers. */ c/e 2693390 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} {{wikibox|align=center|text align=left |left padding=20px|right padding=10px|<u>Further reading</u> * [[:w:Optical fiber connector|Optical fiber connector]] * [[:w:Fiber-optic cable|Fiber-optic cable]] * [[:w:Fiber media converter|Fiber media converter]] * [[:w:D-subminiature|D-sub]] * [[:w:Networking cable|Networking cable]] * [[:w:Broadband over power lines|Broadband over power lines]] * [[:w:Crimp connection|Crimp connection]] * [[:w:Punch down tool|Punch down tool]] * [[:w:Wire stripper|Wire stripper]] * [[:w:Cable tester|Cable tester]] }} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Network architecture|Network architecture]] * [[:w:Multitier architecture|Multitier architecture]] * [[:w:Client–server model|Client–server model]] * [[:w:Peer-to-peer|Peer-to-peer model]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> 6ax0pch8l5hb8d8lnkop1k73d5op971 2693391 2693390 2024-12-26T22:04:26Z Tule-hog 2984180 /* 1.5 Compare and contrast transmission media and transceivers. */ c/e 2693391 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} {{wikibox|wide=yes|<u>Further reading</u> * [[:w:Optical fiber connector|Optical fiber connector]] * [[:w:Fiber-optic cable|Fiber-optic cable]] * [[:w:Fiber media converter|Fiber media converter]] * [[:w:D-subminiature|D-sub]] * [[:w:Networking cable|Networking cable]] * [[:w:Broadband over power lines|Broadband over power lines]] * [[:w:Crimp connection|Crimp connection]] * [[:w:Punch down tool|Punch down tool]] * [[:w:Wire stripper|Wire stripper]] * [[:w:Cable tester|Cable tester]] }} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Network architecture|Network architecture]] * [[:w:Multitier architecture|Multitier architecture]] * [[:w:Client–server model|Client–server model]] * [[:w:Peer-to-peer|Peer-to-peer model]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> 9hk1a2qq3bmykwm4olfceewjiz6gi14 2693393 2693391 2024-12-26T22:06:46Z Tule-hog 2984180 /* 1.5 Compare and contrast transmission media and transceivers. */ rm dup item 2693393 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} {{wikibox|wide=yes|<u>Further reading</u> * [[:w:Optical fiber connector|Optical fiber connector]] * [[:w:Fiber-optic cable|Fiber-optic cable]] * [[:w:Fiber media converter|Fiber media converter]] * [[:w:D-subminiature|D-sub]] * [[:w:Networking cable|Networking cable]] * [[:w:Broadband over power lines|Broadband over power lines]] * [[:w:Crimp connection|Crimp connection]] * [[:w:Punch down tool|Punch down tool]] * [[:w:Wire stripper|Wire stripper]] }} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Network architecture|Network architecture]] * [[:w:Multitier architecture|Multitier architecture]] * [[:w:Client–server model|Client–server model]] * [[:w:Peer-to-peer|Peer-to-peer model]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> f7sz8pal5w3lp6knn7kjq1jd28gna0s 2693394 2693393 2024-12-26T22:09:21Z Tule-hog 2984180 /* 1.5 Compare and contrast transmission media and transceivers. */ c/e 2693394 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} {{wikibox|wide=yes|<u>Further reading</u> * [[:w:Optical fiber connector|Optical fiber connector]] * [[:w:Fiber-optic cable|Fiber-optic cable]] * [[:w:Fiber media converter|Fiber media converter]] * [[:w:D-subminiature|D-sub]] * [[:w:Networking cable|Networking cable]] * [[:w:Broadband over power lines|Broadband over power lines]] * [[:w:Patch panel|Patch panel]] * [[:w:Wire stripper|Wire stripper]] * [[:w:Punch down tool|Punch down tool]] * [[:w:Punch down block|Punch down block]] * [[:w:110 block|110 block]] * [[:w:Crimp connection|Crimp connection]] }} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Network architecture|Network architecture]] * [[:w:Multitier architecture|Multitier architecture]] * [[:w:Client–server model|Client–server model]] * [[:w:Peer-to-peer|Peer-to-peer model]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> 98t4npt2vkfdnyduogyr3zlw39dkj5b 2693406 2693394 2024-12-26T22:46:39Z Tule-hog 2984180 /* 1.3 Summarize cloud concepts and connectivity options. */ add further reading box 2693406 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} {{wikibox|<u>Further reading</u> * [[:w:Web service|Web service]] * [[:w:Unified communications|Unified communications]] }} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} {{wikibox|wide=yes|<u>Further reading</u> * [[:w:Optical fiber connector|Optical fiber connector]] * [[:w:Fiber-optic cable|Fiber-optic cable]] * [[:w:Fiber media converter|Fiber media converter]] * [[:w:D-subminiature|D-sub]] * [[:w:Networking cable|Networking cable]] * [[:w:Broadband over power lines|Broadband over power lines]] * [[:w:Patch panel|Patch panel]] * [[:w:Wire stripper|Wire stripper]] * [[:w:Punch down tool|Punch down tool]] * [[:w:Punch down block|Punch down block]] * [[:w:110 block|110 block]] * [[:w:Crimp connection|Crimp connection]] }} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Network architecture|Network architecture]] * [[:w:Multitier architecture|Multitier architecture]] * [[:w:Client–server model|Client–server model]] * [[:w:Peer-to-peer|Peer-to-peer model]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> c15rhyo2uhs6tqwlnv3vrdc8mxgr9he 2693407 2693406 2024-12-26T22:49:03Z Tule-hog 2984180 /* 1.2 Compare and contrast networking appliances, applications, and functions. */ add further reading box 2693407 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{wikibox|<u>Further reading</u> * [[:w:Networking hardware|Networking hardware]] }} {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} {{wikibox|<u>Further reading</u> * [[:w:Web service|Web service]] * [[:w:Unified communications|Unified communications]] }} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} {{wikibox|wide=yes|<u>Further reading</u> * [[:w:Optical fiber connector|Optical fiber connector]] * [[:w:Fiber-optic cable|Fiber-optic cable]] * [[:w:Fiber media converter|Fiber media converter]] * [[:w:D-subminiature|D-sub]] * [[:w:Networking cable|Networking cable]] * [[:w:Broadband over power lines|Broadband over power lines]] * [[:w:Patch panel|Patch panel]] * [[:w:Wire stripper|Wire stripper]] * [[:w:Punch down tool|Punch down tool]] * [[:w:Punch down block|Punch down block]] * [[:w:110 block|110 block]] * [[:w:Crimp connection|Crimp connection]] }} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Network architecture|Network architecture]] * [[:w:Multitier architecture|Multitier architecture]] * [[:w:Client–server model|Client–server model]] * [[:w:Peer-to-peer|Peer-to-peer model]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> po0w8piwlkwyu8axlpnnijij1c8tbmu 2693415 2693407 2024-12-26T23:05:39Z Tule-hog 2984180 /* 1.2 Compare and contrast networking appliances, applications, and functions. */ add further item 2693415 wikitext text/x-wiki ===1.1 Explain concepts related to the Open Systems Interconnection (OSI) reference model.=== * [[:w:OSI model|OSI model]] * [[:w:Physical layer|Layer 1 - Physical]] * [[:w:Data link layer|Layer 2 - Data link]] * [[:w:Network layer|Layer 3 - Network]] * [[:w:Transport layer|Layer 4 - Transport]] * [[:w:Session layer|Layer 5 - Session]] * [[:w:Presentation layer|Layer 6 - Presentation]] * [[:w:Application layer|Layer 7 - Application]] ===1.2 Compare and contrast networking appliances, applications, and functions.=== {{col-begin}} {{col-break}} * Physical and virtual appliances ** [[:w:Router (computing)|Router]] ** [[:w:Network switch|Switch]] ** [[:w:Firewall (computing)|Firewall]] ** [[:w:Intrusion detection system|Intrusion detection system]] (IDS)/[[:w:Intrusion prevention system|intrusion prevention system]] (IPS) ** [[:w:Load balancer|Load balancer]] ** [[:w:Proxy server|Proxy]] ** [[:w:Network-attached storage|Network-attached storage]] (NAS) ** [[:w:Storage area network|Storage area network]] (SAN) ** Wireless *** [[:w:Access point|Access point]] (AP) *** Controller {{col-break}} * Applications ** [[:w:Content delivery network|Content delivery network]] (CDN) * Functions ** [[:w:Virtual private network|Virtual private network]] (VPN) ** [[:w:Quality of service|Quality of service]] (QoS) ** [[:w:Time to live|Time to live]] (TTL) {{wikibox|<u>Further reading</u> * [[:w:Networking hardware|Networking hardware]] * [[:w:Comparison of firewalls|Comparison of firewalls]] }} {{col-end}} ===1.3 Summarize cloud concepts and connectivity options.=== {{col-begin}} {{col-break}} * [[:w:Network functions virtualization|Network functions virtualization]] (NFV) * Virtual private cloud (VPC) * Network security groups * Network security lists * Cloud gateways ** Internet gateway ** Network address translation (NAT) gateway * Cloud connectivity options ** VPN ** Direct Connect {{col-break}} * [[:w:Cloud deployment|Deployment models]] ** Public ** Private ** Hybrid * Service models ** [[:w:Software as a service|Software as a service]] (SaaS) ** [[:w:Infrastructure as a service|Infrastructure as a service]] (IaaS) ** [[:w:Platform as a service|Platform as a service]] (PaaS) * [[:w:Scalability|Scalability]] * [[:w:Elasticity (computing)|Elasticity]] * [[:w:Multitenancy|Multitenancy]] {{col-end}} {{wikibox|<u>Further reading</u> * [[:w:Web service|Web service]] * [[:w:Unified communications|Unified communications]] }} ===1.4 Explain common networking ports, protocols, services, and traffic types.=== {| class="wikitable" |+ style="font-weight:normal;" | See [https://quizlet.com/960350833/common-ports-flash-cards Quizlet]. |- ! Protocols !! Ports |- | [[:w:File Transfer Protocol|File Transfer Protocol]] (FTP) || 20/21 |- | [[:w:SSH File Transfer Protocol|Secure File Transfer Protocol]] (SFTP) || 22 |- | [[:w:Secure Shell|Secure Shell]] (SSH) || 22 |- | [[:w:Telnet|Telnet]] || 23 |- | [[:w:Simple Mail Transfer Protocol|Simple Mail Transfer Protocol]] (SMTP) || 25 |- | [[:w:Domain Name System|Domain Name System]] (DNS) || 53 |- | [[:w:Dynamic Host Configuration Protocol|Dynamic Host Configuration Protocol]] (DHCP) || 67/68 |- | [[:w:Trivial File Transfer Protocol|Trivial File Transfer Protocol]] (TFTP) || 69 |- | [[:w:Hypertext Transfer Protocol|Hypertext Transfer Protocol]] (HTTP) || 80 |- | [[:w:Network Time Protocol|Network Time Protocol]] (NTP) || 123 |- | [[:w:Simple Network Management Protocol|Simple Network Management Protocol]] (SNMP) || 161/162 |- | [[:w:Lightweight Directory Access Protocol|Lightweight Directory Access Protocol]] (LDAP) || 389 |- | [[:w:Hypertext Transfer Protocol Secure|Hypertext Transfer Protocol Secure]] (HTTPS) || 443 |- | [[:w:Server Message Block|Server Message Block]] (SMB) || 445 |- | [[:w:Syslog|Syslog]] || 514 |- | [[:w:Simple Mail Transfer Protocol Secure|Simple Mail Transfer Protocol Secure]] (SMTPS) || 587 |- | [[:w:Lightweight Directory Access Protocol over SSL|Lightweight Directory Access Protocol over SSL]] (LDAPS) || 636 |- | Structured Query Language (SQL) Server || 1433 |- | [[:w:Remote Desktop Protocol|Remote Desktop Protocol]] (RDP) || 3389 |- | [[:w:Session Initiation Protocol|Session Initiation Protocol]] (SIP) || 5060/5061 |} * [[:w:Internet protocol suite|Internet Protocol]] (IP) types ** [[:w:Internet Control Message Protocol|Internet Control Message Protocol]] (ICMP) ** [[:w:Transmission Control Protocol|Transmission Control Protocol]] (TCP) ** [[:w:User Datagram Protocol|User Datagram Protocol]] (UDP) ** [[:w:Generic Routing Encapsulation|Generic Routing Encapsulation]] (GRE) ** [[:w:Internet Protocol Security|Internet Protocol Security]] (IPSec) *** [[:w:Authentication Header|Authentication Header]] (AH) *** [[:w:Encapsulating Security Payload|Encapsulating Security Payload]] (ESP) *** [[:w:Internet Key Exchange|Internet Key Exchange]] (IKE) * [[:w:Internet_Protocol#Addressing_methods|Traffic types]] ** [[:w:Unicast|Unicast]] ** [[:w:Multicast|Multicast]] ** [[:w:Anycast|Anycast]] ** [[:w:Broadcasting (networking)|Broadcast]] ===1.5 Compare and contrast transmission media and transceivers.=== {{col-begin}} {{col-break}} * Wireless ** [[:w:802.11 standards|802.11 standards]] ** Cellular ** Satellite * Wired ** [[:w:802.3 standards|802.3 standards]] ** Single-mode vs. multimode fiber ** [[:w:DAC cable|Direct attach copper (DAC) cable]] *** [[:w:Twinaxial cable|Twinaxial cable]] ** [[:w:Coaxial cable|Coaxial cable]] ** Cable speeds ** [[:w:Plenum cable|Plenum]] vs. non-plenum cable {{col-break}} * [[:w:Transceivers|Transceivers]] ** Protocol *** [[:w:Ethernet|Ethernet]] *** [[:w:Fibre Channel|Fibre Channel]] (FC) ** Form factors *** [[:w:Small form-factor pluggable|Small form-factor pluggable]] (SFP) *** [[:w:Quad small form-factor pluggable|Quad small form-factor pluggable]] (QSFP) * Connector types ** [[:w:Subscriber connector|Subscriber connector]] (SC) ** [[:w:Local connector|Local connector]] (LC) ** [[:w:Straight tip connector|Straight tip]] (ST) ** [[:w:Multi-fiber push on|Multi-fiber push on]] (MPO) ** [[:w:Registered jack|Registered jack]] [[:w:RJ11|(RJ)11]] ** [[:w:RJ45|RJ45]] ** [[:w:F-type connector|F-type]] ** [[:w:Bayonet Neill–Concelman|Bayonet Neill–Concelman]] (BNC) {{col-end}} {{wikibox|wide=yes|<u>Further reading</u> * [[:w:Optical fiber connector|Optical fiber connector]] * [[:w:Fiber-optic cable|Fiber-optic cable]] * [[:w:Fiber media converter|Fiber media converter]] * [[:w:D-subminiature|D-sub]] * [[:w:Networking cable|Networking cable]] * [[:w:Broadband over power lines|Broadband over power lines]] * [[:w:Patch panel|Patch panel]] * [[:w:Wire stripper|Wire stripper]] * [[:w:Punch down tool|Punch down tool]] * [[:w:Punch down block|Punch down block]] * [[:w:110 block|110 block]] * [[:w:Crimp connection|Crimp connection]] }} ===1.6 Compare and contrast network topologies, architectures, and types.=== {{col-begin}} {{col-break}} * [[:w:Mesh networking|Mesh]] * [[:w:Network topology#Hybrid|Hybrid]] * [[:w:Star topology|Star/hub and spoke]] * Spine and leaf * [[:w:Point-to-point (telecommunications)|Point to point]] * [[:w:Three-tier hierarchical model|Three-tier hierarchical model]] ** Core ** Distribution ** Access {{col-break}} * Collapsed core * Traffic flows ** North-south ** East-west {{wikibox|<u>Further reading</u> * [[:w:Network topology|Network topology]] * [[:w:Network architecture|Network architecture]] * [[:w:Multitier architecture|Multitier architecture]] * [[:w:Client–server model|Client–server model]] * [[:w:Peer-to-peer|Peer-to-peer model]] }} {{col-end}} ===1.7 Given a scenario, use appropriate IPv4 network addressing.=== {{col-begin}} {{col-break}} * Public vs. private ** [[:w:APIPA|Automatic Private IP Addressing]] (APIPA) ** [[:w:RFC1918|RFC1918]] ** [[:w:Loopback|Loopback]]/[[:w:localhost|localhost]] * [[:w:Subnetting|Subnetting]] ** [[:w:Variable Length Subnet Mask|Variable Length Subnet Mask]] (VLSM) ** [[:w:Classless Inter-domain Routing|Classless Inter-domain Routing]] (CIDR) {{col-break}} * [[:w:Classful network|IPv4 address classes]] ** Class A ** Class B ** Class C ** Class D ** Class E {{wikibox|<u>Further reading</u> * [[:w:IP address|IP address]] }} {{col-end}} ===1.8 Summarize evolving use cases for modern network environments=== {{col-begin}} {{col-break}} * [[:w:Software-defined network|Software-defined network]] (SDN) and [[:w:SD-WAN|software-defined wide area network]] (SD-WAN) ** Application aware ** [[:w:Zero-touch provisioning|Zero-touch provisioning]] ** Transport agnostic ** Central policy management * [[:w:Virtual Extensible Local Area Network|Virtual Extensible Local Area Network]] (VXLAN) ** [[:w:Data center interconnect|Data center interconnect]] (DCI) ** Layer 2 encapsulation * [[:w:Zero trust architecture|Zero trust architecture]] (ZTA) ** Policy-based authentication ** Authorization ** [[:w:Principle of least privilege|Least privilege access]] * [[:w:Secure Access Secure Edge|Secure Access Secure Edge]] (SASE)/[[:w:Security Service Edge|Security Service Edge]] (SSE) {{col-break}} * [[:w:Infrastructure as code|Infrastructure as code]] (IaC) ** Automation *** Playbooks/templates/reusable tasks *** Configuration drift/compliance *** Upgrades *** Dynamic inventories ** [[:w:Source control|Source control]] *** Version control *** [[:w:Central repository|Central repository]] *** Conflict identification *** Branching * [[:w:IPv6 addressing|IPv6 addressing]] ** Mitigating [[:w:address exhaustion|address exhaustion]] ** Compatibility requirements *** Tunneling *** [[:w:Dual stack|Dual stack]] *** [[:w:NAT64|NAT64]] {{col-end}} {{wikibox|align=center|left padding=20px |right padding=10px|text align=left|<u>Further reading</u> * [[:w:IPv6|IPv6]] * [[:w:Computer network|Computer network]] * [[:w:Private network|Private network]] * [[:w:Template:Area networks|Area networks]] - [[:w:Local area network|Local area network]] (LAN)<br/> - [[:w:Wide area network|Wide area network]] (WAN)<br/> - [[:w:Wireless LAN|Wireless LAN]] (WLAN)<br/> - [[:w:Personal area network|Personal area network]] (PAN) - [[:w:Metropolitan area network|Metropolitan area network]] (MAN) }} <noinclude> {{BookCat}} </noinclude> 8yv767em50qkx391iyu0520uqt3yvxk 0 308111 2693360 2668599 2024-12-26T19:49:06Z Tule-hog 2984180 /* 2.1 Explain characteristics of routing technologies. */ add further reading from [[Network+/Architecture/Routing]] 2693360 wikitext text/x-wiki ====2.1 Explain characteristics of routing technologies.==== {{col-begin}} {{col-break}} * [[:w:Static routing|Static routing]] * [[:w:Dynamic routing|Dynamic routing]] ** [[:w:Border Gateway Protocol|Border Gateway Protocol]] (BGP) ** [[:w:Enhanced Interior Gateway Routing Protocol|Enhanced Interior Gateway Routing Protocol]] (EIGRP) ** [[:w:Open Shortest Path First|Open Shortest Path First]] (OSPF) * Route selection ** Administrative distance ** Prefix length ** [[:w:Metrics (networking)|Metric]] * Address translation ** [[:w:Network address translation|NAT]] ** [[:w:Port address translation|Port address translation]] (PAT) {{col-break}} * [[:w:First Hop Redundancy Protocol|First Hop Redundancy Protocol]] (FHRP) * [[:w:Virtual IP|Virtual IP]] (VIP) * Subinterfaces {{wikibox|<u>Further reading</u> * [[:w:Routing|Routing]] * [[:w:Distance-vector routing protocol|Distance-vector routing protocol]] * [[:w:Link-state routing protocol|Link-state routing protocol]] * [[:w:Routing Information Protocol|Routing Information Protocol]] (RIP) * [[:w:Virtual Router Redundancy Protocol|Virtual Router Redundancy Protocol]] (VRRP) * [[:w:Hot Standby Router Protocol|Hot Standby Router Protocol]] (HSRP) }} {{col-end}} ====2.2 Given a scenario, configure switching technologies and features.==== {{col-begin}} {{col-break}} * [[:w:Virtual Local Area Network|Virtual Local Area Network]] (VLAN) ** VLAN database ** [[:w:Switch virtual interface|Switch Virtual Interface]] (SVI) * [[:w:Spanning Tree Protocol|Spanning tree]] * [[:w:Maximum transmission unit|Maximum transmission unit]] (MTU) ** [[:w:Jumbo frame|Jumbo frames]] {{col-break}} * Interface configuration ** [[:w:Native VLAN|Native VLAN]] ** Voice VLAN ** [[:w:802.1Q tagging|802.1Q tagging]] ** [[:w:Link aggregation|Link aggregation]] ** Speed ** Duplex {{col-end}} ====2.3 Given a scenario, select and configure wireless devices and technologies.==== {{col-begin}} {{col-break}} {{see|:w:IEEE 802.11#Channels and frequencies}} * Channels ** Channel width ** Non-overlapping channels ** Regulatory impacts *** [[:w:802.11h|802.11h]] * Frequency options ** 2.4GHz ** 5GHz ** [[:w: Wi-Fi 6|6GHz]] ** [[:w:Band steering|Band steering]] {{see also|:w:Service set (802.11 network)}} * [[:w:Service set identifier|Service set identifier]] (SSID) ** [[:w:Basic service set identifier|Basic service set identifier]] (BSSID) ** [[:w:Extended service set identifier|Extended service set identifier]] (ESSID) {{col-break}} {{see also|:w:Wireless network}} * Network types ** [[:w: Wireless mesh network |Mesh networks]] ** [[:w:Wireless ad hoc network|Ad hoc]] ** [[:w:Point-to-point_(telecommunications)#Modern_links|Point to point]] ** Infrastructure * Encryption ** [[:w:Wi-Fi Protected Access 2|Wi-Fi Protected Access 2]] (WPA2) ** [[:w:WPA3|WPA3]] * Guest networks ** [[:w:Captive portal|Captive portals]] * [[:w:Authentication|Authentication]] ** [[:w:Pre-shared key|Pre-shared key]] (PSK) vs. [[:w:Enterprise key|Enterprise]] * [[:w:Antennas|Antennas]] ** [[:w:Omnidirectional antenna|Omnidirectional]] vs. [[:w:Directional antenna|directional]] * Autonomous vs. lightweight access point {{col-end}} ====2.4 Explain important factors of physical installations.==== {{col-begin}} {{col-break}} * Important installation implications ** Locations *** [[:w:Intermediate distribution frame|Intermediate distribution frame]] (IDF) *** [[:w:Main distribution frame|Main distribution frame]] (MDF) ** Rack size ** Port-side exhaust/intake ** Cabling *** [[:w:Patch panel|Patch panel]] *** [[:w:Fiber distribution panel|Fiber distribution panel]] ** Lockable {{col-break}} * Power ** [[:w:Uninterruptible power supply|Uninterruptible power supply]] (UPS) ** [[:w:Power distribution unit|Power distribution unit]] (PDU) ** Power load ** Voltage * Environmental factors ** Humidity ** Fire suppression ** Temperature {{col-end}} <noinclude> {{BookCat}} </noinclude> jwo4jx2q4w59wzfd6a9uy5x79zwmyqc 2693362 2693360 2024-12-26T19:51:39Z Tule-hog 2984180 /* 2.3 Given a scenario, select and configure wireless devices and technologies. */ mk further reading box 2693362 wikitext text/x-wiki ====2.1 Explain characteristics of routing technologies.==== {{col-begin}} {{col-break}} * [[:w:Static routing|Static routing]] * [[:w:Dynamic routing|Dynamic routing]] ** [[:w:Border Gateway Protocol|Border Gateway Protocol]] (BGP) ** [[:w:Enhanced Interior Gateway Routing Protocol|Enhanced Interior Gateway Routing Protocol]] (EIGRP) ** [[:w:Open Shortest Path First|Open Shortest Path First]] (OSPF) * Route selection ** Administrative distance ** Prefix length ** [[:w:Metrics (networking)|Metric]] * Address translation ** [[:w:Network address translation|NAT]] ** [[:w:Port address translation|Port address translation]] (PAT) {{col-break}} * [[:w:First Hop Redundancy Protocol|First Hop Redundancy Protocol]] (FHRP) * [[:w:Virtual IP|Virtual IP]] (VIP) * Subinterfaces {{wikibox|<u>Further reading</u> * [[:w:Routing|Routing]] * [[:w:Distance-vector routing protocol|Distance-vector routing protocol]] * [[:w:Link-state routing protocol|Link-state routing protocol]] * [[:w:Routing Information Protocol|Routing Information Protocol]] (RIP) * [[:w:Virtual Router Redundancy Protocol|Virtual Router Redundancy Protocol]] (VRRP) * [[:w:Hot Standby Router Protocol|Hot Standby Router Protocol]] (HSRP) }} {{col-end}} ====2.2 Given a scenario, configure switching technologies and features.==== {{col-begin}} {{col-break}} * [[:w:Virtual Local Area Network|Virtual Local Area Network]] (VLAN) ** VLAN database ** [[:w:Switch virtual interface|Switch Virtual Interface]] (SVI) * [[:w:Spanning Tree Protocol|Spanning tree]] * [[:w:Maximum transmission unit|Maximum transmission unit]] (MTU) ** [[:w:Jumbo frame|Jumbo frames]] {{col-break}} * Interface configuration ** [[:w:Native VLAN|Native VLAN]] ** Voice VLAN ** [[:w:802.1Q tagging|802.1Q tagging]] ** [[:w:Link aggregation|Link aggregation]] ** Speed ** Duplex {{col-end}} ====2.3 Given a scenario, select and configure wireless devices and technologies.==== {{col-begin}} {{col-break}} * Channels ** Channel width ** Non-overlapping channels ** Regulatory impacts *** [[:w:802.11h|802.11h]] * Frequency options ** 2.4GHz ** 5GHz ** [[:w: Wi-Fi 6|6GHz]] ** [[:w:Band steering|Band steering]] * [[:w:Service set identifier|Service set identifier]] (SSID) ** [[:w:Basic service set identifier|Basic service set identifier]] (BSSID) ** [[:w:Extended service set identifier|Extended service set identifier]] (ESSID) * Encryption ** [[:w:Wi-Fi Protected Access 2|Wi-Fi Protected Access 2]] (WPA2) ** [[:w:WPA3|WPA3]] {{col-break}} * Network types ** [[:w: Wireless mesh network |Mesh networks]] ** [[:w:Wireless ad hoc network|Ad hoc]] ** [[:w:Point-to-point_(telecommunications)#Modern_links|Point to point]] ** Infrastructure * Guest networks ** [[:w:Captive portal|Captive portals]] * [[:w:Authentication|Authentication]] ** [[:w:Pre-shared key|Pre-shared key]] (PSK) vs. [[:w:Enterprise key|Enterprise]] * [[:w:Antennas|Antennas]] ** [[:w:Omnidirectional antenna|Omnidirectional]] vs. [[:w:Directional antenna|directional]] * Autonomous vs. lightweight access point {{wikibox|<u>Further reading</u> * [[:w:Service set (802.11 network)|Service set (802.11 network)]] * [[:w:Wireless network|Wireless network]] * [[:w:IEEE 802.11#Channels and frequencies|IEEE 802.11#Channels and frequencies]] }} {{col-end}} ====2.4 Explain important factors of physical installations.==== {{col-begin}} {{col-break}} * Important installation implications ** Locations *** [[:w:Intermediate distribution frame|Intermediate distribution frame]] (IDF) *** [[:w:Main distribution frame|Main distribution frame]] (MDF) ** Rack size ** Port-side exhaust/intake ** Cabling *** [[:w:Patch panel|Patch panel]] *** [[:w:Fiber distribution panel|Fiber distribution panel]] ** Lockable {{col-break}} * Power ** [[:w:Uninterruptible power supply|Uninterruptible power supply]] (UPS) ** [[:w:Power distribution unit|Power distribution unit]] (PDU) ** Power load ** Voltage * Environmental factors ** Humidity ** Fire suppression ** Temperature {{col-end}} <noinclude> {{BookCat}} </noinclude> fufe38nk0vv25nrt5x1e9a2bm9wb9sm 2693403 2693362 2024-12-26T22:38:30Z Tule-hog 2984180 /* 2.1 Explain characteristics of routing technologies. */ add further item 2693403 wikitext text/x-wiki ====2.1 Explain characteristics of routing technologies.==== {{col-begin}} {{col-break}} * [[:w:Static routing|Static routing]] * [[:w:Dynamic routing|Dynamic routing]] ** [[:w:Border Gateway Protocol|Border Gateway Protocol]] (BGP) ** [[:w:Enhanced Interior Gateway Routing Protocol|Enhanced Interior Gateway Routing Protocol]] (EIGRP) ** [[:w:Open Shortest Path First|Open Shortest Path First]] (OSPF) * Route selection ** Administrative distance ** Prefix length ** [[:w:Metrics (networking)|Metric]] * Address translation ** [[:w:Network address translation|NAT]] ** [[:w:Port address translation|Port address translation]] (PAT) {{col-break}} * [[:w:First Hop Redundancy Protocol|First Hop Redundancy Protocol]] (FHRP) * [[:w:Virtual IP|Virtual IP]] (VIP) * Subinterfaces {{wikibox|<u>Further reading</u> * [[:w:Routing|Routing]] * [[:w:Distance-vector routing protocol|Distance-vector routing protocol]] * [[:w:Link-state routing protocol|Link-state routing protocol]] * [[:w:Routing Information Protocol|Routing Information Protocol]] (RIP) * [[:w:Virtual Router Redundancy Protocol|Virtual Router Redundancy Protocol]] (VRRP) * [[:w:Hot Standby Router Protocol|Hot Standby Router Protocol]] (HSRP) * [[:w:Port forwarding|Port forwarding]] }} {{col-end}} ====2.2 Given a scenario, configure switching technologies and features.==== {{col-begin}} {{col-break}} * [[:w:Virtual Local Area Network|Virtual Local Area Network]] (VLAN) ** VLAN database ** [[:w:Switch virtual interface|Switch Virtual Interface]] (SVI) * [[:w:Spanning Tree Protocol|Spanning tree]] * [[:w:Maximum transmission unit|Maximum transmission unit]] (MTU) ** [[:w:Jumbo frame|Jumbo frames]] {{col-break}} * Interface configuration ** [[:w:Native VLAN|Native VLAN]] ** Voice VLAN ** [[:w:802.1Q tagging|802.1Q tagging]] ** [[:w:Link aggregation|Link aggregation]] ** Speed ** Duplex {{col-end}} ====2.3 Given a scenario, select and configure wireless devices and technologies.==== {{col-begin}} {{col-break}} * Channels ** Channel width ** Non-overlapping channels ** Regulatory impacts *** [[:w:802.11h|802.11h]] * Frequency options ** 2.4GHz ** 5GHz ** [[:w: Wi-Fi 6|6GHz]] ** [[:w:Band steering|Band steering]] * [[:w:Service set identifier|Service set identifier]] (SSID) ** [[:w:Basic service set identifier|Basic service set identifier]] (BSSID) ** [[:w:Extended service set identifier|Extended service set identifier]] (ESSID) * Encryption ** [[:w:Wi-Fi Protected Access 2|Wi-Fi Protected Access 2]] (WPA2) ** [[:w:WPA3|WPA3]] {{col-break}} * Network types ** [[:w: Wireless mesh network |Mesh networks]] ** [[:w:Wireless ad hoc network|Ad hoc]] ** [[:w:Point-to-point_(telecommunications)#Modern_links|Point to point]] ** Infrastructure * Guest networks ** [[:w:Captive portal|Captive portals]] * [[:w:Authentication|Authentication]] ** [[:w:Pre-shared key|Pre-shared key]] (PSK) vs. [[:w:Enterprise key|Enterprise]] * [[:w:Antennas|Antennas]] ** [[:w:Omnidirectional antenna|Omnidirectional]] vs. [[:w:Directional antenna|directional]] * Autonomous vs. lightweight access point {{wikibox|<u>Further reading</u> * [[:w:Service set (802.11 network)|Service set (802.11 network)]] * [[:w:Wireless network|Wireless network]] * [[:w:IEEE 802.11#Channels and frequencies|IEEE 802.11#Channels and frequencies]] }} {{col-end}} ====2.4 Explain important factors of physical installations.==== {{col-begin}} {{col-break}} * Important installation implications ** Locations *** [[:w:Intermediate distribution frame|Intermediate distribution frame]] (IDF) *** [[:w:Main distribution frame|Main distribution frame]] (MDF) ** Rack size ** Port-side exhaust/intake ** Cabling *** [[:w:Patch panel|Patch panel]] *** [[:w:Fiber distribution panel|Fiber distribution panel]] ** Lockable {{col-break}} * Power ** [[:w:Uninterruptible power supply|Uninterruptible power supply]] (UPS) ** [[:w:Power distribution unit|Power distribution unit]] (PDU) ** Power load ** Voltage * Environmental factors ** Humidity ** Fire suppression ** Temperature {{col-end}} <noinclude> {{BookCat}} </noinclude> 49y25u75fnnftiswlvfrnsxjajvpz1k 0 308113 2693405 2653082 2024-12-26T22:46:17Z Tule-hog 2984180 /* 4.1 Explain the importance of basic network security concepts. */ add further reading bx 2693405 wikitext text/x-wiki ====4.1 Explain the importance of basic network security concepts.==== {{col-begin}} {{col-break}} * [[:w:Logical security|Logical security]] ** Encryption *** [[:w:Data in transit|Data in transit]] *** [[:w:Data at rest|Data at rest]] ** [[:w:Public key certificate|Certificates]] *** [[:w:Public key infrastructure|Public key infrastructure]] (PKI) *** Self-signed ** [[:w:Identity and access management|Identity and access management]] (IAM) *** Authentication *** [[:w:Multi-factor authentication|Multifactor authentication]] (MFA) *** [[:w:Single sign-on|Single sign-on]] (SSO) *** [[:w:Remote Authentication Dial-in User Service|Remote Authentication Dial-in User Service]] (RADIUS) *** [[:w:LDAP|LDAP]] *** [[:w:Security Assertion Markup Language|Security Assertion Markup Language]] (SAML) *** [[:w:Terminal Access Controller Access Control System Plus|Terminal Access Controller Access Control System Plus]] (TACACS+) *** Time-based authentication *** Authorization *** [[:w:Principle of least privilege|Least privilege]] *** [[:w:Role-based access control|Role-based access control]] ** [[:w:Geofencing|Geofencing]] {{col-break}} * [[:w:Physical security|Physical security]] ** Camera ** Locks * Deception technologies ** [[:w:Honeypot (computing)|Honeypot]] ** [[:w:Honeynet|Honeynet]] * Common security terminology ** Risk ** [[:w: Vulnerability (computer security)|Vulnerability]] ** [[:w:Exploit (computer security)|Exploit]] ** [[:w:Threat (computer security)|Threat]] ** [[:w:CIA triad|Confidentiality, Integrity, and Availability (CIA) triad]] * Audits and regulatory compliance ** Data locality ** [[:w:Payment Card Industry Data Security Standard|Payment Card Industry Data Security Standards]] (PCI DSS) ** [[:w:General Data Protection Regulation|General Data Protection Regulation]] (GDPR) * Network segmentation enforcement ** [[:w:Internet of things|Internet of Things]] (IoT) and [[:w:Industrial internet of things|Industrial Internet of Things]] (IIoT) ** [[:w:Supervisory control and data acquisition|Supervisory control and data acquisition]] (SCADA), [[:w:Industrial control system|industrial control system]] (ICS), [[:w:Operational technology|operational technology]] (OT) ** Guest ** [[:w:Bring your own device|Bring your own device]] (BYOD) {{col-end}} {{wikibox|<u>Further reading</u> * [[:w:Remote access service|Remote access service]] (RAS) * [[:w:Wireless LAN controller|Wireless LAN controller]] (WLC) }} ====4.2 Summarize various types of attacks and their impact to the network.==== {{col-begin}} {{col-break}} * [[:w:DoS|Denial-of-service]] (DoS)/[[:w:DDoS|distributed denial-of-service]] (DDoS) * [[:w:VLAN hopping|VLAN hopping]] * [[:w:MAC flooding|Media Access Control (MAC) flooding]] * Address Resolution Protocol (ARP) poisoning * [[:w:ARP spoofing|ARP spoofing]] * DNS poisoning * [[:w:DNS spoofing|DNS spoofing]] {{col-break}} * Rogue devices and services ** [[:w:Rogue DHCP|DHCP]] ** [[:w:Rouge AP|AP]] * [[:w:Evil twin (wireless networks)|Evil twin]] * [[:w:On-path attack|On-path attack]] * [[:w:Social engineering (security)|Social engineering]] ** [[:w:Phishing|Phishing]] ** [[:w:Dumpster diving#Dumpster diving with criminal intentions (Garbage theft)|Dumpster diving]] ** [[:w:Shoulder surfing (computer security)|Shoulder surfing]] ** [[:w:Piggybacking (security)|Tailgating]] * [[:w:Malware|Malware]] {{col-end}} ====4.3 Given a scenario, apply network security features, defense techniques, and solutions.==== {{col-begin}} {{col-break}} * [[:w:Device hardening|Device hardening]] ** Disable unused ports and services ** Change default passwords * [[:w:Network access control|Network access control]] (NAC) ** [[:w:Port security (networking)|Port security]] ** [[:w:802.1X|802.1X]] ** [[:w:MAC filtering|MAC filtering]] {{col-break}} * [[:w:Key management|Key management]] * Security rules ** [[:w:Access control list|Access control list]] (ACL) ** Uniform Resource Locator (URL) filtering ** Content filtering * Zones ** Trusted vs. untrusted ** [[:w:Screened subnet|Screened subnet]] {{col-end}} <noinclude> {{BookCat}} </noinclude> [[Category:Security]] 0bagq2w48j10lfohq0h3jxepkopnhaz 0 316933 2693458 2692324 2024-12-26T23:45:55Z MathXplore 2888076 added [[Category:Engineering]] using [[Help:Gadget-HotCat|HotCat]] 2693458 wikitext text/x-wiki ==== [[Introduction to telecommunications: the struggle of nationalists|1.Introduction to telecommunications: the struggle of nationalists.]] ==== ==== [[2. wiring and telephone office exchanges.]] ==== ==== [[Integration and ethernet protocol|3. Integration and ethernet protocol.]] ==== [[Category:Engineering]] dke91i71omsys1op2f21wlaiumnjj4m 0 316944 2693586 2692962 2024-12-27T09:25:36Z UniBambergIslamicStudies 2987517 2693586 wikitext text/x-wiki [[The Bamberg Introduction to the History of Islam (BIHI) 02|2 <<<]] — [[The Bamberg Introduction to the History of Islam (BIHI) 04|>>> 4]] = 3. The Prophet of Yathrib and the New Polity (622-630) = The center of the new religion shifts to the oasis of Yathrib, with warfare taking center stage. Muḥammad and his followers engage in battles against pagan Mecca and increasingly come into conflict with the Jews of Yathrib, who are ultimately expelled from the oasis. As the leader of the nascent community, Muḥammad implements a series of legal, social, and ritual reforms. == 3.1. Maghāzī – The Military Expeditions of Muḥammad == === 3.1.1. The Provocation of the Quraysh === Arab sources consistently report that Muḥammad arrived at the oasis of Yathrib on September 24, 622, following his emigration from Mecca. Having been expelled from his hometown, he considered it justifiable to engage in conflict against his former hometown. This is clearly reflected in two Qur'anic verses, widely recognized as the earliest revelations on the subject of warfare: {{quote|Sanction is given unto those who fight because they have been wronged; and Allah is indeed able to give them victory; Those who have been driven from their homes unjustly only because they said: Our Lord is Allah. [https://corpuscoranicum.de/en/verse-navigator/sura/22/verse/39/print – Q 22:39f]}} The war with Mecca, which Muḥammad waged from his new base in Yathrib, began with minor pinpricks. According to the chronology of [[w:Al-Waqidi|al-Wāqidī]], who composed a detailed account of Muḥammad's military expeditions ([https://de.wikipedia.org/wiki/Magh%C4%81z%C4%AB maghāzī]) in the early 9th century, Muḥammad dispatched his uncle [[w:Hamza_ibn_Abd_al-Muttalib|Ḥamza]] with a group of warriors seven months after his arrival in Yathrib to intercept a Meccan trade caravan returning from Syria under the leadership of [[w:Amr_ibn_Hisham|Abū Jahl]]. However, no combat occurred because a man from the [[w:Juhaynah|Juhaynah]] tribe, allied with both sides, intervened. During a [[w:Expedition_of_Ubaydah_ibn_al-Harith|second expedition]] in April 623, "the first arrow of Islam" was launched. The conflict with the Meccans soon disregarded traditional Arab religious norms, such as the obligation to maintain peace during the sacred months (see above [[The Bamberg Introduction to the History of Islam (BIHI) 01#1.3.3. Ancient Arabian Paganism and the Sacred Sites of Mecca|1.3.3.]]). For example, a unit commissioned by Muḥammad [[w:Raid_on_Nakhla|raided]] a Meccan caravan during the sacred month of [[w:Rajab|Rajab]] near [[w:Nakhla_(Saudi_Arabia)|Nakhla]], south of Mecca. According to tradition, this event prompted the following revelation: {{quote|They question [you] (O Muhammad) with regard to warfare in the sacred month. Say: Warfare therein is a great (transgression), but to turn (men) from the way of Allah, and to disbelieve in Him and in the Inviolable Place of Worship, and to expel His people thence, is a greater with Allah; for persecution is worse than killing. And they will not cease from fighting against you till they have made you renegades from your religion, if they can. [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/217/print – Q 2:217]}} <!-- linked German article for Maghāzī as there isn't an English one. -linked Raid on Nakhla, Nakhla_(Saudi_Arabia), and Expedition_of_Ubaydah_ibn_al-Harith articles -My understanding is that many early Muslims were tortured, killed, or socially ostracized for their faith. They were not only expelled but lost their homes and wealth during the migration, as their properties were taken by the Quraysh. And in Medina, they were under threat from Quraysh as well, no? So besides 'reclaiming' some of these things or gaining economic power, didn’t the raids and expeditions target Meccan trade routes to weaken Quraysh’s economic base as well? Here, to me, it reads as the Prophet was the provocateur (as evidenced also by the title) and initiated hostilities out of vengeance (b/c he was expelled) and for economic gain. --> From this Qur'anic verse, it is evident that the continued existence of the old religion in Mecca posed a constant temptation for Muḥammad’s followers to abandon their faith. Since many of them apparently found military combat (''qitāl'') undesirable, Muḥammad now declared it a duty (cf. Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/216/print 2:216]) and elevated it to a religious level by designating it as ''jihād fī sabīl Allāh'' (“striving [[w:Fi_sabilillah|in the way of God]]”, as stated in the subsequent verse Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/218/print 2:218]). This term has also been adopted into the English language in the form of [[w:Jihad|Jihad]]. <!-- "erklärte ihn Muhammad nun zur Pflicht (vgl. Q 2:216)": could this be stated as "Qur'an" instead of Muhammad here for neutral language? --> [[File:Balami - Tarikhnama - The Battle of Badr - The death of Abu Jahl, and the casting of the Meccan dead into dry wells (cropped).jpg|thumb|Illustration of the [[w:Battle_of_Badr|Battle of Badr]] in a Persian manuscript, early 14th century]] The first major confrontation between the Meccans and Muḥammad’s followers took place in March 624 near the site of [[w:Badr,_Saudi_Arabia|Badr]], approximately 130 kilometers southwest of Yathrib. Muḥammad had received information about a wealthy Meccan caravan returning from Syria. With 300 men, including members of the [[w:Banu_Muzaina|Muzaynah]] tribe allied with the [[w:Banu_Aws|Aws]], he set out for Badr, situated along the coastal road, to intercept the caravan. A battle ensued between Muḥammad's forces and a Meccan army of approximately 950 men, which had rushed to the caravan's aid under the command of Muḥammad’s bitter adversary Abū Jahl. Muḥammad's forces achieved an unexpected victory. The Meccans suffered between 45 and 70 fatalities, with a similar number taken prisoner. Among the fallen Meccans were several prominent figures, including Abū Jahl. In contrast, Muḥammad’s followers lost only 14 men and captured substantial spoils of war. <!-- linked city of Badr --> Following the battle, Muḥammad had some of the prisoners beheaded, including his former adversary [[w:Nadr_ibn_al-Harith|al-Naḍr ibn al-Ḥārith]]. The [[w:Battle_of_Badr|victory at Badr]] was of immense military and religious significance for Muḥammad's followers. Apparently, however, not all of them contributed to this victory. This is evident from verses revealed after Badr, which clarify that those among the believers who “sit still” at home without a valid excuse are not equal in rank before God to the Mujāhidūn – those who engage in jihad (strive in the way of Allah) with their wealth and their lives (cf. Q [https://corpuscoranicum.de/en/verse-navigator/sura/4/verse/95/print 4:95f]). <!-- -ließ Muhammad einige der Gefangenen enthaupten: "ordered the execution of several prisoners" would sound more academic and formal in English, but I translated it as "had some of the prisoners beheaded" to align with German wording. -Pickthall (and others similarly) translate the verse as “those who strive in the way of Allah with their wealth and lives”. I translated it as 'engage in jihad' to align with the German but put 'strive in the way of Allah' in parentheses. --> === 3.1.2. The Defense Against the Meccan Counterattack === The defeat at Badr dealt a severe blow to the Quraysh of Mecca. They had long been regarded as one of the most powerful tribes in Arabia, and to some extent, their commercial success relied on this reputation. Their trade depended on cooperation with many other tribes, and now, insubordination from some of these tribes was to be anticipated. It was therefore of critical importance for the Quraysh to demonstrate that they still possessed the strength to exact revenge for the wrongs they had suffered. Ten weeks after the Battle of Badr, [[w:Abu_Sufyan_ibn_Harb|Abū Sufyān ibn Ḥarb]], who had assumed leadership of Mecca following the battle, carried out a swift raid on Yathrib. After setting fire to two houses, however, he quickly withdrew. [[File:The Prophet Muhammad and the Muslim Army at the Battle of Uhud, from the Siyer-i Nebi, 1595.jpg|thumb|A depiction of the Battle of Uhud in a [[w:Siyer-i_Nebi|Siyer-i Nebi]] from 1594, now part of the David Collection in Copenhagen.]] In the months that followed, Abū Sufyān succeeded in recruiting 3,000 well-equipped warriors. In March 625, he advanced toward Yathrib with this force, penetrating the oasis from its northwestern corner. At Mount Uhud, a battle ensued, with the momentum shifting back and forth between the two sides for a long time. As the tide of war began to shift in favor of Muḥammad’s followers, they started gathering the spoils. This prompted a group of Muḥammad’s archers to abandon their positions to turn their attention to the spoils. On the Meccan side, [[w:Khalid_ibn_al-Walid|Khālid ibn al-Walīd]], a prominent warrior, exploited the situation to sow confusion among the ranks of Muḥammad's followers and ultimately overpower them. However, in the end, Muḥammad’s followers succeeded in regaining critical positions, causing the Meccans to withdraw without permanently eliminating their adversary, Muḥammad. For Muḥammad’s followers, the [[w:Battle_of_Uhud|Battle of Uhud]] was nevertheless a bitter disappointment: not only because they had lost 50 to 70 men, including Muḥammad’s uncle [[w:Hamza_ibn_Abd_al-Muttalib|Ḥamza]], and Muḥammad himself had been injured, but also because they came to realize that divine support was not as assured as it had seemed after their victory at Badr. Several Qur’anic verses from this period affirm that those who are killed “in the way of God” are not truly dead but living (Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/154/print 2:154]), are provided for by their Lord (Q [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/169/print 3:169]), have their sins forgiven (Q [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/157/print 3:157]), and are admitted directly into Paradise (Q [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/195/print 3:195]). <!-- Q 3:165-168: is said to deal with the reason as to why they lost at Uhud. --> The conflict between Muḥammad and the Meccans was by no means concluded with the Battle of Uhud. As Muḥammad continued to disrupt Meccan trade and found an increasing number of allies among the Arabian Bedouins, the Meccans felt compelled to take action against him once more. In turn, they sought to recruit a number of Bedouin tribes to their side. These alliances demonstrate that the conflict between Mecca and Yathrib had by then extended to the surrounding regions of both cities. In July 625, the [[w:Banu_Sulaym|Banū Sulaym]], a tribe allied with the Quraysh, massacred a large number of Muslims at [[w:Massacre_of_Bi%27r_Ma%27una|Biʾr Maʿūna]], located between Mecca and Yathrib. In response, Muḥammad is said to have cursed the Banū Sulaym for an entire month. This practice has been preserved in a modified form as part of the [[w:Qunut|Qunūt]], a supplication recited during the morning prayer or the nightly [[w:Witr|Witr]] prayer. <!-- linked the article on the massacre & Witr prayer -the first paragraph mentions the wrongs Quraysh has suffered, whereas this paragraph shows the ongoing reasons as to why the Quraysh felt they had to attack. Can the Medinan side also be described? My understanding is that among other things Quraysh continued to oppress Muslims who had not migrated to Medina. There are documented cases of torture and economic deprivations. Quraysh also would cut off trade relationships with tribes that supported the Muslim community and use families of migrants as leverage, no? --> At the beginning of 627, the Meccans and their allies advanced to Yathrib with a force of 10,000 men. Muḥammad, however, had a trench (''khandaq'') excavated around the less fortified areas of the oasis settlement, making it wide enough that a horse could not leap across. This move took the Meccans by such surprise that they were unable to devise an effective strategic response. What had been intended as an assault instead turned into a siege. Due to intrigues, however, the Meccan alliance collapsed after only 14 days, forcing an end to the [[w:Battle_of_the_Trench|siege of Yathrib]]. The Meccans ultimately withdrew without having achieved anything. === 3.1.3. The Military and Political Breakthrough === [[File:Dinar, Khusro II, 590, 591-628 AD, year 31 - Bode-Museum - DSC02737.JPG|thumb|Sasanian ruler [[w:Khosrow_II|Khosrow II]] (r. 590–628) depicted on a gold coin, [[w:Bode_Museum|Bode Museum]]]] The Battle of the Trench was, essentially, Muḥammad’s final defensive campaign. From that point onward, his life entered an offensive phase, marking the beginning of an era of conquests for the community he had established. To understand Muḥammad's subsequent military success, it is necessary to contextualize the political dynamics of the Middle East during that period. At the beginning of the 7th century, a prolonged conflict erupted between the [[w:Byzantine_Empire|Byzantine Empire]] and the [[w:Sasanian_Empire|Sasanian Empire]]. Between 603 and 619, Sasanian forces initially conquered Syria, Palestine, and Egypt. In 622, however, the Byzantine emperor launched a counteroffensive. The conflict led to intense clashes in which the Sasanians suffered several defeats. It concluded in 628 with a peace treaty requiring [[w:Khosrow_II|Khosrow II]] to return all conquered territories. Subsequently, Khosrow was overthrown by his officers, initiating a period of political turmoil in the Sasanian Empire that persisted until 633. During this time, the Sasanian alliance network on the Arabian Peninsula collapsed. It was precisely during this five-year power vacuum that Muḥammad transformed his newly established state into a military and political success. <!-- Thronwirren: translated it as ‘political turmoil’. --> In the year following the [[w:Battle_of_the_Trench|Battle of the Trench]], he led several smaller military expeditions, the most significant being those against the oasis of [[w:Dumat_al-Jandal|Dumat al-Jandal]] and the [[w:Banu_Mustaliq|Muṣṭaliq tribe]], situated west of Yathrib. In March 628, accompanied by a group of believers, he set out for Mecca to perform the [[w:Umrah|ʿUmrah]] pilgrimage. The Meccans, suspecting hostile intentions, ensured that he did not approach the city. From his encampment at al-Ḥudaybiya, on the outskirts of the [[w:Haram_(site)|Ḥaram]], Muḥammad initiated negotiations with the Meccans, resulting in a [[w:Treaty_of_al-Hudaybiya|treaty]]. The treaty imposed what appeared on the surface to be humiliations, which in turn created tensions among his followers. For instance, the Meccan envoy refused to recognize him as “Muḥammad, the Messenger of God,” acknowledging him only as “Muḥammad ibn ʿAbdallāh.” However, the terms of the agreement were of greater significance: they included a ten-year truce and a promise from the Meccans to allow Muḥammad and his followers to enter the city the following year for a three-day ʿUmrah. In return, Muḥammad refrained from performing the ʿUmrah that year and withdrew with his men to Yathrib. <!-- linked Dumat_al-Jandal, Banu_Mustaliq, and Treaty_of_al-Hudaybiya --> The Treaty of Ḥudaybiya was a triumph for the Prophet and his followers. The Qur'an reports that God sent down His [[w:Sakina|''sakīna'']] into the hearts of the believers, increasing their faith (Q [https://corpuscoranicum.de/en/verse-navigator/sura/48/verse/4/print 48:4], [https://corpuscoranicum.de/en/verse-navigator/sura/48/verse/18/print 18]). The term sakīna originates from the Jewish concept of [[w:Shekhinah|Shekhinah]], which denotes the “presence” of God among His people. In this context, however, it also refers to a psychological state of tranquility and serenity. Following the Treaty of Ḥudaybiya, several Arabs from other regions of Arabia who had already pledged allegiance to Muḥammad previously completed their [[w:Hijrah|Hijrah]]—that is, they migrated to Yathrib—to provide military support to Muḥammad. Among them were, for example, the two Yemenis, [[w:Abu_Hurayra|Abū Hurayra]] and [[w:Abu_Musa_al-Ash%27ari|Abū Mūsā al-Ashʿarī]]. The following year, in March 629, Muḥammad traveled to Mecca with approximately 2,000 followers to perform the planned [[w:First_Pilgrimage|ʿUmrah]]. On this occasion, he married [[w:Maymunah_bint_al-Harith|Maymūnah]], the sister-in-law of his uncle [[w:Abbas_ibn_Abd_al-Muttalib|ʿAbbās]], who at that time had assumed the leadership of the [[w:Banu_Hashim|Banū Hāshim]] in Mecca. An increasing number of Meccans began to acknowledge Muḥammad as a prophet and left the city to join him, including those who had fought against him only a short time earlier, such as Khālid ibn al-Walīd, who had been on the opposing side during the Battle of Uhud (see above [[#3.1.2. The Defense Against the Meccan Counterattack|3.1.2.]]). The Qur'an specifies a distinct procedure for women who sought to join the Muslim camp: They were to be examined, and if recognized as true believers, they were not to be sent back to the disbelievers; the Muslim community was required to reimburse the disbelievers for their dowries, after which it was permissible to marry these women (Q [https://corpuscoranicum.de/en/verse-navigator/sura/60/verse/10/print 60:10]). <!-- linked the First Pilgrimage and Maymunah bint al-Harith. -“mit etwa 2.000 Mann”: translated with “followers”, as I think you used ‘Mann’ as a figure of speech here (instead of Männer), assuming there were also women? Otherwise, it would be translated as “2000 men”. --> In the course of 629, Muḥammad oversaw additional military campaigns. In September, he [[w:Battle_of_Mu%27tah|dispatched]] his former slave and adopted son, [[w:Zayd_ibn_Haritha_al-Kalbi|Zayd ibn Ḥāritha]], with an army to [[w:Mu%27tah|Muʿtah]], in present-day Jordan, east of the southern tip of the Dead Sea. A series of events then unfolded, ultimately leading to the peaceful capitulation of Mecca. Muḥammad married [[w:Umm_Habiba|Umm Ḥabība]], the daughter of [[w:Abu_Sufyan_ibn_Harb|Abū Sufyān]], who had embraced Islam years earlier and whose Muslim husband had passed away. Shortly thereafter, a clan of the [[w:Banu_Khuza%27ah|Khuzāʿah]] tribe, which had allied with Muḥammad after Ḥudaybiya, was attacked by a clan of the [[w:Kinana|Kināna tribe]], who were allied with the Meccans. Under duress, the Khuzāʿah clan appealed to Muḥammad, who regarded the Treaty of Ḥudaybiya as breached due to this incident. <!-- linked Mu'tah, Battle of Mu'tah, and Umm Habiba --> To avoid a military confrontation, Abū Sufyān traveled to Yathrib under the pretext of visiting his daughter and conducted negotiations with Muḥammad. Although the exact course of the subsequent events remains unclear, it is certain that gifts were exchanged between Muḥammad and Abū Sufyān following the latter's return to Mecca. In the matter itself, however, Muḥammad was unwilling to make any concessions and gave the command to prepare for a campaign to [[w:Conquest_of_Mecca|capture Mecca]]. With an army of approximately 10,000 men, comprising not only his followers from Mecca and Yathrib but also fighters from neighboring tribes such as the [[w:Banu_Sulaym|Banū Sulaym]] and [[w:Banu_Muzaina|Muzayna]], he marched toward Mecca. Abū Sufyān met him on the way and engaged in negotiations. In return for his conversion to Islam, he was granted a guarantee of safety for all Meccan residents who refrained from armed resistance. These extensive assurances resulted in Muḥammad's army facing only minimal resistance as they advanced into the city from multiple directions in January 630. In Arabic sources, the conquest of Mecca is referred to as ''fatḥ'', “opening”, serving as an archetype for subsequent Muslim conquests ([[w:Futuh|futūḥ]]) of cities and lands under Muḥammad's successors. Separate texts and works were later dedicated to documenting these events. <!-- linked the Conquest of Mecca -Kontaktgespräche: translated as negotiations. --> == 3.2. The Internal Development of the Community in Yathrib == === 3.2.1. The So-Called “Constitution of Medina” === Upon Muḥammad's arrival in Yathrib, his followers primarily consisted of two main groups: the members of the Quraysh who had undertaken the Hijrah with him from Mecca, and the clans of [[w:Banu_Aws|Aws]] and [[w:Banu_Khazraj|Khazraj]], who had received these emigrants in Yathrib. Establishing a bond of loyalty between these two groups was an urgent necessity to establish a cohesive community. This very issue is addressed in a verse of the Qur'an, which states: “Those who believed and left their homes and strove with their wealth and their lives for the cause of Allah, and those who took them in and helped them: these are protecting friends one of another.” (Q [https://corpuscoranicum.de/en/verse-navigator/sura/8/verse/72/print 8:72]). Evidently, practical measures were undertaken to achieve this objective, as there are reports suggesting the establishment of a “[[w:Brotherhood_among_the_Sahabah|brotherhood]]” (''muʾākhāh'') between members of the two groups. The typical form of this bond of brotherhood involved pairing an emigrant (''muhājir'') with one of the “helpers” (''anṣār''), with both declaring themselves as brothers. If one of them fell in battle, the other would inherit from him. The primary purpose of this brotherhood was to achieve greater solidarity in warfare. However, this measure did not entirely overcome the division among Muḥammad's followers. Over time, the distinction between the Meccan “Emigrants” ([[w:Muhajirun|muhājirūn]]) and the “Helpers” ([[w:Ansar_(Islam)|anṣār]]) from Yathrib seems to have become further entrenched, as suggested by two later Qur'anic verses (Q [https://corpuscoranicum.de/en/verse-navigator/sura/9/verse/100/print 9:100], [https://corpuscoranicum.de/en/verse-navigator/sura/9/verse/117/print 117]), in which the two groups are juxtaposed. <!-- Verbrüderung (muʾākhāh): I used the translation “brotherhood” here, but also found it translated it as a “system of brotherhood” or “bond of brotherhood”. linked Brotherhood_among_the_Sahabah --> A historical document offers a much more detailed account of the political circumstances in Yathrib following Muḥammad’s arrival than the Qur’an: the so-called [[w:Constitution_of_Medina|Constitution of Medina]]. This document, transmitted by Ibn Hishām, contains the earliest recorded instance of the Arabic term ''al-Madīna'' (“the city” or “the place of jurisdiction”) as a designation for Yathrib. The term later became widely used as the city’s name and also appears in the final chapters of the Qur’an (e.g., Q [https://corpuscoranicum.de/en/verse-navigator/sura/9/verse/102/print 9:102]; [https://corpuscoranicum.de/en/verse-navigator/sura/63/verse/8/print 63:8]). Regarding the political order, the document begins by establishing itself as “a compact from Muḥammad the Prophet between the Believers and Muslims of Quraysh and Yathrib, and those who follow them, join them, and fight alongside them” (§ 1). They are to form “one Ummah, distinct from all others” (§ 2). The term [[w:Ummah|Ummah]] is referenced in many other sections of the Qur'an, to denote various communities led by prophets. However, the Ummah of Medina was more oriented toward tribal concepts. The nine primary signatories of the pact were the "Emigrants of the Quraysh," who were evidently considered a single clan, along with eight clans from Yathrib. As stated, each group was to preserve its tribal structure and bear responsibility for paying [[w:Blood_money_in_Islam|blood money]] and ransoms on behalf of its members. However, this obligation of solidarity was restricted to the believers (''muʾminūn'') within each group. The second part of the document focuses on relations with the Jewish tribes of Yathrib and their Bedouin allies. The document concludes by declaring that the Valley of Yathrib is sacred (''ḥarām'') for all treaty partners (§ 39). The document demonstrates that the concept of Ummah at that time was understood to encompass relationships with members of other religions. Essentially, it was a treaty of alliance consistent with traditional Arab legal concepts. However, by restricting the obligations of solidarity in the first section to “believers,” a religious dimension was introduced. Muḥammad himself was assigned a judicial role within the framework of the Constitution. Thus, it was stipulated that those engaged in a dispute over a matter should bring it before God and Muḥammad (§§ 23, 42). === 3.2.2 The Conflict with the Jews === By the end of the Meccan period, Muḥammad maintained a highly positive view of Judaism. Qur'anic texts from this period call for avoiding disputes with the “[[w:People_of_the_Book|People of the Book]]” (Ahl al-Kitāb), referring to the Jews of Yathrib, and instead emphasize highlighting the shared elements between the two revealed religions (Q [https://corpuscoranicum.de/en/verse-navigator/sura/29/verse/46/print 29:46]). During the early Medinan period, the positive relationship with the Jewish community led Muḥammad and his followers to further align themselves with Jewish provisions. For instance, a third daily prayer was introduced following the Jewish model (cf. Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/238/print 2:238f]), and the consumption of pork as well as sexual relations during [[w:Menstruation|menstruation]] were prohibited (cf. Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/173/print 2:173], [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/222/print 222]). <!-- Pickthall translates Ahl al-Kitāb as “People of the Scripture” in this verse. I mirrored your translation as People of the Book is more commonly used. --> <!-- This section frames Prophet’s alignment with Jewish provisions as a result of “good relations.” This interpretation (including similar ones in the coming paragraphs) then portrays the Qur'an as a political tool created by him rather than as the word of God. Can both views be presented throughout? --> During the early Medinan period, Muḥammad was also referred to as a ''nabī '', a title used for prophets in the Bible and Jewish tradition. This title may have been introduced to persuade the Jews of Yathrib to acknowledge Muḥammad's religious leadership. Particularly noteworthy is the adjective ''ummī '', which accompanies the term ''nabī'' in its first Qur'anic occurrence (Q [https://corpuscoranicum.de/en/verse-navigator/sura/7/verse/157/print 7:157f]). It is likely derived from the Hebrew expression ''ummot ha-ʿolam'', used in the [[w:Talmud|Talmud]] to refer to non-Jewish nations where [[w:Righteous_Among_the_Nations|righteous individuals]] may emerge. The reference to this pagan-friendly tradition was significant, the prevailing belief among the Jews of Yathrib held that no prophecy could arise among the Arabs. However, this rapprochement with Judaism did not imply that Muḥammad sought to subsume his religion into it. The Constitution of Medina explicitly states, “The Jews have their religion and the Believers have theirs.” He identified a shared religious foundation in monotheism, as reflected in the following Qur'anic verse: {{quote|Say: O People of the Scripture! Come to an agreement between us and you: that we shall worship none but Allah, and that we shall ascribe no partner unto Him, and that none of us shall take others for lords beside Allah. [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/64/print – Q 3:64]}} Muḥammad seemingly envisioned that the Jews would eventually recognize his message as a reaffirmation of their own religion. This ultimately led him to expect that they would acknowledge his religious authority. Several Qur'anic verses call upon the [[w:Israelites|Israelites]] to believe in what was revealed to him as a confirmation of their own scripture (Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/41/print 2:41]) and to regard him as a confirmer of the [[w:Torah|Torah]] (Q [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/50/print 3:50]). <!-- Die Annäherung an das Judentum bedeutete allerdings nicht, dass Muhammad seine Religion darin aufgehen lassen wollte: This reflects a specific interpretation that emphasizes Prophet Muhammad's actions as those of a man without divine guidance negotiating religious alignment. --> It was likely in the context of the conflict with the Jews of Medina that Muḥammad first referred to the [[w:Archangel|Archangel]] [[w:Gabriel|Gabriel]] as the mediator of the Qur'anic revelation. Gabriel had already played a prominent role in Jewish traditions concerning Abraham and Moses in pre-Islamic times. According to Surah [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/97/print 2:97], Gabriel is the one who has revealed the Qur’an to Muḥammad's heart “by God’s [permission], confirming that which was (revealed) before it, and a guidance and glad tidings to believers.” Qur'anic commentators explain that these verses were revealed when the Jews of Medina inquired of Muḥammad about the angel who brought him the revelations. When he responded that it was Gabriel—regarded as the friend of all prophets—they reportedly replied that they could not acknowledge him, as their friend was the angel [[w:Michael_(archangel)|Michael]], while Gabriel was their enemy. In Surah [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/97/print 2:97-98], this “enmity” is condemned, emphasizing that both angels, Gabriel and Michael, are servants of God. <!-- "zum ersten Mal auf den Erzengel Gabriel”: you mean mentioned him for the first time by name? As the Meccan-period surah 26:193 is the first mention, no? -Pickthall translates “by God’s permission” as “by God’s leave,” which is apparently an archaic usage. So I replaced it with permission in square brackets to indicate the change. - “dass beide Engel, Gabriel und Michael, Diener Gottes seien”: The above quoted verses do not state that Michael and Gabriel are servants of God. I could not find a verse that directly states that, but Surah 43:19 refers to the angels in general as servants of God. --> Such Qur'anic passages clearly demonstrate that the Jews of Medina were unwilling to recognize Muḥammad as a prophet within their tradition. Other verses reference recurring disputes with them (e.g., Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/139/print 2:139]). Historiographical sources report that, at that time in Medina, there was only one prominent Jew who converted to Islam: [[w:Abd_Allah_ibn_Salam|ʿAbdallāh ibn Salām]]. His conversion is surrounded by numerous legends in Islamic tradition. Disappointed by the lack of support from Medina's Jewish community, Muḥammad severed ties with Judaism following the Battle of Badr. He urged the Jews to no longer call on him as an arbitrator, as they did not even abide by the rules of their own Torah (Q [https://corpuscoranicum.de/en/verse-navigator/sura/5/verse/43/print 5:43-45]). Verbal attacks against the Jews became increasingly frequent. In various passages, they are accused of obduracy (e.g. Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/87/print 2:87-90]) and their strict dietary prohibitions are explained as a consequence of their grave sins from the past (Q [https://corpuscoranicum.de/en/verse-navigator/sura/4/verse/160/print 4:160]). In other passages, they are accused of concealing parts of their scriptures (cf. Q [https://corpuscoranicum.de/en/verse-navigator/sura/11/verse/18/print 11:18]). Finally, there are four passages that state the Jews deliberately “distorted” or “corrupted” their scripture (Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/75/print 2:75-79], [https://corpuscoranicum.de/en/verse-navigator/sura/4/verse/46/print 4:46], [https://corpuscoranicum.de/en/verse-navigator/sura/5/verse/13/print 5:13], [https://corpuscoranicum.de/en/verse-navigator/sura/5/verse/41/print 5:41]). <!-- Aus Enttäuschung darüber, dass ihn die Juden von Medina nicht unterstützen, brach Muhammad nach der Badr-Schlacht mit dem Judentum: that he severed ties and especially due to disappointment seems speculative, but the statement that he severed ties with “Judaism” denotes a rejection of Judaism as a religion, which is not my understanding of Islam’s stance -can the depictions be a bit more balanced? I really worry about these being used by extremists. There were many positive interactions and relations. If the Jewish population didn’t trust the Prophet, then they would never ask him to arbitrate among them. Even the fact that the Prophet would arbitrate between Jews using the Torah at times… Or his positive interactions with his Jewish neighbors, etc… --> Verbal attacks were soon followed by physical attacks on Yathrib's Jewish tribes, which did not constitute a unified community but were part of three different tribes: the [[w:Banu_Qaynuqa|Banū Qaynuqāʿ]], [[w:Banu_Nadir|Banū al-Naḍīr]], and [[w:Banu_Qurayza|Banū Qurayẓa]]. The Banū Qaynuqāʿ, who maintained their own market in Yathrib, were the victims of the first attack. According to tradition, the cause is said to have been an incident that occurred on their market following the Battle of Badr in 624, where an Arab woman was harassed by a group of Jews. A Muslim who witnessed the incident killed one of the Jews and was subsequently killed himself. This event served as the catalyst for Muḥammad to lay [[w:Siege_of_Banu_Qaynuqa|siege to the Banū Qaynuqāʿ]] in their fortified quarter in April 624. They were forced to leave their possessions behind and migrate to Adhriʿāt (modern-day [[w:Daraa|Darʿā]] in [[w: Hauran|Ḥaurān]], Syria). The expulsion of the Banū Qaynuqāʿ also had an economic dimension, as Muḥammad was seeking to establish his own marketplace at the time, and the Qaynuqāʿ's market was an obstacle to this effort. <!-- linked Siege of Banu Qaynuqa’ -This paragraph depicts the Muslim community of the time as sole aggressors (verbalen Attacken, reelle Angriffe) and the Jewish community of the time as the sole victims (das Opfer). Even the Muslim women was merely “beleidigt” (insulted/harassed) even though, according to sources, either her hijab was taken off or she was stripped naked, either of which would be assault today. In general, other historical accounts suggest that tensions were mutual, rooted in breaches of alliances, political rivalries, etc… And the market incident was not an isolated event but there was a list of violations, threats, and attacks before and after this incident. -The economic dimension connotates that actions were motivated by economic goals, which is once again speculative. --> [[File:Khaybar - deserted houses.jpg|thumb|250px| The Oasis of Khaybar Today]] In August 625, Muḥammad took action against the Jewish tribe of Banū al-Naḍīr. According to tradition, the conflict was triggered by an attempted assassination of Muḥammad and accusations of collaboration with the hostile Quraysh. Muḥammad besieged them in their fortified houses and ordered the destruction of their palm plantations, thereby breaking an unwritten law of Arab warfare. According to Islamic tradition, verse Q [https://corpuscoranicum.de/en/verse-navigator/sura/59/verse/5/print 59:5] serves as a justification for this action. Like the Banū Qaynuqāʿ, the Banū al-Naḍīr were also expelled, with the majority seeking refuge in the Jewish oasis of [[w:Khaybar|Khaybar]], where they possessed land and fortresses. Following the Battle of the Trench in 627, a confrontation arose between the Muslims and the Banū Qurayẓa, the third and last remaining Jewish tribe in Medina, who also owned various other oases in the northern part of the peninsula. They were accused of breaching their agreement with Muḥammad and collaborating with the Quraysh during the Battle of the Trench. A siege of their fortified district lasting several weeks ensued. As allies of the Banū Qurayẓa, the Banū Aws interceded on their behalf, leading Muḥammad to leave the decision regarding their fate to [[w:Sa%27d_ibn_Mu%27adh|Saʿd ibn Muʿādh]], a member of the Banū Aws who had been gravely injured during the Battle of the Trench and was near death. He decreed that all men be executed, the women and children taken into captivity, and their property divided. This was carried out immediately despite protests from other allies. <!-- Saʿd ibn Muʿādh, einem Angehörigen der Banū Aus: the Wikipedia articles as well as other sources name Saʿd ibn Muʿādh as a chief or Anführer of Banu Aws, which seems important in this context as he was chosen as an arbitrator. -It also seems that his ruling aligns with both Jewish legal provisions and the broader norms of 7th-century Arabia, as several sources suggest. I believe it is important to provide such context, given -once again- such instances are misinterpreted and misused, especially by extremists. --> Even after this incident, the conflict with the Jews persisted. After their expulsion from Medina, the Banū al-Naḍīr began preparing for war against Muḥammad in Khaybar, in coordination with neighboring Arab tribes. In the spring of 628, Muḥammad launched a preemptive [[w:Battle_of_Khaybar|campaign]] against them. Quran [https://corpuscoranicum.de/en/verse-navigator/sura/5/verse/33/print 5:33] prescribes cruel punishments (killing, crucifixion, or the amputation of a hand and a foot on opposite sides) for those who wage war against God and His Messenger, and it is associated with this historical context. Khaybar capitulated in June 628, and its Jewish inhabitants were compelled to surrender their wealth and to surrender half of their annual harvest to the Muslim conquerors. Shortly thereafter, two other Jewish-inhabited oases, [[w:Fadak|Fadak]] and [[w:Tayma|Taymāʾ]], voluntarily submitted to Muḥammad's authority. <!-- linked Fadak & Tayma --> [[Category:Islamic Studies]] mni2yqt90lt2x9nc57hcxv4zs5zi49u 3 317198 2693424 2691675 2024-12-26T23:20:43Z Tule-hog 2984180 welcome! 2693424 wikitext text/x-wiki == Course:Complex Analysis/Curve == You created page [[Course:Complex Analysis/Curve]], starting with "Course:", which I find unexpected. There is [[Complex Analysis/Curves]] page already. What are you trying to do with the page? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:18, 12 December 2024 (UTC) ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Eshaa2024!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Tule-hog|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. 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See you around Wikiversity! --[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:20, 26 December 2024 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 844323k3c9asuz90hk54fqpg66bdp6u 2 317234 2693446 2693122 2024-12-26T23:35:46Z Atcovi 276019 +ch11 2693446 wikitext text/x-wiki * [[User:Atcovi/Health Psychology/Chapter 1 - What is Health?]] * [[User:Atcovi/Health Psychology/Chapter 5 - Diverse Understandings of Stress]] * [[User:Atcovi/Health Psychology/Chapter 6 - Coping and Social Support]] * [[User:Atcovi/Health Psychology/Chapter 7 -Why Don’t We Do What We Need to?|User:Atcovi/Health Psychology/Chapter 7 - Why Don’t We Do What We Need to?]] * [[User:Atcovi/Health Psychology/Chapter 8 - Health Behaviors]] * SEPERATION * [[User:Atcovi/Health Psychology/Chapter 9 - Illness Cognitions, Adherence, and Patient–Practitioner Interactions: Introduction]] * [[User:Atcovi/Health Psychology/Chapter 10: Diverse Approaches to Pain]] * [[User:Atcovi/Health Psychology/Chapter 11: Disability, Terminal Illness, and Death]] [[Category:Psychology]] [[Category:Atcovi's Work]] mmpg1bk0bitakz0n5maujhiutqxzplu 0 317254 2693339 2693285 2024-12-26T19:09:20Z Watchduck 137431 2693339 wikitext text/x-wiki <templatestyles src="Boolf prop/props.css" /> {{boolf header}} {| class="wikitable sortable boolf-props" style="text-align: center;" |- ! <abbr title="number of blocks">#</abbr> ! integer partition ! properties |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[16, 1, 240, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">240</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_noble|is noble]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[16, 1, 240, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">240</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_linear|is linear]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[24, 1, 232, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">24</span> + <span class="count">1</span>⋅<span class="size">232</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_dense|is dense]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[57, 1, 199, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">57</span> + <span class="count">1</span>⋅<span class="size">199</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_honest|is honest]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[62, 1, 194, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">62</span> + <span class="count">1</span>⋅<span class="size">194</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_blotless|is blotless]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[64, 1, 192, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">64</span> + <span class="count">1</span>⋅<span class="size">192</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[64, 1, 192, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">64</span> + <span class="count">1</span>⋅<span class="size">192</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[66, 1, 190, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">66</span> + <span class="count">1</span>⋅<span class="size">190</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[96, 1, 160, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">96</span> + <span class="count">1</span>⋅<span class="size">160</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_blightless|is blightless]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[97, 1, 159, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">97</span> + <span class="count">1</span>⋅<span class="size">159</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_male|is male]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_acute|is acute]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_odd|is odd]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_odious|is odious]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_ugly|is ugly]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_rough|is rough]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_sharp|is sharp]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_solid|is solid]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#zhegalkin_deviation_patron|zhegalkin deviation patron]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#zhegalkin_deviation_is_odious|zhegalkin deviation is odious]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_rude|is rude]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[16, 1, 96, 1, 144, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">96</span> + <span class="count">1</span>⋅<span class="size">144</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/faction size|faction size]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[16, 1, 112, 1, 128, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">112</span> + <span class="count">1</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/nonlinearity|nonlinearity]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[32, 1, 96, 1, 128, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">32</span> + <span class="count">1</span>⋅<span class="size">96</span> + <span class="count">1</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/village weight|village weight]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[40, 1, 57, 1, 159, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">40</span> + <span class="count">1</span>⋅<span class="size">57</span> + <span class="count">1</span>⋅<span class="size">159</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/honesty and gender|honesty and gender]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[80, 2, 96, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">80</span> + <span class="count">1</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/greater guild|greater guild]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 1, 6, 1, 30, 1, 218, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">6</span> + <span class="count">1</span>⋅<span class="size">30</span> + <span class="count">1</span>⋅<span class="size">218</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/valency|valency]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 1, 14, 1, 56, 1, 184, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">14</span> + <span class="count">1</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">184</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/senior village weight|senior village weight]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 1, 14, 1, 56, 1, 184, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">14</span> + <span class="count">1</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">184</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/family size|family size]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 2, 12, 1, 240, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">1</span>⋅<span class="size">240</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/adicity|adicity]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great patron dominion|great patron dominion]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great patron principality|great patron principality]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/consul weight|consul weight]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/prefect weight|prefect weight]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great patron|great patron]]</span><span class="prop other">patron tiling and slatting</span><span class="prop other">patron symmetry perm</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great quaestor|great quaestor]]</span><span class="prop other">quaestor tiling and slatting</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great prefect|great prefect]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great praetor|great praetor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quadrant|quadrant]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/lictor|lictor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/lictor sword|lictor sword]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/praetor shield|praetor shield]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/reverse lictor|reverse lictor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main nameless">[[Boolf prop/3-ary/nameless 5|nameless 5]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/weight quadrant|weight quadrant]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/guardian|guardian]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/patron index consul|patron index consul]]</span> |- |class="number-of-blocks"| 5 |class="intpart"| <span class="sortkey">[16, 1, 48, 3, 96, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">48</span> + <span class="count">1</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/symmetry perm|symmetry perm]]</span> |- |class="number-of-blocks"| 5 |class="intpart"| <span class="sortkey">[16, 2, 48, 2, 128, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">16</span> + <span class="count">2</span>⋅<span class="size">48</span> + <span class="count">1</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/tribe|tribe]]</span> |- |class="number-of-blocks"| 5 |class="intpart"| <span class="sortkey">[16, 2, 64, 2, 96, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">16</span> + <span class="count">2</span>⋅<span class="size">64</span> + <span class="count">1</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quaestor weight|quaestor weight]]</span> |- |class="number-of-blocks"| 5 |class="intpart"| <span class="sortkey">[16, 2, 64, 2, 96, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">16</span> + <span class="count">2</span>⋅<span class="size">64</span> + <span class="count">1</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/praetor weight|praetor weight]]</span> |- |class="number-of-blocks"| 5 |class="intpart"| <span class="sortkey">[16, 2, 64, 2, 96, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">16</span> + <span class="count">2</span>⋅<span class="size">64</span> + <span class="count">1</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/patron index weight|patron index weight]]</span> |- 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class="formula"><span class="count">4</span>⋅<span class="size">16</span> + <span class="count">4</span>⋅<span class="size">48</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span class="sortkey">[16, 4, 48, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">16</span> + <span class="count">4</span>⋅<span class="size">48</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/patron dominion|patron dominion]]</span><span class="prop other">patron principality</span><span class="prop other">patron king index and quadrant</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span class="sortkey">[32, 8]</span><span class="formula"><span class="count">8</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quaestor sword|quaestor sword]]</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span class="sortkey">[32, 8]</span><span class="formula"><span class="count">8</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/greater twin mentor|greater twin mentor]]</span><span class="prop other">leveled praetor sword</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span class="sortkey">[32, 8]</span><span class="formula"><span class="count">8</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/octant|octant]]</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span class="sortkey">[32, 8]</span><span class="formula"><span class="count">8</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/consul|consul]]</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span class="sortkey">[32, 8]</span><span class="formula"><span class="count">8</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great sub-prefect|great sub-prefect]]</span> |- |class="number-of-blocks"| 9 |class="intpart"| <span class="sortkey">[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">1</span> + <span class="count">2</span>⋅<span class="size">8</span> + <span class="count">2</span>⋅<span class="size">28</span> + <span class="count">2</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">70</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/weight|weight]]</span> |- |class="number-of-blocks"| 9 |class="intpart"| <span class="sortkey">[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">1</span> + <span class="count">2</span>⋅<span class="size">8</span> + <span class="count">2</span>⋅<span class="size">28</span> + <span class="count">2</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">70</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/zhegalkin weight|zhegalkin weight]]</span> |- |class="number-of-blocks"| 11 |class="intpart"| <span class="sortkey">[4, 2, 8, 2, 16, 2, 32, 3, 40, 1, 64, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">4</span> + <span class="count">2</span>⋅<span class="size">8</span> + <span class="count">2</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">32</span> + <span class="count">1</span>⋅<span class="size">40</span> + <span class="count">1</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/ultra clan|ultra clan]]</span> |- |class="number-of-blocks"| 11 |class="intpart"| <span class="sortkey">[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]</span><span class="formula"><span class="count">2</span>⋅<span class="size">4</span> + <span class="count">2</span>⋅<span 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class="count">8</span>⋅<span class="size">16</span> + <span class="count">4</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/patron symmetry negperm|patron symmetry negperm]]</span> |- |class="number-of-blocks"| 13 |class="intpart"| <span class="sortkey">[4, 2, 8, 1, 12, 2, 24, 7, 48, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">8</span> + <span class="count">2</span>⋅<span class="size">12</span> + <span class="count">7</span>⋅<span class="size">24</span> + <span class="count">1</span>⋅<span class="size">48</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/platoon|platoon]]</span> |- |class="number-of-blocks"| 13 |class="intpart"| <span class="sortkey">[4, 2, 12, 6, 24, 2, 36, 2, 56, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">4</span> + <span class="count">6</span>⋅<span 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nameless">[[Boolf prop/3-ary/nameless 1|nameless 1]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/prefect|prefect]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/praetor|praetor]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quaestor|quaestor]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/patron|patron]]</span><span class="prop other">patron index</span><span class="prop other">praetor sword</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main nameless">[[Boolf prop/3-ary/nameless 3|nameless 3]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main nameless">[[Boolf prop/3-ary/nameless 4|nameless 4]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great twin mentor|great twin mentor]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/zhegalkin linear|zhegalkin linear]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/zhegalkin deviation|zhegalkin deviation]]</span> |- |class="number-of-blocks"| 18 |class="intpart"| <span class="sortkey">[4, 4, 8, 6, 16, 4, 32, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">4</span> + <span class="count">6</span>⋅<span class="size">8</span> + <span class="count">4</span>⋅<span class="size">16</span> + <span class="count">4</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop 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class="size">8</span> + <span class="count">8</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/super family|super family]]</span> |- |class="number-of-blocks"| 32 |class="intpart"| <span class="sortkey">[8, 32]</span><span class="formula"><span class="count">32</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/sub-prefect|sub-prefect]]</span> |- |class="number-of-blocks"| 32 |class="intpart"| <span class="sortkey">[8, 32]</span><span class="formula"><span class="count">32</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/super chunk|super chunk]]</span> |- |class="number-of-blocks"| 37 |class="intpart"| <span class="sortkey">[2, 12, 4, 12, 8, 3, 10, 4, 20, 6]</span><span class="formula"><span class="count">12</span>⋅<span class="size">2</span> + <span class="count">12</span>⋅<span class="size">4</span> + <span class="count">3</span>⋅<span class="size">8</span> + <span class="count">4</span>⋅<span class="size">10</span> + <span class="count">6</span>⋅<span class="size">20</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/symmetry negperm|symmetry negperm]]</span> |- |class="number-of-blocks"| 38 |class="intpart"| <span class="sortkey">[1, 8, 3, 14, 9, 8, 12, 4, 20, 2, 23, 2]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">14</span>⋅<span class="size">3</span> + <span class="count">8</span>⋅<span class="size">9</span> + <span class="count">4</span>⋅<span class="size">12</span> + <span class="count">2</span>⋅<span class="size">20</span> + <span class="count">2</span>⋅<span class="size">23</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/burden|burden]]</span> |- |class="number-of-blocks"| 44 |class="intpart"| <span class="sortkey">[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">8</span>⋅<span class="size">3</span> + <span class="count">8</span>⋅<span class="size">4</span> + <span class="count">8</span>⋅<span class="size">6</span> + <span class="count">12</span>⋅<span class="size">12</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/principality|principality]]</span> |- |class="number-of-blocks"| 44 |class="intpart"| <span class="sortkey">[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">8</span>⋅<span class="size">3</span> + <span class="count">8</span>⋅<span class="size">4</span> + <span class="count">8</span>⋅<span class="size">6</span> + <span class="count">12</span>⋅<span class="size">12</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/dominion|dominion]]</span> |- |class="number-of-blocks"| 46 |class="intpart"| <span class="sortkey">[1, 2, 2, 7, 4, 14, 8, 23]</span><span class="formula"><span class="count">2</span>⋅<span class="size">1</span> + <span class="count">7</span>⋅<span class="size">2</span> + <span class="count">14</span>⋅<span class="size">4</span> + <span class="count">23</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/family|family]]</span> |- |class="number-of-blocks"| 46 |class="intpart"| <span class="sortkey">[1, 2, 2, 7, 4, 14, 8, 23]</span><span class="formula"><span class="count">2</span>⋅<span class="size">1</span> + <span class="count">7</span>⋅<span class="size">2</span> + <span class="count">14</span>⋅<span class="size">4</span> + <span class="count">23</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/reverse family|reverse family]]</span><span class="prop other">senior village</span> |- |class="number-of-blocks"| 64 |class="intpart"| <span class="sortkey">[4, 64]</span><span class="formula"><span class="count">64</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/twin mentor|twin mentor]]</span> |- |class="number-of-blocks"| 64 |class="intpart"| <span class="sortkey">[4, 64]</span><span class="formula"><span class="count">64</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chunk|chunk]]</span> |- |class="number-of-blocks"| 66 |class="intpart"| <span class="sortkey">[1, 28, 2, 18, 6, 8, 12, 12]</span><span class="formula"><span class="count">28</span>⋅<span class="size">1</span> + <span class="count">18</span>⋅<span class="size">2</span> + <span class="count">8</span>⋅<span class="size">6</span> + <span class="count">12</span>⋅<span class="size">12</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/noble symmetry negperm|noble symmetry negperm]]</span> |- |class="number-of-blocks"| 80 |class="intpart"| <span class="sortkey">[1, 16, 3, 48, 6, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">1</span> + <span class="count">48</span>⋅<span class="size">3</span> + <span class="count">16</span>⋅<span class="size">6</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/faction|faction]]</span> |- |class="number-of-blocks"| 184 |class="intpart"| <span class="sortkey">[1, 124, 2, 48, 3, 12]</span><span class="formula"><span class="count">124</span>⋅<span class="size">1</span> + <span class="count">48</span>⋅<span class="size">2</span> + <span class="count">12</span>⋅<span class="size">3</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/splinter|splinter]]</span> |} [[Category:Boolf prop/3-ary| ]] 7h0j7nn65prpeljmkrtqse1zj3v0ebd 2693342 2693339 2024-12-26T19:12:59Z Watchduck 137431 2693342 wikitext text/x-wiki <templatestyles src="Boolf prop/props.css" /> {{boolf header}} {| class="wikitable sortable boolf-props" style="text-align: center;" |- ! <abbr title="number of blocks">#</abbr> ! integer partition ! properties |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[16, 1, 240, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">240</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_noble|is noble]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[16, 1, 240, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">240</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_linear|is linear]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[24, 1, 232, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">24</span> + <span class="count">1</span>⋅<span class="size">232</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_dense|is dense]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[57, 1, 199, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">57</span> + <span class="count">1</span>⋅<span class="size">199</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_honest|is honest]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[62, 1, 194, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">62</span> + <span class="count">1</span>⋅<span class="size">194</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_blotless|is blotless]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[64, 1, 192, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">64</span> + <span class="count">1</span>⋅<span class="size">192</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[64, 1, 192, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">64</span> + <span class="count">1</span>⋅<span class="size">192</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[66, 1, 190, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">66</span> + <span class="count">1</span>⋅<span class="size">190</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[96, 1, 160, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">96</span> + <span class="count">1</span>⋅<span class="size">160</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_blightless|is blightless]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[97, 1, 159, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">97</span> + <span class="count">1</span>⋅<span class="size">159</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_male|is male]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_acute|is acute]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_odd|is odd]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_odious|is odious]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_ugly|is ugly]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_rough|is rough]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_sharp|is sharp]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_solid|is solid]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#zhegalkin_deviation_patron|zhegalkin deviation patron]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#zhegalkin_deviation_is_odious|zhegalkin deviation is odious]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_rude|is rude]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[16, 1, 96, 1, 144, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">96</span> + <span class="count">1</span>⋅<span class="size">144</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/faction size|faction size]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[16, 1, 112, 1, 128, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">112</span> + <span class="count">1</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/nonlinearity|nonlinearity]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[32, 1, 96, 1, 128, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">32</span> + <span class="count">1</span>⋅<span class="size">96</span> + <span class="count">1</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/village weight|village weight]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[40, 1, 57, 1, 159, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">40</span> + <span class="count">1</span>⋅<span class="size">57</span> + <span class="count">1</span>⋅<span class="size">159</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/honesty and gender|honesty and gender]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[80, 2, 96, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">80</span> + <span class="count">1</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/greater guild|greater guild]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 1, 6, 1, 30, 1, 218, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">6</span> + <span class="count">1</span>⋅<span class="size">30</span> + <span class="count">1</span>⋅<span class="size">218</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/valency|valency]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 1, 14, 1, 56, 1, 184, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">14</span> + <span class="count">1</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">184</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/senior village weight|senior village weight]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 1, 14, 1, 56, 1, 184, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">14</span> + <span class="count">1</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">184</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/family size|family size]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 2, 12, 1, 240, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">1</span>⋅<span class="size">240</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/adicity|adicity]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great patron dominion|great patron dominion]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great patron principality|great patron principality]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/consul weight|consul weight]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/prefect weight|prefect weight]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great patron|great patron]]</span><span class="prop other">patron tiling and slatting</span><span class="prop other">patron symmetry perm</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great quaestor|great quaestor]]</span><span class="prop other">quaestor tiling and slatting</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great prefect|great prefect]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great praetor|great praetor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quadrant|quadrant]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/lictor|lictor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/lictor sword|lictor sword]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/praetor shield|praetor shield]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/reverse lictor|reverse lictor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main nameless">[[Boolf prop/3-ary/nameless 5|nameless 5]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/weight quadrant|weight quadrant]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/guardian|guardian]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/patron index consul|patron index consul]]</span> |- |class="number-of-blocks"| 5 |class="intpart"| <span class="sortkey">[16, 1, 48, 3, 96, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">48</span> + <span class="count">1</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/symmetry 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class="count">4</span>⋅<span class="size">16</span> + <span class="count">4</span>⋅<span class="size">48</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/patron dominion|patron dominion]]</span><span class="prop other">patron principality</span><span class="prop other">patron king index and quadrant</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span class="sortkey">[32, 8]</span><span class="formula"><span class="count">8</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quaestor sword|quaestor sword]]</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span class="sortkey">[32, 8]</span><span class="formula"><span class="count">8</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/super great twin mentor|super great twin mentor]]</span><span class="prop other">leveled praetor sword</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span 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|class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[2, 8, 8, 7, 184, 1]</span><span class="formula"><span class="count">8</span>⋅<span class="size">2</span> + <span class="count">7</span>⋅<span class="size">8</span> + <span class="count">1</span>⋅<span class="size">184</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/symmetry neg|symmetry neg]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main nameless">[[Boolf prop/3-ary/nameless 1|nameless 1]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/prefect|prefect]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span 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|class="props"| <span class="prop main nameless">[[Boolf prop/3-ary/nameless 3|nameless 3]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main nameless">[[Boolf prop/3-ary/nameless 4|nameless 4]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great twin mentor|great twin mentor]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/zhegalkin linear|zhegalkin linear]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span 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|class="props"| <span class="prop main">[[Boolf prop/3-ary/squad|squad]]</span> |- |class="number-of-blocks"| 20 |class="intpart"| <span class="sortkey">[8, 8, 16, 12]</span><span class="formula"><span class="count">8</span>⋅<span class="size">8</span> + <span class="count">12</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/noble symmetry perm|noble symmetry perm]]</span> |- |class="number-of-blocks"| 20 |class="intpart"| <span class="sortkey">[10, 16, 24, 4]</span><span class="formula"><span class="count">16</span>⋅<span class="size">10</span> + <span class="count">4</span>⋅<span class="size">24</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/guild|guild]]</span> |- |class="number-of-blocks"| 22 |class="intpart"| <span class="sortkey">[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]</span><span class="formula"><span class="count">2</span>⋅<span class="size">1</span> + <span class="count">1</span>⋅<span class="size">2</span> + <span class="count">2</span>⋅<span class="size">4</span> + <span class="count">2</span>⋅<span class="size">6</span> + <span class="count">5</span>⋅<span class="size">8</span> + <span class="count">4</span>⋅<span class="size">12</span> + <span class="count">6</span>⋅<span class="size">24</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/clan|clan]]</span> |- |class="number-of-blocks"| 30 |class="intpart"| <span class="sortkey">[2, 8, 8, 14, 16, 8]</span><span class="formula"><span class="count">8</span>⋅<span class="size">2</span> + <span class="count">14</span>⋅<span class="size">8</span> + <span class="count">8</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/super family|super family]]</span> |- |class="number-of-blocks"| 32 |class="intpart"| <span class="sortkey">[8, 32]</span><span class="formula"><span class="count">32</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/sub-prefect|sub-prefect]]</span> |- |class="number-of-blocks"| 32 |class="intpart"| <span class="sortkey">[8, 32]</span><span class="formula"><span class="count">32</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/super chunk|super chunk]]</span> |- |class="number-of-blocks"| 37 |class="intpart"| <span class="sortkey">[2, 12, 4, 12, 8, 3, 10, 4, 20, 6]</span><span class="formula"><span class="count">12</span>⋅<span class="size">2</span> + <span class="count">12</span>⋅<span class="size">4</span> + <span class="count">3</span>⋅<span class="size">8</span> + <span class="count">4</span>⋅<span class="size">10</span> + <span class="count">6</span>⋅<span class="size">20</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/symmetry negperm|symmetry negperm]]</span> |- |class="number-of-blocks"| 38 |class="intpart"| <span class="sortkey">[1, 8, 3, 14, 9, 8, 12, 4, 20, 2, 23, 2]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">14</span>⋅<span class="size">3</span> + <span class="count">8</span>⋅<span class="size">9</span> + <span class="count">4</span>⋅<span class="size">12</span> + <span class="count">2</span>⋅<span class="size">20</span> + <span class="count">2</span>⋅<span class="size">23</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/burden|burden]]</span> |- |class="number-of-blocks"| 44 |class="intpart"| <span class="sortkey">[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">8</span>⋅<span class="size">3</span> + <span class="count">8</span>⋅<span class="size">4</span> + <span class="count">8</span>⋅<span class="size">6</span> + <span class="count">12</span>⋅<span class="size">12</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/principality|principality]]</span> |- |class="number-of-blocks"| 44 |class="intpart"| <span class="sortkey">[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">8</span>⋅<span class="size">3</span> + <span class="count">8</span>⋅<span class="size">4</span> + <span class="count">8</span>⋅<span class="size">6</span> + <span class="count">12</span>⋅<span class="size">12</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/dominion|dominion]]</span> |- |class="number-of-blocks"| 46 |class="intpart"| <span class="sortkey">[1, 2, 2, 7, 4, 14, 8, 23]</span><span class="formula"><span class="count">2</span>⋅<span class="size">1</span> + <span class="count">7</span>⋅<span class="size">2</span> + <span class="count">14</span>⋅<span class="size">4</span> + <span class="count">23</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/family|family]]</span> |- |class="number-of-blocks"| 46 |class="intpart"| <span class="sortkey">[1, 2, 2, 7, 4, 14, 8, 23]</span><span class="formula"><span class="count">2</span>⋅<span class="size">1</span> + <span class="count">7</span>⋅<span class="size">2</span> + <span class="count">14</span>⋅<span class="size">4</span> + <span class="count">23</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/reverse family|reverse family]]</span><span class="prop other">senior village</span> |- |class="number-of-blocks"| 64 |class="intpart"| <span class="sortkey">[4, 64]</span><span class="formula"><span class="count">64</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/twin mentor|twin mentor]]</span> |- |class="number-of-blocks"| 64 |class="intpart"| <span class="sortkey">[4, 64]</span><span class="formula"><span class="count">64</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chunk|chunk]]</span> |- |class="number-of-blocks"| 66 |class="intpart"| <span class="sortkey">[1, 28, 2, 18, 6, 8, 12, 12]</span><span class="formula"><span class="count">28</span>⋅<span class="size">1</span> + <span class="count">18</span>⋅<span class="size">2</span> + <span class="count">8</span>⋅<span class="size">6</span> + <span class="count">12</span>⋅<span class="size">12</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/noble symmetry negperm|noble symmetry negperm]]</span> |- |class="number-of-blocks"| 80 |class="intpart"| <span class="sortkey">[1, 16, 3, 48, 6, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">1</span> + <span class="count">48</span>⋅<span class="size">3</span> + <span class="count">16</span>⋅<span class="size">6</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/faction|faction]]</span> |- |class="number-of-blocks"| 184 |class="intpart"| <span class="sortkey">[1, 124, 2, 48, 3, 12]</span><span class="formula"><span class="count">124</span>⋅<span class="size">1</span> + <span class="count">48</span>⋅<span class="size">2</span> + <span class="count">12</span>⋅<span class="size">3</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/splinter|splinter]]</span> |} [[Category:Boolf prop/3-ary| ]] s8b1o21zf2gyq1c5l8wl033vqmvemeq 2693345 2693342 2024-12-26T19:16:44Z Watchduck 137431 2693345 wikitext text/x-wiki <templatestyles src="Boolf prop/props.css" /> {{boolf header}} {| class="wikitable sortable boolf-props" style="text-align: center;" |- ! <abbr title="number of blocks">#</abbr> ! integer partition ! properties |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[16, 1, 240, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">240</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_noble|is noble]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[16, 1, 240, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">240</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_linear|is linear]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[24, 1, 232, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">24</span> + <span class="count">1</span>⋅<span class="size">232</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_dense|is dense]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[57, 1, 199, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">57</span> + <span class="count">1</span>⋅<span class="size">199</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_honest|is honest]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[62, 1, 194, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">62</span> + <span class="count">1</span>⋅<span class="size">194</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_blotless|is blotless]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[64, 1, 192, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">64</span> + <span class="count">1</span>⋅<span class="size">192</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[64, 1, 192, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">64</span> + <span class="count">1</span>⋅<span class="size">192</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[66, 1, 190, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">66</span> + <span class="count">1</span>⋅<span class="size">190</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[96, 1, 160, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">96</span> + <span class="count">1</span>⋅<span class="size">160</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_blightless|is blightless]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[97, 1, 159, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">97</span> + <span class="count">1</span>⋅<span class="size">159</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_male|is male]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_acute|is acute]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_odd|is odd]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_odious|is odious]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_ugly|is ugly]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_rough|is rough]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_sharp|is sharp]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_solid|is solid]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#zhegalkin_deviation_patron|zhegalkin deviation patron]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#zhegalkin_deviation_is_odious|zhegalkin deviation is odious]]</span> |- |class="number-of-blocks"| 2 |class="intpart"| <span class="sortkey">[128, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/binary#is_rude|is rude]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[16, 1, 96, 1, 144, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">96</span> + <span class="count">1</span>⋅<span class="size">144</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/faction size|faction size]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[16, 1, 112, 1, 128, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">1</span>⋅<span class="size">112</span> + <span class="count">1</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/nonlinearity|nonlinearity]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[32, 1, 96, 1, 128, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">32</span> + <span class="count">1</span>⋅<span class="size">96</span> + <span class="count">1</span>⋅<span class="size">128</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/village weight|village weight]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[40, 1, 57, 1, 159, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">40</span> + <span class="count">1</span>⋅<span class="size">57</span> + <span class="count">1</span>⋅<span class="size">159</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/honesty and gender|honesty and gender]]</span> |- |class="number-of-blocks"| 3 |class="intpart"| <span class="sortkey">[80, 2, 96, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">80</span> + <span class="count">1</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/super great guild|super great guild]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 1, 6, 1, 30, 1, 218, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">6</span> + <span class="count">1</span>⋅<span class="size">30</span> + <span class="count">1</span>⋅<span class="size">218</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/valency|valency]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 1, 14, 1, 56, 1, 184, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">14</span> + <span class="count">1</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">184</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/senior village weight|senior village weight]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 1, 14, 1, 56, 1, 184, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">14</span> + <span class="count">1</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">184</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/family size|family size]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[2, 2, 12, 1, 240, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">1</span>⋅<span class="size">240</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/adicity|adicity]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great patron dominion|great patron dominion]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great patron principality|great patron principality]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/consul weight|consul weight]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[32, 2, 96, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/prefect weight|prefect weight]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great patron|great patron]]</span><span class="prop other">patron tiling and slatting</span><span class="prop other">patron symmetry perm</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great quaestor|great quaestor]]</span><span class="prop other">quaestor tiling and slatting</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great prefect|great prefect]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great praetor|great praetor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/quadrant|quadrant]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/lictor|lictor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/lictor sword|lictor sword]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/praetor shield|praetor shield]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/reverse lictor|reverse lictor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main nameless">[[Boolf prop/3-ary/nameless 5|nameless 5]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/weight quadrant|weight quadrant]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[64, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/guardian|guardian]]</span> |- 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main">[[Boolf prop/3-ary/noble symmetry perm|noble symmetry perm]]</span> |- |class="number-of-blocks"| 20 |class="intpart"| <span class="sortkey">[10, 16, 24, 4]</span><span class="formula"><span class="count">16</span>⋅<span class="size">10</span> + <span class="count">4</span>⋅<span class="size">24</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/guild|guild]]</span> |- |class="number-of-blocks"| 22 |class="intpart"| <span class="sortkey">[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]</span><span class="formula"><span class="count">2</span>⋅<span class="size">1</span> + <span class="count">1</span>⋅<span class="size">2</span> + <span class="count">2</span>⋅<span class="size">4</span> + <span class="count">2</span>⋅<span class="size">6</span> + <span class="count">5</span>⋅<span class="size">8</span> + <span class="count">4</span>⋅<span class="size">12</span> + <span class="count">6</span>⋅<span class="size">24</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/clan|clan]]</span> |- |class="number-of-blocks"| 30 |class="intpart"| <span class="sortkey">[2, 8, 8, 14, 16, 8]</span><span class="formula"><span class="count">8</span>⋅<span class="size">2</span> + <span class="count">14</span>⋅<span class="size">8</span> + <span class="count">8</span>⋅<span class="size">16</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/super family|super family]]</span> |- |class="number-of-blocks"| 32 |class="intpart"| <span class="sortkey">[8, 32]</span><span class="formula"><span class="count">32</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/sub-prefect|sub-prefect]]</span> |- |class="number-of-blocks"| 32 |class="intpart"| <span class="sortkey">[8, 32]</span><span class="formula"><span class="count">32</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/super chunk|super chunk]]</span> |- |class="number-of-blocks"| 37 |class="intpart"| <span class="sortkey">[2, 12, 4, 12, 8, 3, 10, 4, 20, 6]</span><span class="formula"><span class="count">12</span>⋅<span class="size">2</span> + <span class="count">12</span>⋅<span class="size">4</span> + <span class="count">3</span>⋅<span class="size">8</span> + <span class="count">4</span>⋅<span class="size">10</span> + <span class="count">6</span>⋅<span class="size">20</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/symmetry negperm|symmetry negperm]]</span> |- |class="number-of-blocks"| 38 |class="intpart"| <span class="sortkey">[1, 8, 3, 14, 9, 8, 12, 4, 20, 2, 23, 2]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">14</span>⋅<span class="size">3</span> + <span class="count">8</span>⋅<span class="size">9</span> + <span class="count">4</span>⋅<span class="size">12</span> + <span class="count">2</span>⋅<span class="size">20</span> + <span class="count">2</span>⋅<span class="size">23</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/burden|burden]]</span> |- |class="number-of-blocks"| 44 |class="intpart"| <span class="sortkey">[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">8</span>⋅<span class="size">3</span> + <span class="count">8</span>⋅<span class="size">4</span> + <span class="count">8</span>⋅<span class="size">6</span> + <span class="count">12</span>⋅<span class="size">12</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/principality|principality]]</span> |- |class="number-of-blocks"| 44 |class="intpart"| <span class="sortkey">[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">8</span>⋅<span class="size">3</span> + <span class="count">8</span>⋅<span class="size">4</span> + <span class="count">8</span>⋅<span class="size">6</span> + <span class="count">12</span>⋅<span class="size">12</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/dominion|dominion]]</span> |- |class="number-of-blocks"| 46 |class="intpart"| <span class="sortkey">[1, 2, 2, 7, 4, 14, 8, 23]</span><span class="formula"><span class="count">2</span>⋅<span class="size">1</span> + <span class="count">7</span>⋅<span class="size">2</span> + <span class="count">14</span>⋅<span class="size">4</span> + <span class="count">23</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/family|family]]</span> |- |class="number-of-blocks"| 46 |class="intpart"| <span class="sortkey">[1, 2, 2, 7, 4, 14, 8, 23]</span><span class="formula"><span class="count">2</span>⋅<span class="size">1</span> + <span class="count">7</span>⋅<span class="size">2</span> + <span class="count">14</span>⋅<span class="size">4</span> + <span class="count">23</span>⋅<span class="size">8</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/reverse family|reverse family]]</span><span class="prop other">senior village</span> |- |class="number-of-blocks"| 64 |class="intpart"| <span class="sortkey">[4, 64]</span><span class="formula"><span class="count">64</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/twin mentor|twin mentor]]</span> |- |class="number-of-blocks"| 64 |class="intpart"| <span class="sortkey">[4, 64]</span><span class="formula"><span class="count">64</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chunk|chunk]]</span> |- |class="number-of-blocks"| 66 |class="intpart"| <span class="sortkey">[1, 28, 2, 18, 6, 8, 12, 12]</span><span class="formula"><span class="count">28</span>⋅<span class="size">1</span> + <span class="count">18</span>⋅<span class="size">2</span> + <span class="count">8</span>⋅<span class="size">6</span> + <span class="count">12</span>⋅<span class="size">12</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/noble symmetry negperm|noble symmetry negperm]]</span> |- |class="number-of-blocks"| 80 |class="intpart"| <span class="sortkey">[1, 16, 3, 48, 6, 16]</span><span class="formula"><span class="count">16</span>⋅<span class="size">1</span> + <span class="count">48</span>⋅<span class="size">3</span> + <span class="count">16</span>⋅<span class="size">6</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/faction|faction]]</span> |- |class="number-of-blocks"| 184 |class="intpart"| <span class="sortkey">[1, 124, 2, 48, 3, 12]</span><span class="formula"><span class="count">124</span>⋅<span class="size">1</span> + <span class="count">48</span>⋅<span class="size">2</span> + <span class="count">12</span>⋅<span class="size">3</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/splinter|splinter]]</span> |} [[Category:Boolf prop/3-ary| ]] bgyurewd80v2op13gcnrwqbxhkxi3g5 0 317274 2693346 2692155 2024-12-26T19:16:52Z Watchduck 137431 Watchduck moved page [[Boolf prop/3-ary/greater guild]] to [[Boolf prop/3-ary/super great guild]] 2692155 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">3</span></span> Integer partition: &nbsp; <span class="count">2</span>⋅<span class="size">80</span> + <span class="count">1</span>⋅<span class="size">96</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| greater guild !class="block"| block |- |class="size"| 80 |class="prop"| 0 |class="block"| <span class="block-list small">[0, 1, 6, 7, 18, 19, 20, 21, 24, 25, 30, 31, 36, 37, 40, 41, 54, 55, 60, 61, 66, 67, 72, 73, 86, 87, 90, 91, 96, 97, 102, 103, 106, 107, 108, 109, 120, 121, 126, 127, 128, 129, 134, 135, 146, 147, 148, 149, 152, 153, 158, 159, 164, 165, 168, 169, 182, 183, 188, 189, 194, 195, 200, 201, 214, 215, 218, 219, 224, 225, 230, 231, 234, 235, 236, 237, 248, 249, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_88201424837050089877109022730090942228518787393015200721429556080100081664195.svg|420px]] |- |class="size"| 80 |class="prop"| 2 |class="block"| <span class="block-list small">[2, 3, 4, 5, 8, 9, 14, 15, 16, 17, 22, 23, 32, 33, 42, 43, 50, 51, 62, 63, 64, 65, 76, 77, 84, 85, 94, 95, 104, 105, 110, 111, 112, 113, 118, 119, 122, 123, 124, 125, 130, 131, 132, 133, 136, 137, 142, 143, 144, 145, 150, 151, 160, 161, 170, 171, 178, 179, 190, 191, 192, 193, 204, 205, 212, 213, 222, 223, 232, 233, 238, 239, 240, 241, 246, 247, 250, 251, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_27484651953405175618223986524966191219557218455653235424269787484354240561980.svg|420px]] |- |class="size"| 96 |class="prop"| guildless |class="block"| <span class="block-list small">[10, 11, 12, 13, 26, 27, 28, 29, 34, 35, 38, 39, 44, 45, 46, 47, 48, 49, 52, 53, 56, 57, 58, 59, 68, 69, 70, 71, 74, 75, 78, 79, 80, 81, 82, 83, 88, 89, 92, 93, 98, 99, 100, 101, 114, 115, 116, 117, 138, 139, 140, 141, 154, 155, 156, 157, 162, 163, 166, 167, 172, 173, 174, 175, 176, 177, 180, 181, 184, 185, 186, 187, 196, 197, 198, 199, 202, 203, 206, 207, 208, 209, 210, 211, 216, 217, 220, 221, 226, 227, 228, 229, 242, 243, 244, 245]</span>[[File:Set_of_3-ary_Boolean_functions_106012446860929928237975753630774405193978816972127893758240443458807413760.svg|420px]] |} [[Category:Boolf prop/3-ary|greater guild]] e9hijlm5dsxwhv5nzuuipxvzeoa3hvb 2693348 2693346 2024-12-26T19:17:46Z Watchduck 137431 2693348 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">3</span></span> Integer partition: &nbsp; <span class="count">2</span>⋅<span class="size">80</span> + <span class="count">1</span>⋅<span class="size">96</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| super great guild !class="block"| block |- |class="size"| 80 |class="prop"| 0 |class="block"| [[File:Set_of_3-ary_Boolean_functions_88201424837050089877109022730090942228518787393015200721429556080100081664195.svg|420px]] |- |class="size"| 80 |class="prop"| 2 |class="block"| [[File:Set_of_3-ary_Boolean_functions_27484651953405175618223986524966191219557218455653235424269787484354240561980.svg|420px]] |- |class="size"| 96 |class="prop"| guildless |class="block"| [[File:Set_of_3-ary_Boolean_functions_106012446860929928237975753630774405193978816972127893758240443458807413760.svg|420px]] |} [[Category:Boolf prop/3-ary|super great guild]] 7v9zaei4tynky7farz6z0x8h693j1qk 0 317303 2693340 2693067 2024-12-26T19:12:26Z Watchduck 137431 Watchduck moved page [[Boolf prop/3-ary/greater twin mentor]] to [[Boolf prop/3-ary/super great twin mentor]] 2693067 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">8</span></span> Integer partition: &nbsp; <span class="count">8</span>⋅<span class="size">32</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| greater twin mentor !class="prop"| leveled praetor sword !class="block"| block |- |class="size"| 32 |class="prop"| 0 |class="prop"| 0 |class="block"| <span class="block-list small">[0, 1, 30, 31, 40, 41, 54, 55, 72, 73, 86, 87, 96, 97, 126, 127, 128, 129, 158, 159, 168, 169, 182, 183, 200, 201, 214, 215, 224, 225, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844067008945975608714099877531015663368387777468789899675262155586789179395.svg|420px]] |- |class="size"| 32 |class="prop"| 190 |class="prop"| 2 |class="block"| <span class="block-list small">[2, 3, 28, 29, 42, 43, 52, 53, 74, 75, 84, 85, 98, 99, 124, 125, 130, 131, 156, 157, 170, 171, 180, 181, 202, 203, 212, 213, 226, 227, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_21711017055535911985676224782151989008715061916228543421554093693005209796620.svg|420px]] |- |class="size"| 32 |class="prop"| 222 |class="prop"| 4 |class="block"| <span class="block-list small">[4, 5, 26, 27, 44, 45, 50, 51, 76, 77, 82, 83, 100, 101, 122, 123, 132, 133, 154, 155, 172, 173, 178, 179, 204, 205, 210, 211, 228, 229, 250, 251]</span>[[File:Set_of_3-ary_Boolean_functions_5427755477081650330409855446614937623670625366502519641929636039685352456240.svg|420px]] |- |class="size"| 32 |class="prop"| 96 |class="prop"| 6 |class="block"| <span class="block-list small">[6, 7, 24, 25, 46, 47, 48, 49, 78, 79, 80, 81, 102, 103, 120, 121, 134, 135, 152, 153, 174, 175, 176, 177, 206, 207, 208, 209, 230, 231, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1356943722061101918565660865961495891885095891407165056646859475657538142400.svg|420px]] |- |class="size"| 32 |class="prop"| 150 |class="prop"| 8 |class="block"| <span class="block-list small">[8, 9, 22, 23, 32, 33, 62, 63, 64, 65, 94, 95, 104, 105, 118, 119, 136, 137, 150, 151, 160, 161, 190, 191, 192, 193, 222, 223, 232, 233, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339255361896450186873466782499328282172849015222100644151734878385054155520.svg|420px]] |- |class="size"| 32 |class="prop"| 40 |class="prop"| 10 |class="block"| <span class="block-list small">[10, 11, 20, 21, 34, 35, 60, 61, 66, 67, 92, 93, 106, 107, 116, 117, 138, 139, 148, 149, 162, 163, 188, 189, 194, 195, 220, 221, 234, 235, 244, 245]</span>[[File:Set_of_3-ary_Boolean_functions_84891485120514011178369876170335680218058130139046442904049563119987461120.svg|420px]] |- |class="size"| 32 |class="prop"| 72 |class="prop"| 12 |class="block"| <span class="block-list small">[12, 13, 18, 19, 36, 37, 58, 59, 68, 69, 90, 91, 108, 109, 114, 115, 140, 141, 146, 147, 164, 165, 186, 187, 196, 197, 218, 219, 236, 237, 242, 243]</span>[[File:Set_of_3-ary_Boolean_functions_21533449865734360634605191224598358753898037868846738190475764874892554240.svg|420px]] |- |class="size"| 32 |class="prop"| 246 |class="prop"| 14 |class="block"| <span class="block-list small">[14, 15, 16, 17, 38, 39, 56, 57, 70, 71, 88, 89, 110, 111, 112, 113, 142, 143, 144, 145, 166, 167, 184, 185, 198, 199, 216, 217, 238, 239, 240, 241]</span>[[File:Set_of_3-ary_Boolean_functions_6625676808857021518702186534207344486008530803552194405472437598305894400.svg|420px]] |} [[Category:Boolf prop/3-ary|greater twin mentor]] 7w1ez9n031jm4xv5rbxuj2jkjaxgics 2693343 2693340 2024-12-26T19:14:07Z Watchduck 137431 2693343 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">8</span></span> Integer partition: &nbsp; <span class="count">8</span>⋅<span class="size">32</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| super great twin mentor !class="prop"| leveled praetor sword !class="block"| block |- |class="size"| 32 |class="prop"| 0 |class="prop"| 0 |class="block"| <span class="block-list small">[0, 1, 30, 31, 40, 41, 54, 55, 72, 73, 86, 87, 96, 97, 126, 127, 128, 129, 158, 159, 168, 169, 182, 183, 200, 201, 214, 215, 224, 225, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844067008945975608714099877531015663368387777468789899675262155586789179395.svg|420px]] |- |class="size"| 32 |class="prop"| 190 |class="prop"| 2 |class="block"| <span class="block-list small">[2, 3, 28, 29, 42, 43, 52, 53, 74, 75, 84, 85, 98, 99, 124, 125, 130, 131, 156, 157, 170, 171, 180, 181, 202, 203, 212, 213, 226, 227, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_21711017055535911985676224782151989008715061916228543421554093693005209796620.svg|420px]] |- |class="size"| 32 |class="prop"| 222 |class="prop"| 4 |class="block"| <span class="block-list small">[4, 5, 26, 27, 44, 45, 50, 51, 76, 77, 82, 83, 100, 101, 122, 123, 132, 133, 154, 155, 172, 173, 178, 179, 204, 205, 210, 211, 228, 229, 250, 251]</span>[[File:Set_of_3-ary_Boolean_functions_5427755477081650330409855446614937623670625366502519641929636039685352456240.svg|420px]] |- |class="size"| 32 |class="prop"| 96 |class="prop"| 6 |class="block"| <span class="block-list small">[6, 7, 24, 25, 46, 47, 48, 49, 78, 79, 80, 81, 102, 103, 120, 121, 134, 135, 152, 153, 174, 175, 176, 177, 206, 207, 208, 209, 230, 231, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1356943722061101918565660865961495891885095891407165056646859475657538142400.svg|420px]] |- |class="size"| 32 |class="prop"| 150 |class="prop"| 8 |class="block"| <span class="block-list small">[8, 9, 22, 23, 32, 33, 62, 63, 64, 65, 94, 95, 104, 105, 118, 119, 136, 137, 150, 151, 160, 161, 190, 191, 192, 193, 222, 223, 232, 233, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339255361896450186873466782499328282172849015222100644151734878385054155520.svg|420px]] |- |class="size"| 32 |class="prop"| 40 |class="prop"| 10 |class="block"| <span class="block-list small">[10, 11, 20, 21, 34, 35, 60, 61, 66, 67, 92, 93, 106, 107, 116, 117, 138, 139, 148, 149, 162, 163, 188, 189, 194, 195, 220, 221, 234, 235, 244, 245]</span>[[File:Set_of_3-ary_Boolean_functions_84891485120514011178369876170335680218058130139046442904049563119987461120.svg|420px]] |- |class="size"| 32 |class="prop"| 72 |class="prop"| 12 |class="block"| <span class="block-list small">[12, 13, 18, 19, 36, 37, 58, 59, 68, 69, 90, 91, 108, 109, 114, 115, 140, 141, 146, 147, 164, 165, 186, 187, 196, 197, 218, 219, 236, 237, 242, 243]</span>[[File:Set_of_3-ary_Boolean_functions_21533449865734360634605191224598358753898037868846738190475764874892554240.svg|420px]] |- |class="size"| 32 |class="prop"| 246 |class="prop"| 14 |class="block"| <span class="block-list small">[14, 15, 16, 17, 38, 39, 56, 57, 70, 71, 88, 89, 110, 111, 112, 113, 142, 143, 144, 145, 166, 167, 184, 185, 198, 199, 216, 217, 238, 239, 240, 241]</span>[[File:Set_of_3-ary_Boolean_functions_6625676808857021518702186534207344486008530803552194405472437598305894400.svg|420px]] |} [[Category:Boolf prop/3-ary|super great twin mentor]] 4po96brwnpb644slq1sk5o0aniyeg84 0 317376 2693330 2692739 2024-12-26T18:57:28Z 2A01:CDE0:102:1CEE:FB9B:AE92:ADBE:125C Undo all revisions. Resource is empty, but not [[Wikiversity:Deletions|deleted]]. 2693330 wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 2693352 2693330 2024-12-26T19:28:16Z Tule-hog 2984180 Undo revision [[Special:Diff/2693330|2693330]] by [[Special:Contributions/2A01:CDE0:102:1CEE:FB9B:AE92:ADBE:125C|2A01:CDE0:102:1CEE:FB9B:AE92:ADBE:125C]] ([[User talk:2A01:CDE0:102:1CEE:FB9B:AE92:ADBE:125C|talk]]) - unexplained blank 2693352 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">4</span></span> Integer partition: &nbsp; <span class="count">2</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">96</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| prefect weight !class="block"| block |- |class="size"| 32 |class="prop"| 0 |class="block"| <span class="block-list small">[0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 232, 233, 234, 235, 236, 237, 238, 239, 248, 249, 250, 251, 252, 253, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_115341536334051360628963456711242601354708523820828309034750019264059304509695.svg|420px]] |- |class="size"| 96 |class="prop"| 1 |class="block"| [[File:Set_of_3-ary_Boolean_functions_450552903159932278392934366827363174655893616276452539978402249012076936960.svg|420px]] |- |class="size"| 96 |class="prop"| 2 |class="block"| [[File:Set_of_3-ary_Boolean_functions_104902516214593930612256546768980043855800194642683360636263464960.svg|420px]] |- |class="size"| 32 |class="prop"| 3 |class="block"| <span class="block-list small">[104, 105, 106, 107, 108, 109, 110, 111, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 144, 145, 146, 147, 148, 149, 150, 151]</span>[[File:Set_of_3-ary_Boolean_functions_5686777136587184680002270086479134205484728320.svg|420px]] |} [[Category:Boolf prop/3-ary|prefect weight]] nxejgwpex88mmu3ydp4xyt57quuabsb 0 317411 2693506 2692933 2024-12-27T00:39:12Z Watchduck 137431 2693506 wikitext text/x-wiki {{Boolf header}} Boolean functions belong to the same family, when they can be transformed into each other by negating arguments.<br> The size of a family is always a power of two. The maximal size is 2^{''{{Boolf-hard|adicity}}''}, i.e. the period length of the truth table.<br> <small style="opacity: .5;">(While the size of {{Boolf-hard|faction}} and {{Boolf-hard|clan}} increases with the chosen arity, the size of a family is fixed.)</small> The simplest property of a family is its weight, i.e. the number of true entries in each truth table.<br> An important property is the parity of the weight. Those with odd weight shall be called '''sharp''', those with even weight '''dull'''.<br> <small style="opacity: .5;">(It is the same as the parity of the quaestor weight.)</small> Sharp families always have the maximal size.<br> Each member of sharp family has a unique {{boolf-soft|consul}}, while all members of a dull family have the same consul.<br> This makes it easy to calculate a family representative for sharp Boolean functions. <small>(For [[Gender of Boolean functions|males]] to be precise.)</small> A family belongs to a {{boolf-hard|clan}} (where the arguments are not just negated, but also permuted).<br> <small>Together with its complement, it forms a super-family.</small> <small style="opacity: .5;">Adding a half-complement forms an {{Boolf-soft|ultra-family}}.</small><br> Clans belong to a {{Boolf-soft|tribe}}. Sharp BF form a tribe on their own. The tribe of a dull BF is the binary weight of the consul. ==sequences== {{Families of Boolean functions/table of sequences}} {{Collapsible START|family representatives|collapsed gap-below}} The representative is the [[smallest Zhegalkin index]].<br> <small>The length of the sequence for arity ''n'' is {{oeislink|A000231}}(''n''), the number of ''n''-ary families.</small> {{Integer sequences related to Boolean functions/reps of families/nested}} {{Collapsible END}} {{oeislink|A227725}} <math>T(n, k)</math> is the number of <math>n</math>-ary families of '''size''' <math>2^k</math>.<br> {{oeislink|A227724}} <math>T(n, k)</math> is the number of '''balanced''' <math>n</math>-ary families of '''size''' <math>2^k</math>.<br> {{oeislink|A054724}} <math>T(n, k)</math> is the number of <math>n</math>-ary families with '''weight''' <math>k</math>.<br> ==super-clans== The following table shows the 46 3-ary families within the 14 super-clans. That means, each family is shown in its {{Boolf-hard|clan}}, and together with its complement.<br> In each matrix a family has a distinct color. Complements have the same base color (RGB or beige). Clans have the same shade (light or dark).<br> <small style="opacity: .5;">(The number of families in a super-clan is 1, 2, 3 or 6.)</small> {{Collapsible START|table of super-clans|collapsed wide}} {{Families of Boolean functions/box with sharp families by quaestor weight}} {{Collapsible END}} {{Families of Boolean functions/table of super-clans}} {{Collapsible END}} ==super-families== The following table shows the 46 3-ary families within the 30 super-families. That means, each family is shown together with its complement.<br> An important property <small>(that links the 3-ary to 2-ary families)</small> is the village &ndash; seen as columns in the matrices of Zhegalkin indices.<br> Blunt families have a consul and a {{Boolf prop 3-ary binary|is solid|solidity}}. <small>(Sharp families have four solid and four fluid members.)</small> {{Collapsible START|table of super-families|collapsed wide}} {{Families of Boolean functions/table of super-families}} {{Collapsible END}} ==misc.== '''3-ary partitions:''' {{Boolf prop 3-ary|family}} (46), {{Boolf prop 3-ary|reverse family}} (46), {{Boolf prop 3-ary|super family}} (30), {{Boolf prop 3-ary|ultra family}} (18), {{Boolf prop 3-ary|family size}} (4), {{Boolf prop 3-ary|quaestor weight}} (5), {{Boolf prop 3-ary|tribe}} (5), {{Boolf prop 3-ary binary|is sharp}} (2) * [[c:Category:3-ary truth tables in octeract matrix; ultra-families]] * [[c:Category:3-ary Boolean functions in octeract matrix; super-clans]] * [[c:Category:3-ary Boolean functions in octeract matrix; reverse super-clans]] * [[c:Category:3-ary Boolean functions; small equivalence classes; cubes]] * [[c:Category:Walsh spectra of 3-ary Boolean functions; families]] * [[c:Category:Family to senior nobles to prefects (image set)]] [[Category:Families of Boolean functions]] 42ai7dnljw9otd2gbzq4i1uenf4sq5x 2693600 2693506 2024-12-27T10:52:08Z Watchduck 137431 /* super-families */ 2693600 wikitext text/x-wiki {{Boolf header}} Boolean functions belong to the same family, when they can be transformed into each other by negating arguments.<br> The size of a family is always a power of two. The maximal size is 2^{''{{Boolf-hard|adicity}}''}, i.e. the period length of the truth table.<br> <small style="opacity: .5;">(While the size of {{Boolf-hard|faction}} and {{Boolf-hard|clan}} increases with the chosen arity, the size of a family is fixed.)</small> The simplest property of a family is its weight, i.e. the number of true entries in each truth table.<br> An important property is the parity of the weight. Those with odd weight shall be called '''sharp''', those with even weight '''dull'''.<br> <small style="opacity: .5;">(It is the same as the parity of the quaestor weight.)</small> Sharp families always have the maximal size.<br> Each member of sharp family has a unique {{boolf-soft|consul}}, while all members of a dull family have the same consul.<br> This makes it easy to calculate a family representative for sharp Boolean functions. <small>(For [[Gender of Boolean functions|males]] to be precise.)</small> A family belongs to a {{boolf-hard|clan}} (where the arguments are not just negated, but also permuted).<br> <small>Together with its complement, it forms a super-family.</small> <small style="opacity: .5;">Adding a half-complement forms an {{Boolf-soft|ultra-family}}.</small><br> Clans belong to a {{Boolf-soft|tribe}}. Sharp BF form a tribe on their own. The tribe of a dull BF is the binary weight of the consul. ==sequences== {{Families of Boolean functions/table of sequences}} {{Collapsible START|family representatives|collapsed gap-below}} The representative is the [[smallest Zhegalkin index]].<br> <small>The length of the sequence for arity ''n'' is {{oeislink|A000231}}(''n''), the number of ''n''-ary families.</small> {{Integer sequences related to Boolean functions/reps of families/nested}} {{Collapsible END}} {{oeislink|A227725}} <math>T(n, k)</math> is the number of <math>n</math>-ary families of '''size''' <math>2^k</math>.<br> {{oeislink|A227724}} <math>T(n, k)</math> is the number of '''balanced''' <math>n</math>-ary families of '''size''' <math>2^k</math>.<br> {{oeislink|A054724}} <math>T(n, k)</math> is the number of <math>n</math>-ary families with '''weight''' <math>k</math>.<br> ==super-clans== The following table shows the 46 3-ary families within the 14 super-clans. That means, each family is shown in its {{Boolf-hard|clan}}, and together with its complement.<br> In each matrix a family has a distinct color. Complements have the same base color (RGB or beige). Clans have the same shade (light or dark).<br> <small style="opacity: .5;">(The number of families in a super-clan is 1, 2, 3 or 6.)</small> {{Collapsible START|table of super-clans|collapsed wide}} {{Families of Boolean functions/box with sharp families by quaestor weight}} {{Collapsible END}} {{Families of Boolean functions/table of super-clans}} {{Collapsible END}} ==super-families== The following table shows the 46 3-ary families within the 30 super-families. That means, each family is shown together with its complement.<br> An important property <small>(that links the 3-ary to 2-ary families)</small> is the {{Boolf prop 3-ary|village}} &ndash; seen as columns in the matrices of Zhegalkin indices.<br> Blunt families have a consul and a {{Boolf prop 3-ary binary|is solid|solidity}}. <small>(Sharp families have four solid and four fluid members.)</small> {{Collapsible START|table of super-families|collapsed wide}} {{Families of Boolean functions/table of super-families}} {{Collapsible END}} ==misc.== '''3-ary partitions:''' {{Boolf prop 3-ary|family}} (46), {{Boolf prop 3-ary|reverse family}} (46), {{Boolf prop 3-ary|super family}} (30), {{Boolf prop 3-ary|ultra family}} (18), {{Boolf prop 3-ary|family size}} (4), {{Boolf prop 3-ary|quaestor weight}} (5), {{Boolf prop 3-ary|tribe}} (5), {{Boolf prop 3-ary binary|is sharp}} (2) * [[c:Category:3-ary truth tables in octeract matrix; ultra-families]] * [[c:Category:3-ary Boolean functions in octeract matrix; super-clans]] * [[c:Category:3-ary Boolean functions in octeract matrix; reverse super-clans]] * [[c:Category:3-ary Boolean functions; small equivalence classes; cubes]] * [[c:Category:Walsh spectra of 3-ary Boolean functions; families]] * [[c:Category:Family to senior nobles to prefects (image set)]] [[Category:Families of Boolean functions]] tdpztwxp3sj5iw1aj33otb7lj9ge4qi 10 317412 2693432 2693175 2024-12-26T23:23:09Z Watchduck 137431 2693432 wikitext text/x-wiki <templatestyles src="Template:Families of Boolean functions/table of super-clans/style.css" /> {| class="wikitable sortable" id="super-clans" !class="super-clan-size"| <abbr title="super-clan">s.-c.</abbr><br>size !class="family-size"| family<br>size !class="weight"| weight !class="tribe"| tribe !class="quaestor-weight"| quaestor<br>weight !class="rep"| <abbr title="representative">rep.</abbr> !class="matrix"| truth<br>tables !class="matrix"| Zhegalkin<br>indices !class="matrix"| reverse |- |class="super-clan-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' | 0 | 0<br>[[File:Venn 0000 0000.svg|30px]]<br><br>Ж 0<br>[[File:Venn 0000 0000.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 0.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 0 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 0.svg|200px]] |- |class="super-clan-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 4 | 150<br>[[File:Venn 0110 1001.svg|30px]]<br><br>Ж 22<br>[[File:Venn 0110 1000.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 22.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 22 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 22.svg|200px]] |- |class="super-clan-size"| 6 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 4 | 170<br>[[File:Venn 0101 0101.svg|30px]]<br><br>Ж 2<br>[[File:Venn 0100 0000.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 2.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 2 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 2.svg|200px]] |- |class="super-clan-size"| 6 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 0 | 102<br>[[File:Venn 0110 0110.svg|30px]]<br><br>Ж 6<br>[[File:Venn 0110 0000.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 6.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 6 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 6.svg|200px]] |- |class="super-clan-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' | 0 | 66<br>[[File:Venn 0100 0010.svg|30px]]<br><br>Ж 106<br>[[File:Venn 0101 0110.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 106.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 106 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 106.svg|200px]] |- |class="super-clan-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' | 4 | 232<br>[[File:Venn 0001 0111.svg|30px]]<br><br>Ж 104<br>[[File:Venn 0001 0110.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 104.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 104 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 104.svg|200px]] |- |class="super-clan-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp | 1 | 128<br>[[File:Venn 0000 0001.svg|30px]]<br><br>Ж 128<br>[[File:Venn 0000 0001.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 128.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 128 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 128.svg|200px]] |- |class="super-clan-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 3 | 22<br>[[File:Venn 0110 1000.svg|30px]]<br><br>Ж 150<br>[[File:Venn 0110 1001.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 150.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 150 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 150.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' | 2 | 40<br>[[File:Venn 0001 0100.svg|30px]]<br><br>Ж 40<br>[[File:Venn 0001 0100.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 40.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 40 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 40.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' | 2 | 136<br>[[File:Venn 0001 0001.svg|30px]]<br><br>Ж 8<br>[[File:Venn 0001 0000.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 8.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 8 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 8.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' | 2 | 120<br>[[File:Venn 0001 1110.svg|30px]]<br><br>Ж 24<br>[[File:Venn 0001 1000.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 24.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 24 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 24.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' | 2 | 228<br>[[File:Venn 0010 0111.svg|30px]]<br><br>Ж 44<br>[[File:Venn 0011 0100.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 44.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 44 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 44.svg|200px]] |- |class="super-clan-size"| 48 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 3 | 42<br>[[File:Venn 0101 0100.svg|30px]]<br><br>Ж 130<br>[[File:Venn 0100 0001.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 130.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 130 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 130.svg|200px]] |- |class="super-clan-size"| 48 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 1 | 230<br>[[File:Venn 0110 0111.svg|30px]]<br><br>Ж 134<br>[[File:Venn 0110 0001.svg|30px]] | [[File:3-ary Boolean functions; super-clan of Zhe 134.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 134 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 134.svg|200px]] |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> fqxwebz4l7buq7j58wy29ngns9ueyrc 2693475 2693432 2024-12-27T00:10:42Z Watchduck 137431 2693475 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-clans/style.css" /> {| class="wikitable sortable" id="super-clans" !class="super-clan-size"| <abbr title="super-clan">s.-c.</abbr><br>size !class="family-size"| family<br>size !class="weight"| weight !class="tribe"| tribe !class="quaestor-weight"| quaestor<br>weight !class="rep"| <abbr title="representative">rep.</abbr> !class="matrix"| truth<br>tables !class="matrix"| Zhegalkin<br>indices !class="matrix"| reverse |- |class="super-clan-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' | 0 | <span class="sortkey">0</span>0<br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0 | [[File:3-ary Boolean functions; super-clan of Zhe 0.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 0 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 0.svg|200px]] |- |class="super-clan-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 4 | <span class="sortkey">22</span>150<br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22 | [[File:3-ary Boolean functions; super-clan of Zhe 22.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 22 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 22.svg|200px]] |- |class="super-clan-size"| 6 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 4 | <span class="sortkey">2</span>170<br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2 | [[File:3-ary Boolean functions; super-clan of Zhe 2.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 2 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 2.svg|200px]] |- |class="super-clan-size"| 6 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 0 | <span class="sortkey">6</span>102<br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6 | [[File:3-ary Boolean functions; super-clan of Zhe 6.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 6 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 6.svg|200px]] |- |class="super-clan-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' | 0 | <span class="sortkey">106</span>66<br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106 | [[File:3-ary Boolean functions; super-clan of Zhe 106.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 106 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 106.svg|200px]] |- |class="super-clan-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' | 4 | <span class="sortkey">104</span>232<br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104 | [[File:3-ary Boolean functions; super-clan of Zhe 104.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 104 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 104.svg|200px]] |- |class="super-clan-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp | 1 | <span class="sortkey">128</span>128<br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128 | [[File:3-ary Boolean functions; super-clan of Zhe 128.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 128 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 128.svg|200px]] |- |class="super-clan-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 3 | <span class="sortkey">150</span>22<br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150 | [[File:3-ary Boolean functions; super-clan of Zhe 150.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 150 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 150.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' | 2 | <span class="sortkey">40</span>40<br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40 | [[File:3-ary Boolean functions; super-clan of Zhe 40.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 40 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 40.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' | 2 | <span class="sortkey">8</span>136<br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8 | [[File:3-ary Boolean functions; super-clan of Zhe 8.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 8 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 8.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' | 2 | <span class="sortkey">24</span>120<br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24 | [[File:3-ary Boolean functions; super-clan of Zhe 24.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 24 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 24.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' | 2 | <span class="sortkey">44</span>228<br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44 | [[File:3-ary Boolean functions; super-clan of Zhe 44.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 44 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 44.svg|200px]] |- |class="super-clan-size"| 48 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 3 | <span class="sortkey">130</span>42<br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130 | [[File:3-ary Boolean functions; super-clan of Zhe 130.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 130 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 130.svg|200px]] |- |class="super-clan-size"| 48 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 1 | <span class="sortkey">134</span>230<br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134 | [[File:3-ary Boolean functions; super-clan of Zhe 134.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 134 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 134.svg|200px]] |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> dx3b8wypfurf7cmb4lhghzfgrc7gpfs 2693514 2693475 2024-12-27T00:45:46Z Watchduck 137431 2693514 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-clans/style.css" /> {| class="wikitable sortable" id="super-clans" !class="super-clan-size"| <abbr title="super-clan">s.-c.</abbr><br>size !class="family-size"| family<br>size !class="weight"| weight !class="tribe"| tribe !class="quaestor-weight"| quaestor<br>weight !class="rep"| <abbr title="representative">rep.</abbr> !class="matrix"| truth<br>tables !class="matrix"| Zhegalkin<br>indices !class="matrix"| reverse |- |class="super-clan-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' | 0 | <span class="sortkey">0</span><small>0</small><br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0 | [[File:3-ary Boolean functions; super-clan of Zhe 0.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 0 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 0.svg|200px]] |- |class="super-clan-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 4 | <span class="sortkey">22</span><small>150</small><br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22 | [[File:3-ary Boolean functions; super-clan of Zhe 22.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 22 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 22.svg|200px]] |- |class="super-clan-size"| 6 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 4 | <span class="sortkey">2</span><small>170</small><br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2 | [[File:3-ary Boolean functions; super-clan of Zhe 2.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 2 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 2.svg|200px]] |- |class="super-clan-size"| 6 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 0 | <span class="sortkey">6</span><small>102</small><br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6 | [[File:3-ary Boolean functions; super-clan of Zhe 6.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 6 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 6.svg|200px]] |- |class="super-clan-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' | 0 | <span class="sortkey">106</span><small>66</small><br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106 | [[File:3-ary Boolean functions; super-clan of Zhe 106.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 106 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 106.svg|200px]] |- |class="super-clan-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' | 4 | <span class="sortkey">104</span><small>232</small><br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104 | [[File:3-ary Boolean functions; super-clan of Zhe 104.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 104 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 104.svg|200px]] |- |class="super-clan-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp | 1 | <span class="sortkey">128</span><small>128</small><br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128 | [[File:3-ary Boolean functions; super-clan of Zhe 128.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 128 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 128.svg|200px]] |- |class="super-clan-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 3 | <span class="sortkey">150</span><small>22</small><br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150 | [[File:3-ary Boolean functions; super-clan of Zhe 150.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 150 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 150.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' | 2 | <span class="sortkey">40</span><small>40</small><br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40 | [[File:3-ary Boolean functions; super-clan of Zhe 40.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 40 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 40.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' | 2 | <span class="sortkey">8</span><small>136</small><br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8 | [[File:3-ary Boolean functions; super-clan of Zhe 8.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 8 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 8.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' | 2 | <span class="sortkey">24</span><small>120</small><br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24 | [[File:3-ary Boolean functions; super-clan of Zhe 24.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 24 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 24.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' | 2 | <span class="sortkey">44</span><small>228</small><br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44 | [[File:3-ary Boolean functions; super-clan of Zhe 44.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 44 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 44.svg|200px]] |- |class="super-clan-size"| 48 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 3 | <span class="sortkey">130</span><small>42</small><br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130 | [[File:3-ary Boolean functions; super-clan of Zhe 130.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 130 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 130.svg|200px]] |- |class="super-clan-size"| 48 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 1 | <span class="sortkey">134</span><small>230</small><br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134 | [[File:3-ary Boolean functions; super-clan of Zhe 134.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 134 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 134.svg|200px]] |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> 3l4ipe5jo6744bjeaa7v1vpx38tninq 2693535 2693514 2024-12-27T00:51:54Z Watchduck 137431 2693535 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-clans/style.css" /> {| class="wikitable sortable" id="super-clans" !class="super-clan-size"| <abbr title="super-clan">s.-c.</abbr><br>size !class="family-size"| family<br>size !class="weight"| weight !class="tribe"| tribe !class="quaestor-weight"| quaestor<br>weight !class="rep"| <abbr title="representative">rep.</abbr> !class="matrix"| truth<br>tables !class="matrix"| Zhegalkin<br>indices !class="matrix"| reverse |- |class="super-clan-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' | 0 | <span class="sortkey">0<br></span><small>0</small><br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0 | [[File:3-ary Boolean functions; super-clan of Zhe 0.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 0 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 0.svg|200px]] |- |class="super-clan-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 4 | <span class="sortkey">22<br></span><small>150</small><br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22 | [[File:3-ary Boolean functions; super-clan of Zhe 22.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 22 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 22.svg|200px]] |- |class="super-clan-size"| 6 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 4 | <span class="sortkey">2<br></span><small>170</small><br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2 | [[File:3-ary Boolean functions; super-clan of Zhe 2.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 2 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 2.svg|200px]] |- |class="super-clan-size"| 6 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' | 0 | <span class="sortkey">6<br></span><small>102</small><br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6 | [[File:3-ary Boolean functions; super-clan of Zhe 6.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 6 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 6.svg|200px]] |- |class="super-clan-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' | 0 | <span class="sortkey">106<br></span><small>66</small><br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106 | [[File:3-ary Boolean functions; super-clan of Zhe 106.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 106 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 106.svg|200px]] |- |class="super-clan-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' | 4 | <span class="sortkey">104<br></span><small>232</small><br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104 | [[File:3-ary Boolean functions; super-clan of Zhe 104.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 104 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 104.svg|200px]] |- |class="super-clan-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp | 1 | <span class="sortkey">128<br></span><small>128</small><br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128 | [[File:3-ary Boolean functions; super-clan of Zhe 128.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 128 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 128.svg|200px]] |- |class="super-clan-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 3 | <span class="sortkey">150<br></span><small>22</small><br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150 | [[File:3-ary Boolean functions; super-clan of Zhe 150.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 150 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 150.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' | 2 | <span class="sortkey">40<br></span><small>40</small><br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40 | [[File:3-ary Boolean functions; super-clan of Zhe 40.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 40 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 40.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' | 2 | <span class="sortkey">8<br></span><small>136</small><br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8 | [[File:3-ary Boolean functions; super-clan of Zhe 8.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 8 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 8.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' | 2 | <span class="sortkey">24<br></span><small>120</small><br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24 | [[File:3-ary Boolean functions; super-clan of Zhe 24.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 24 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 24.svg|200px]] |- |class="super-clan-size"| 24 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' | 2 | <span class="sortkey">44<br></span><small>228</small><br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44 | [[File:3-ary Boolean functions; super-clan of Zhe 44.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 44 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 44.svg|200px]] |- |class="super-clan-size"| 48 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 3 | <span class="sortkey">130<br></span><small>42</small><br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130 | [[File:3-ary Boolean functions; super-clan of Zhe 130.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 130 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 130.svg|200px]] |- |class="super-clan-size"| 48 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp | 1 | <span class="sortkey">134<br></span><small>230</small><br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134 | [[File:3-ary Boolean functions; super-clan of Zhe 134.svg|200px]] | [[File:3-ary Boolean functions; super-clan of Zhe 134 (indices).svg|200px]] | [[File:3-ary Boolean functions; transposed super-clan of Zhe 134.svg|200px]] |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> ct1s9dtudg6dnzakma5xjei8ybmr5ec 10 317418 2693476 2692909 2024-12-27T00:10:52Z Watchduck 137431 2693476 sanitized-css text/css table#super-clans {text-align: center;} table#super-clans td.super-clan-size {color: gray; font-size: 110%;} table#super-clans td.family-size {color: gray; font-size: 90%;} table#super-clans td.weight, table#super-clans td.tribe {background-color: #fef4e8;} table#super-clans span.sortkey {display: none;} qycerbvc2x34cmb4h0xxuwdm30pus3y 14 317419 2693480 2692911 2024-12-27T00:12:15Z Watchduck 137431 2693480 wikitext text/x-wiki * {{tl|Families of Boolean functions/table of super-clans/style.css}} * {{tl|Families of Boolean functions/table of super-families/style.css}} [[Category:Studies of Boolean functions]] 3dt7dyuiftvhhg98r0tsouythurpmax 0 317451 2693384 2693134 2024-12-26T21:13:35Z DavidMCEddy 218607 /* Selected comments */ syntax error 2693384 wikitext text/x-wiki :''This is a discussion of a Zoom interview recorded 2024-12-20 with Michael Novick, former interim general manager for KPFK, the second station in the Pacifica Radio Network, about HR 9495, which has been called the "nonprofit-killer bill, by its opponents. 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:HR 9495, the nonprofit-killer bill, per Michael Novick.webm|thumb|Interview recorded 2024-12-20 with Michael Novick regarding HR 9495, called the nonprofit-killer bill, because it is allegedly designed to suppress dissent in the US.]] <!--[[File: ... .ogg|thumb|29:00 mm:ss extract from interview recorded 2024-12-20 with Michael Novick regarding, HR 9495, the nonprofit-killer bill.]]--> Michael Novick discussed HR 9495, the “Stop Terror Financing and Tax Penalties on American Hostages Act”, which passed the House November 21. Its opponents have called it the “nonprofit-killer” bill, because it would give the Secretary of Treasury the authority to designate any nonprofit as a suspected “Terrorist Supporting Organization" and remove their tax-exempt status unless they convince the Secretary of Treasury that they do not support terrorists.<ref name=HR9495>US House (2024).</ref> Mother Jones reported, 'In the bill’s original iteration, it was popular among both Republicans and Democrats, who saw it as an appealing way to police Palestinian rights organizations after protests last year. An earlier version, in April, passed the House easily, with only 11 votes against the bill. It didn’t make it through the Senate ... One of those early no votes was Rep. Rashida Tlaib (D-Mich.), who said on the House floor [November 21], “... This is a dangerous and an unconstitutional bill that would allow unchecked power to target nonprofit organizations as political enemies and shut them down without due process.”'<ref>Hurwitz (2024).</ref> Beth Gazley, Professor of Nonprofit Management and Policy at Indiana University,<ref><!--Beth Gazley-->{{cite Q|Q131542978}}</ref> said, "I believe that this is part of a strategy to preempt opposition to Republican policies and encourage self-censorship. It’s a way for the GOP to try to restrict what activists and nonprofit organizations can say or do. And, essentially, it’s a threat to political opponents of President-elect Donald Trump." On November 21, only 15 Democrats supported it and one Republican opposed it. Rep. [[w:Jamie Raskin|Jamie Raskin]] (D-MD) called the bill “a werewolf in sheep’s clothing." ... An earlier version of this legislation was introduced in December 2023 and passed in the House in April 2024. Based on the timing, it was widely interpreted as an attempt to quell widespread protests by students and other people who were expressing their solidarity with Palestinians and their objections to Israel’s military operations in Gaza. But this legislation could easily do far more than that, because it does not distinguish between foreign and domestic terrorism – whether it’s real or imagined.<ref name=Gazley>Gazley (2024).</ref> Raskin further noted that “rendering support to terrorists is already a felony”,<ref>Raskin was quoted in Gazley (2024). In fact, "[[w:Providing material support for terrorism|Providing material support for terrorism]]" is a felony under the USA Patriot Act of 2001 punishable by fines and imprisonment of up to 15 years or 20 years if human(s) convicted ''know(s)'' they were aiding an organization so classified by the US State Department and life in prison if the "death of a person" has resulted, and 'the term “person” means any individual or entity capable of holding a legal or beneficial interest in property'. In ''[[w:Holder v. the Humanitarian Law Project|Holder v. the Humanitarian Law Project]]'' (2010), the US Supreme Court ruled that teaching nonviolence to someone designated as a "terrorist" was "providing material support for terrorism". [[w:David D. Cole|David D. Cole]], attorney for the [[w:Humanitarian Law Project|Humanitarian Law Project]], said that under that ruling, even asking the State Department to explain why some individual or group was designated as a "terrorist" was similarly "providing material support", a major felony with penalties as just described.</ref> and this bill could end all rights to due process.<ref name=Gazley/> == Michael Novick == Novick has described himself as antiracist, antisexist, anti-imperialist, and anti-authoritarian. Between 2022 and November 2024, he was the interim general manager of [[w:KPFK|KPFK]],<ref>Novick (2022) noted that he began as interim general manager of KKFK in 2022. In this interview, he said he was no longer in that position.</ref> the second [[w:Pacifica Foundation|station in the Pacifica Radio Network]].<ref>The Pacifica Radio Network includes stations owned by the Pacifica Foundation plus over 200 that are "affiliates".</ref> Novick can be reached at antiracistaction_la@yahoo.com or changelinks2@gmail.com. The latter is for the Change Links community calendar.<ref name=Change-Links><!-- Change-Links-->{{cite Q|Q131544553}}</ref> "antiracist.org" is the website for ''Turning the Tide'',<ref><!-- Turning the Tide-->{{cite Q|Q131544806}}</ref> which Novick has been doing since 1988. == HR 9495 == HR 9495 says that "the term ‘terrorist supporting organization’ means any organization which is designated by the Secretary [of Treasury] as having provided, during the 3-year period ending on the date of such designation, material support or resources" to a designated terrorist organization. Before an organization can be so designated, the Secretary is required to mail a written notice of such impending designation and giving them 90 days to "demonstrated to the satisfaction of the Secretary that such organization did not provide the material support or resources".<ref name=HR9495/> == Selected comments == Novik noted that 9495 is not likely to pass the Senate this year, {{blockquote| but they are going to bring it back immediately when the new Congress is seated, which is even prior to the inauguration of the new president. ... It's also related to a separate initiative of [[w:Project 2025|Project 2025]] that Trump, of course, said he never read ... but he's been appointing all sorts of people involved in that Project 2025 to his administration. And he has already said he'll be appointing the person who wrote the section of 2025 on media, who is a current member of the [[w:Federal Communications Commission|FCC]], ... [[w:Brendan Carr|Brendan Carr]]. And in that document, they talk about actually not just defunding the [[w:Corporation for Public Broadcasting|Corporation for Public Broadcasting]] and public media efforts like [[w:NPR|NPR]]. ... But they actually want to remove the non-commercial educational licensing entirely. And they mentioned Pacifica by name in that report 2025. So we're expecting a lot of attacks of this nature to come down the pike. Because ... nonprofits and non-commercial media have been an important avenue for exposing some of the ills of the society and whistleblowing ... . They want to really contain and control the philanthropic sector and the public media sector to carry out some of their other goals ... that have been spelled out pretty clearly in the campaign of mass deportations, ... privatization ... . They want to try to force everything into the commercial enterprises, ... the billionaire owned media." }} Regarding Trump suing [[w:American Broadcasting Company|ABC]] and ''[[w:The Des Moines Register|The Des Moines Register]]'',<ref>Gold (2024).</ref> Novick said, "I think it's part and parcel of this whole attack on on freedom of speech, freedom of the press, the whole thing about fake news. ... [T]his effort internationally and nationally to really control the free flow of information because they understand that an informed public is less likely to sit still for its own rights being violated. And certainly the other end of the freedom of speech is freedom of discourse and freedom of listening. If you can't hear any contrary views to those being expressed by the great leader, that's a violation of your rights, not just the rights of the speakers. And so I think ... that it is an attack on human rights and on people's ability to understand what's going on in the world and do something about it." Graves asked Novick about Trump saying that Liz Cheney, a Republican who represented Wyoming in the US House, should be prosecuted and jailed for her role in a Trump impeachment proceeding during Trump's first term.<ref>Mascaro (2024).</ref> Novick replied, {{blockquote| I think it's a much deeper problem. It it started before him. ... The Biden administration ... prosecuted [[w:Julian Assange|Julian Assange]] and actually won a conviction. They got him to plead guilty to a violation of the Espionage act for releasing data that was not espionage at all. It was whistleblowing about a war crime by the United States killing of reporters in Iraq. And similarly the Biden Administration is the one that pursued the case of the [[w:Uhuru Movement|Uhuru 3]] ... from the [[w:African People's Socialist Party|African People's Socialist party]] ... accusing them of being foreign agents for opposing the war in Iraq and ... the genocide in Palestine. So I think that this is a bipartisan issue. In fact, there were votes on both sides of the aisle for 9495. "One of the first things that Richard Nixon did [as president of the US] was introduce the [[w: Tax Reform Act of 1969|Tax Reform Act of 1969]], as they were trying to dismantle the great society and some of those programs that were about empowering communities, particularly poor people, people of color. The Tax Reform Act of 1969 specifically ... said that any ... organization that wanted nonprofit status could not support voter registration drives or ... activities that ... affected legislation. ... This is a very similar effort, I think," though 9495 has a much more authoritarian stamp. ... 9495 is targeted directly at organizations that are engaged with international solidarity, particularly with the Palestinian cause. But ... the terrorism term has been used extensively as the so-called Communist threat faded ... . I think the head<ref>[[w:Mufid Abdulqader|Mufid Abdulqader]] was released 2024-12-12 after 16 years in federal prison. He was described as a "top fundraiser" and "leader" of the [[w:Holy Land Foundation for Relief and Development|Holy Land Foundation]]</ref> of the [[w:Holy Land Foundation for Relief and Development|Holy Land Foundation ]] was just recently released from prison. ... This was a humanitarian project, ... based in, obviously, the Muslim community ... . [I]t was humanitarian aid, but it was criminalized. And I think that's what they're trying to do here is to really prevent people from any kind of person to person diplomacy outside the bounds of what the State Department is carrying out which is really militarized. The State Department, I think, follows the same dictates as the Department of Defense, so-called, and the CIA. They're basically involved in, you know, trying to identify forces within different societies that will follow the dictates of US Policy. And anybody that does not want to do that is identified as a potential targeted terrorist or someone supporting terrorism. ... I've been associated for many years with the Los Angeles Chapter of the [[w:Anti-Racist Action|Anti-Racist Action Network]] ... . The idea was, be part of the solution. But ... terror is, you know, very widespread in this world, and has mainly been used actually by the right and by State actors including the United States. You know when [[w:George W. Bush|Bush]] [[w:2003 invasion of Iraq|went into Iraq]], ... the second bush and the second [[w:Iraq War|Iraq War]], they talked about "[[w:Shock and awe|shock and awe]]". Well, it's just a polite name for terror. Right? You're bombing people into submission. You're trying to intimidate them through violence. ... it can be classified as terror as well as genocide. They're terrorizing the entire population. Historically, the [[w:Ku Klux Klan|Ku Klux Klan]], and a lot of other organizations have been terrorist organizations. But that's not what they're talking about. They're talking about resistance to white supremacy, resistance to colonialism. ... If there's some action against the German occupation, or in this case the Israeli occupation, they're going to punish people ... in the area where it happened without regard to who is responsible. ... The term is used is to justify that kind of actual terrorism by labelling any resistances as terrorism. ... The case of the [[w:Humanitarian Law Project|Humanitarian Law Project]] ... went all the way to the Supreme Court ... [which] ruled that even providing nonviolence training was a form of material support. Lydia Brazon<ref>Pacifica in Exile (2015).</ref> ... was with the Humanitarian Law Project for a long time. ... She was at one point the executive director of Pacifica. ... Pacifica's mission is to identify the causes of conflict and try to resolve them without violence. ... It was created by a group of pacifists who actually resisted even World War 2. [[w:Lewis Hill (Pacifica Radio)|Lew Hill]] and a group of others ... formed the Pacifica Foundation and launched this project of listener sponsored non-commercial radio. But that's seen as a threat, trying to to solve things, because the State wants to reserve that power of the use of violence. And if you oppose violence by the state, then you're somehow a subversive. When 9495 passed, [[w:Mike Johnson|Mike Johnson]], who is the [[w:Speaker of the United States House of Representatives|Speaker of the House]], sent a tweet to a number of organizations, saying, "We're thinking about you." That included, for example, a [[w:Jewish Voice for Peace|Jewish Voice for Peace]]; ... because they believe in supporting Palestinian rights, they're targeted. We have a program on our station [[w:KPFK|KPFK]], which is Middle Eastern focus for many years, and the current co-host is the head of the LA chapter of a Jewish Voice for Peace, trying to bring about peace in the Middle East. But that's seen as a threat. I also work on a newspaper called ''Change-Links''.<ref name=Change-Links/> It's a community calendar for [[w:Los Angeles|Los Angeles]]. We publish every month with a list of activities that people might be interested in, cultural, political, and otherwise. ... It's not a nonprofit itself. But we have a fiscal sponsor which is the Alliance for Global Justice based in Tucson, Arizona.<ref><!--Alliance for Global Justice-->{{cite Q|Q129502246}}</ref> They serve this purpose for a number of smaller projects around the country. Media and other community service organizations that are not full nonprofits themselves have a fiscal sponsor. The Alliance for Global Justice is also in Mike Johnson's Tweet. He tweeted out to 5 or 6 or maybe 8 organizations. This is something that's very clearly directed at not just the activities but the information. They want everything to be like the right wing echo chamber of right wing talk media. Anything that's outside those bounds becomes a fair target. }} Graves noted that in August he had interviewed Heidi Beirich,<ref name=Beirich><!-- Heidi Beirich-->{{cite Q|Q128844587}}</ref> co-founder and Chief Strategy Officer of the [[Global Project Against Hate & Extremism (GPAHE)]].<ref name=GPAHE><!-- GPAHE-->{{cite Q|Q125952435}}</ref> She noted that recent [[w:National Defense Authorization Act|National Defense Authorization Act]]s have included provisions that explicitly prohibited the Secretary of Defense from attempting to root violent extremists out of the Us. Military.<ref>Donnelly (2022).</ref> Novick replied, {{blockquote| It's not surprising. There is a big struggle in the military. Trump has come in saying he wants to get rid of "[[w:woke|Woke]]" generals.<ref>Axe (2024).</ref> And there is a group that focuses on the question of religious freedom in the military, to be free, free from religion if you want. And there's a number of quite fundamentalist Christian people in high positions of authority in the military that are trying to enforce Christian nationalism. You see a lot of these right wing groups specifically targeting members of the military. We saw that both law enforcement and the military were overrepresented in the people involved in [[w:January 6 United States Capitol attack|January 6 storming of the Capitol]]. It's a longstanding phenomenon. ... I was just reading a book called ''Morningside'' about the [[w:Greensboro massacre|1979 massacre of anti-clan activists in Greensboro, North Carolina]]. The people involved in that were included people in the so-called [[w:White Patriot Party|White Patriot Party]], which was based at a marine base in North Carolina, and law enforcement, ..., the Nazis and the Klan and the United Racist front.<ref>Shetterly (2024).</ref> }} Novick continued, {{blockquote| The militia movement got its start from a couple of sources. One of them was a guy named [[w:John Singlaub|John Singlaub]], who was a general, removed, similar to what happened with [[w:Douglas MacArthur|MacArthur]] in [[w:Korean War|Korea]]. ... He formed something called the [[w:World League for Freedom and Democracy|World Anti-Communist League]] and then proceeded to use right wing Christian forces in the Philippines and Guatemala as a model for organizing similar forces in the United States. They collected money in the United States for these right wing militias in Guatemala and in the Philippines that were involved in terroristic activities of a supposedly anti-communist nature. And once they got that going, they started using it to build up militia groups in the United States on the same model. And you saw some of the repercussions of that with, you know, the [[w:Oklahoma City bombing|Federal building in Oklahoma City]], and other actions that came out of that. So I think that people need to take the threat of terrorism seriously, but that we need to understand where it's coming from. It's not coming from nonprofit humanitarian aid for poor people around the world or poor people in this country. It is coming from very well established and well rooted right wing forces in this country. }} == 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 other interviews regarding "Media & Democracy", available on Wikiversity under [[: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 == * <!--David Axe (2024-12-05) "Trump is planning to rip the guts out of the US armed forces", The Telegraph-->{{cite Q|Q131545681}} * <!--John M. Donnelly (2022-12-14) Final NDAA removes most House provisions on hate groups, Roll Call-->{{cite Q|Q130545466}} * <!--Beth Gazley (2024-11-22) "US House passes measure that could punish nonprofits Treasury Department decides are ‘terrorist’", The Conversation-->{{cite Q|Q131543053}} * <!--Hadas Gold (2024-12-16) "Emboldened by ABC settlement, Trump threatens more lawsuits against the press", CNN-->{{cite Q|Q131545105}} * <!-- Sophie Hurwitz (2024-11-21) "The House Passes Bill Allowing Trump Admin to Declare Nonprofits Terrorist Supporters"-->{{cite Q|Q131540369}} * <!-- Lisa Mascaro (2024-12-16) " After investigating Jan. 6, House GOP sides with Trump and goes after Liz Cheney-->{{cite Q|Q131545154}} * <!--Michael Novick (2022-11-15) "General Manager Report", KPFK-->{{cite Q|Q131543205}} * <!--Pacifica in Exile (2015-09-14) "Lydia Brazon, Executive Director", Pacifica in Exile Newsletter -->{{cite Q|Q131545325|author=Pacifica in Exile}} * <!--Aran Shetterly (2024-10-15) Morningside: The 1979 Greensboro Massacre and the Struggle for an American City's Soul-->{{cite Q|Q131545762}} * <!--US House (2024-11-21) H.R.9495 - Stop Terror-Financing and Tax Penalties on American Hostages Act-->{{cite Q|Q131540249|author = US House of Representatives}} [[Category:Politics]] [[Category:Freedom and abundance]] [[Category:Media reform to improve democracy]] dyxi8wunfnkgi9opih2nohop7xjjunw 0 317506 2693290 2693289 2024-12-26T11:59:26Z Eshaa2024 2993595 /* Definition */ 2693290 wikitext text/x-wiki ==Definition== Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and let <math>f</math> be holomorphic except for isolated singularities <math>S \subset G</math>, i.e., <math>f\colon G \setminus S \to \mathbb{C}</math> is holomorphic. If <math>z_0 \in S \subset G</math> is an [[Complex_analysis/isolated singularity|isolated singularity]] of <math>f</math> with <math>D_r(z_0) \cap S = {z_0}</math>, the residue is defined as: :<math>\mathrm{res}{z_0}(f) := \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{|\xi-z_0|=r} f(\xi), d\xi</math>. == Relation Between Residue and Laurent Series == If <math>f</math> is represented around an isolated singularity <math>z_0 \in S \subset G</math> as a Laurent series, the residue can be computed as follows. With <math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n</math> as the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math>, it holds that: :<math>\mathrm{res}{z_0}(f) = \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} a_{-1}\cdot (\xi-z_0)^{-1}, d\xi = \frac{1}{2\pi i} a_{-1} \cdot \underbrace{\int_{\partial D_r(z_0)} (\xi-z_0)^{-1} , d\xi}{=2\pi i} = a{-1}</math>. It must be taken into account that the closed disk <math>\overline{D_r(z_0)}</math> contains only the singularity <math>z_0 \in S </math>, i.e<math>\overline{D_r(z_0)} \cap s = \{z_0\}</math>. Thus, one can read off the 'residue' <math>\mathrm{res}_{z_0}(f) = a_{-1}</math> from the Laurent expansion of around at the <math>f</math> um <math>z_0</math> an -1-ten coefficient of . ==Considerations== The closed disk <math>\overline{D_r(z_0)}</math> must contain only the singularity <math>z_0 \in S</math>, meaning<math>\overline{D_r(z_0)} \cap S = {z_0}</math>. Thus, the residue <math>\mathrm{res}{z_0}(f) = a{-1}</math> can be read off directly as the coefficient of <math>(z-z_0)^{-1}</math> in the Laurent series expansion of <math>f</math> around <math>z_0</math>. ==Etymology== The term "residue" (from Latin residuere – to remain) is used because in integration along the path <math>\gamma(t) := z_0 + r\cdot e^{it}</math> with <math>t \in [0, 2\pi]</math> around the circle centered at <math>z_0</math>, the following holds: <center><math> \begin{array}{rl} \displaystyle \int_{|w-z_0|=r} f(w)\, dw &= \displaystyle \sum_{n=-\infty}^{+\infty} a_n \int_{|w-z_0|=r} (w-z_0)^n \, dw\\ &= 2\pi i \cdot a_{-1}. \end{array} </math></center> Thus, the residue is what remains after integrating. ==Computation for Poles== If <math>z_0 \in U</math> is a pole of order <math>m</math> of <math>f</math>, the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math> has the form: :<math>f(z) = \sum_{k=-m}^\infty a_k (z-z_0)^k</math> with <math>a_{-m} \neq 0</math>. ===Proof 1: Removing Principal Part by Multiplication=== By multiplying with <math>(z-z_0)^m</math>, we get: :<math>g_m(z) := (z-z_0)^m \cdot f(z) = \sum_{k=0}^\infty a_{k-m} (z-z_0)^k.</math> The residue <math>a_{-1}</math> is then the coefficient of <math>(z-z_0)^{m-1}</math> in the power series of <math>g_m(z)</math>. ===Proof 2: Using (m-1)-fold Differentiation=== By differentiating <math>m-1</math> times, the first <math>m-1</math> terms from <math>n=0</math> to <math>n=m-2</math> vanish. The residue is then found as the coefficient of <math>(z-z_0)^0</math> in: :<math>g_m^{(m-1)}(z) = \sum_{k=m-1}^\infty \frac{k!}{(k-m+1)!} a_{k-m}(z-z_0)^{k-m+1}.</math> ===Proof 3: Limit Process to Compute Coefficient of <math>(z-z_0)^0</math>=== By shifting the index: :<math>g_m^{(m-1)}(z) = \sum_{k=-1}^\infty \frac{(m+k)!}{(k+1)!} a_k (z-z_0)^{k+1}.</math> Taking the limit <math>z \to z_0</math>, all terms with <math>k \geq 0</math> vanish, leaving: :<math>\lim_{z \to z_0} g_m^{(m-1)}(z) = \frac{(m-1)!}{0!} \cdot a_{-1} \cdot (z-z_0)^0 = (m-1)! \cdot a_{-1}.</math> Thus, the residue can be computed <math>z \to z_0</math> using: :<math>\mathrm{res}{z_0}(f) = a{-1} = \frac{1}{(m-1)!} \cdot \lim_{z \to z_0} g_m^{(m-1)}(z).</math> ==Exercises for Students== *Explain why, in the Laurent series expansion, all terms from the principal and outer parts, i.e., <math>n \in \mathbb{Z}</math> with <math>n \neq -1</math>, yield integrals that evaluate to zero: :<math>\int_{\partial D_r(z_0)} a_n \cdot (\xi - z_0)^n, d\xi = 0.</math> *Why can the order of integration and series expansion be interchanged? ::<math>\sum_{n=-\infty}^{+\infty} \int_{\partial D_r(z_0)} a_n (\xi-z_0)^n d\xi = \int_{\partial D_r(z_0)} \sum_{n=-\infty}^{+\infty} a_n (\xi-z_0)^n d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} f(\xi), d\xi = \mathrm{res}_{z_0}(f).</math> *Given the function <math>f:\mathbb{C}\setminus {i} \to \mathbb{C}</math> with <math>z \mapsto f(z) = \frac{e^{z-i}}{(z-i)^5}</math>, calculate the residue <math>\mathrm{res}_{z_0}(f)</math> at <math>z_0 := i</math>. ==See Also== *[[Residue|Residue]] *[[Complex_analysis/Examples_of_Laurent_series|Examples of Laurent Series]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Residuum|Kurs:Funktionentheorie/Residuum ]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum * Date: 12/26/2024<span type="translate" src="Kurs:Funktionentheorie/Residuum " srclang="de" date="12/26/2024" time="11:28" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Residuum ]] </noinclude> [[Category:Wiki2Reveal]] ar313j3rps540zjfjme53l45bg8e0hs 2693291 2693290 2024-12-26T12:01:49Z Eshaa2024 2993595 /* Definition */ 2693291 wikitext text/x-wiki ==Definition== Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and let <math>f</math> be holomorphic except for isolated singularities <math>S \subset G</math>, i.e., <math>f\colon G \setminus S \to \mathbb{C}</math> is holomorphic. If <math>z_0 \in S \subset G</math> is an [[Complex_analysis/Isolated Singularity|Isolated singularity]] of <math>f</math> with <math>D_r(z_0) \cap S = {z_0}</math>, the residue is defined as: :<math>\mathrm{res}{z_0}(f) := \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{|\xi-z_0|=r} f(\xi), d\xi</math>. == Relation Between Residue and Laurent Series == If <math>f</math> is represented around an isolated singularity <math>z_0 \in S \subset G</math> as a Laurent series, the residue can be computed as follows. With <math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n</math> as the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math>, it holds that: :<math>\mathrm{res}{z_0}(f) = \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} a_{-1}\cdot (\xi-z_0)^{-1}, d\xi = \frac{1}{2\pi i} a_{-1} \cdot \underbrace{\int_{\partial D_r(z_0)} (\xi-z_0)^{-1} , d\xi}{=2\pi i} = a{-1}</math>. It must be taken into account that the closed disk <math>\overline{D_r(z_0)}</math> contains only the singularity <math>z_0 \in S </math>, i.e<math>\overline{D_r(z_0)} \cap s = \{z_0\}</math>. Thus, one can read off the 'residue' <math>\mathrm{res}_{z_0}(f) = a_{-1}</math> from the Laurent expansion of around at the <math>f</math> um <math>z_0</math> an -1-ten coefficient of . ==Considerations== The closed disk <math>\overline{D_r(z_0)}</math> must contain only the singularity <math>z_0 \in S</math>, meaning<math>\overline{D_r(z_0)} \cap S = {z_0}</math>. Thus, the residue <math>\mathrm{res}{z_0}(f) = a{-1}</math> can be read off directly as the coefficient of <math>(z-z_0)^{-1}</math> in the Laurent series expansion of <math>f</math> around <math>z_0</math>. ==Etymology== The term "residue" (from Latin residuere – to remain) is used because in integration along the path <math>\gamma(t) := z_0 + r\cdot e^{it}</math> with <math>t \in [0, 2\pi]</math> around the circle centered at <math>z_0</math>, the following holds: <center><math> \begin{array}{rl} \displaystyle \int_{|w-z_0|=r} f(w)\, dw &= \displaystyle \sum_{n=-\infty}^{+\infty} a_n \int_{|w-z_0|=r} (w-z_0)^n \, dw\\ &= 2\pi i \cdot a_{-1}. \end{array} </math></center> Thus, the residue is what remains after integrating. ==Computation for Poles== If <math>z_0 \in U</math> is a pole of order <math>m</math> of <math>f</math>, the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math> has the form: :<math>f(z) = \sum_{k=-m}^\infty a_k (z-z_0)^k</math> with <math>a_{-m} \neq 0</math>. ===Proof 1: Removing Principal Part by Multiplication=== By multiplying with <math>(z-z_0)^m</math>, we get: :<math>g_m(z) := (z-z_0)^m \cdot f(z) = \sum_{k=0}^\infty a_{k-m} (z-z_0)^k.</math> The residue <math>a_{-1}</math> is then the coefficient of <math>(z-z_0)^{m-1}</math> in the power series of <math>g_m(z)</math>. ===Proof 2: Using (m-1)-fold Differentiation=== By differentiating <math>m-1</math> times, the first <math>m-1</math> terms from <math>n=0</math> to <math>n=m-2</math> vanish. The residue is then found as the coefficient of <math>(z-z_0)^0</math> in: :<math>g_m^{(m-1)}(z) = \sum_{k=m-1}^\infty \frac{k!}{(k-m+1)!} a_{k-m}(z-z_0)^{k-m+1}.</math> ===Proof 3: Limit Process to Compute Coefficient of <math>(z-z_0)^0</math>=== By shifting the index: :<math>g_m^{(m-1)}(z) = \sum_{k=-1}^\infty \frac{(m+k)!}{(k+1)!} a_k (z-z_0)^{k+1}.</math> Taking the limit <math>z \to z_0</math>, all terms with <math>k \geq 0</math> vanish, leaving: :<math>\lim_{z \to z_0} g_m^{(m-1)}(z) = \frac{(m-1)!}{0!} \cdot a_{-1} \cdot (z-z_0)^0 = (m-1)! \cdot a_{-1}.</math> Thus, the residue can be computed <math>z \to z_0</math> using: :<math>\mathrm{res}{z_0}(f) = a{-1} = \frac{1}{(m-1)!} \cdot \lim_{z \to z_0} g_m^{(m-1)}(z).</math> ==Exercises for Students== *Explain why, in the Laurent series expansion, all terms from the principal and outer parts, i.e., <math>n \in \mathbb{Z}</math> with <math>n \neq -1</math>, yield integrals that evaluate to zero: :<math>\int_{\partial D_r(z_0)} a_n \cdot (\xi - z_0)^n, d\xi = 0.</math> *Why can the order of integration and series expansion be interchanged? ::<math>\sum_{n=-\infty}^{+\infty} \int_{\partial D_r(z_0)} a_n (\xi-z_0)^n d\xi = \int_{\partial D_r(z_0)} \sum_{n=-\infty}^{+\infty} a_n (\xi-z_0)^n d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} f(\xi), d\xi = \mathrm{res}_{z_0}(f).</math> *Given the function <math>f:\mathbb{C}\setminus {i} \to \mathbb{C}</math> with <math>z \mapsto f(z) = \frac{e^{z-i}}{(z-i)^5}</math>, calculate the residue <math>\mathrm{res}_{z_0}(f)</math> at <math>z_0 := i</math>. ==See Also== *[[Residue|Residue]] *[[Complex_analysis/Examples_of_Laurent_series|Examples of Laurent Series]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Residuum|Kurs:Funktionentheorie/Residuum ]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum * Date: 12/26/2024<span type="translate" src="Kurs:Funktionentheorie/Residuum " srclang="de" date="12/26/2024" time="11:28" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Residuum ]] </noinclude> [[Category:Wiki2Reveal]] ku0kae4gbupktehijczo2yd2i8a7x13 2693292 2693291 2024-12-26T12:04:43Z Eshaa2024 2993595 /* Definition */ 2693292 wikitext text/x-wiki ==Definition== Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and let <math>f</math> be holomorphic except for isolated singularities <math>S \subset G</math>, i.e., <math>f\colon G \setminus S \to \mathbb{C}</math> is holomorphic. If <math>z_0 \in S \subset G</math> is an [[Complex_analysis/isolated Singularity|isolated singularity]] of <math>f</math> with <math>D_r(z_0) \cap S = {z_0}</math>, the residue is defined as: :<math>\mathrm{res}{z_0}(f) := \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{|\xi-z_0|=r} f(\xi), d\xi</math>. == Relation Between Residue and Laurent Series == If <math>f</math> is represented around an isolated singularity <math>z_0 \in S \subset G</math> as a Laurent series, the residue can be computed as follows. With <math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n</math> as the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math>, it holds that: :<math>\mathrm{res}{z_0}(f) = \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} a_{-1}\cdot (\xi-z_0)^{-1}, d\xi = \frac{1}{2\pi i} a_{-1} \cdot \underbrace{\int_{\partial D_r(z_0)} (\xi-z_0)^{-1} , d\xi}{=2\pi i} = a{-1}</math>. It must be taken into account that the closed disk <math>\overline{D_r(z_0)}</math> contains only the singularity <math>z_0 \in S </math>, i.e<math>\overline{D_r(z_0)} \cap s = \{z_0\}</math>. Thus, one can read off the 'residue' <math>\mathrm{res}_{z_0}(f) = a_{-1}</math> from the Laurent expansion of around at the <math>f</math> um <math>z_0</math> an -1-ten coefficient of . ==Considerations== The closed disk <math>\overline{D_r(z_0)}</math> must contain only the singularity <math>z_0 \in S</math>, meaning<math>\overline{D_r(z_0)} \cap S = {z_0}</math>. Thus, the residue <math>\mathrm{res}{z_0}(f) = a{-1}</math> can be read off directly as the coefficient of <math>(z-z_0)^{-1}</math> in the Laurent series expansion of <math>f</math> around <math>z_0</math>. ==Etymology== The term "residue" (from Latin residuere – to remain) is used because in integration along the path <math>\gamma(t) := z_0 + r\cdot e^{it}</math> with <math>t \in [0, 2\pi]</math> around the circle centered at <math>z_0</math>, the following holds: <center><math> \begin{array}{rl} \displaystyle \int_{|w-z_0|=r} f(w)\, dw &= \displaystyle \sum_{n=-\infty}^{+\infty} a_n \int_{|w-z_0|=r} (w-z_0)^n \, dw\\ &= 2\pi i \cdot a_{-1}. \end{array} </math></center> Thus, the residue is what remains after integrating. ==Computation for Poles== If <math>z_0 \in U</math> is a pole of order <math>m</math> of <math>f</math>, the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math> has the form: :<math>f(z) = \sum_{k=-m}^\infty a_k (z-z_0)^k</math> with <math>a_{-m} \neq 0</math>. ===Proof 1: Removing Principal Part by Multiplication=== By multiplying with <math>(z-z_0)^m</math>, we get: :<math>g_m(z) := (z-z_0)^m \cdot f(z) = \sum_{k=0}^\infty a_{k-m} (z-z_0)^k.</math> The residue <math>a_{-1}</math> is then the coefficient of <math>(z-z_0)^{m-1}</math> in the power series of <math>g_m(z)</math>. ===Proof 2: Using (m-1)-fold Differentiation=== By differentiating <math>m-1</math> times, the first <math>m-1</math> terms from <math>n=0</math> to <math>n=m-2</math> vanish. The residue is then found as the coefficient of <math>(z-z_0)^0</math> in: :<math>g_m^{(m-1)}(z) = \sum_{k=m-1}^\infty \frac{k!}{(k-m+1)!} a_{k-m}(z-z_0)^{k-m+1}.</math> ===Proof 3: Limit Process to Compute Coefficient of <math>(z-z_0)^0</math>=== By shifting the index: :<math>g_m^{(m-1)}(z) = \sum_{k=-1}^\infty \frac{(m+k)!}{(k+1)!} a_k (z-z_0)^{k+1}.</math> Taking the limit <math>z \to z_0</math>, all terms with <math>k \geq 0</math> vanish, leaving: :<math>\lim_{z \to z_0} g_m^{(m-1)}(z) = \frac{(m-1)!}{0!} \cdot a_{-1} \cdot (z-z_0)^0 = (m-1)! \cdot a_{-1}.</math> Thus, the residue can be computed <math>z \to z_0</math> using: :<math>\mathrm{res}{z_0}(f) = a{-1} = \frac{1}{(m-1)!} \cdot \lim_{z \to z_0} g_m^{(m-1)}(z).</math> ==Exercises for Students== *Explain why, in the Laurent series expansion, all terms from the principal and outer parts, i.e., <math>n \in \mathbb{Z}</math> with <math>n \neq -1</math>, yield integrals that evaluate to zero: :<math>\int_{\partial D_r(z_0)} a_n \cdot (\xi - z_0)^n, d\xi = 0.</math> *Why can the order of integration and series expansion be interchanged? ::<math>\sum_{n=-\infty}^{+\infty} \int_{\partial D_r(z_0)} a_n (\xi-z_0)^n d\xi = \int_{\partial D_r(z_0)} \sum_{n=-\infty}^{+\infty} a_n (\xi-z_0)^n d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} f(\xi), d\xi = \mathrm{res}_{z_0}(f).</math> *Given the function <math>f:\mathbb{C}\setminus {i} \to \mathbb{C}</math> with <math>z \mapsto f(z) = \frac{e^{z-i}}{(z-i)^5}</math>, calculate the residue <math>\mathrm{res}_{z_0}(f)</math> at <math>z_0 := i</math>. ==See Also== *[[Residue|Residue]] *[[Complex_analysis/Examples_of_Laurent_series|Examples of Laurent Series]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Residuum|Kurs:Funktionentheorie/Residuum ]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum * Date: 12/26/2024<span type="translate" src="Kurs:Funktionentheorie/Residuum " srclang="de" date="12/26/2024" time="11:28" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Residuum ]] </noinclude> [[Category:Wiki2Reveal]] 1alqvl07w4v90oc10602r33qjt8proy 2693293 2693292 2024-12-26T12:06:15Z Eshaa2024 2993595 /* Definition */ 2693293 wikitext text/x-wiki ==Definition== Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and let <math>f</math> be holomorphic except for isolated singularities <math>S \subset G</math>, i.e., <math>f\colon G \setminus S \to \mathbb{C}</math> is holomorphic. If <math>z_0 \in S \subset G</math> is an [[Complex_analysis/Isolated_singularity|isolated singularity]] of <math>f</math> with <math>D_r(z_0) \cap S = {z_0}</math>, the residue is defined as: :<math>\mathrm{res}{z_0}(f) := \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{|\xi-z_0|=r} f(\xi), d\xi</math>. == Relation Between Residue and Laurent Series == If <math>f</math> is represented around an isolated singularity <math>z_0 \in S \subset G</math> as a Laurent series, the residue can be computed as follows. With <math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n</math> as the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math>, it holds that: :<math>\mathrm{res}{z_0}(f) = \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} a_{-1}\cdot (\xi-z_0)^{-1}, d\xi = \frac{1}{2\pi i} a_{-1} \cdot \underbrace{\int_{\partial D_r(z_0)} (\xi-z_0)^{-1} , d\xi}{=2\pi i} = a{-1}</math>. It must be taken into account that the closed disk <math>\overline{D_r(z_0)}</math> contains only the singularity <math>z_0 \in S </math>, i.e<math>\overline{D_r(z_0)} \cap s = \{z_0\}</math>. Thus, one can read off the 'residue' <math>\mathrm{res}_{z_0}(f) = a_{-1}</math> from the Laurent expansion of around at the <math>f</math> um <math>z_0</math> an -1-ten coefficient of . ==Considerations== The closed disk <math>\overline{D_r(z_0)}</math> must contain only the singularity <math>z_0 \in S</math>, meaning<math>\overline{D_r(z_0)} \cap S = {z_0}</math>. Thus, the residue <math>\mathrm{res}{z_0}(f) = a{-1}</math> can be read off directly as the coefficient of <math>(z-z_0)^{-1}</math> in the Laurent series expansion of <math>f</math> around <math>z_0</math>. ==Etymology== The term "residue" (from Latin residuere – to remain) is used because in integration along the path <math>\gamma(t) := z_0 + r\cdot e^{it}</math> with <math>t \in [0, 2\pi]</math> around the circle centered at <math>z_0</math>, the following holds: <center><math> \begin{array}{rl} \displaystyle \int_{|w-z_0|=r} f(w)\, dw &= \displaystyle \sum_{n=-\infty}^{+\infty} a_n \int_{|w-z_0|=r} (w-z_0)^n \, dw\\ &= 2\pi i \cdot a_{-1}. \end{array} </math></center> Thus, the residue is what remains after integrating. ==Computation for Poles== If <math>z_0 \in U</math> is a pole of order <math>m</math> of <math>f</math>, the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math> has the form: :<math>f(z) = \sum_{k=-m}^\infty a_k (z-z_0)^k</math> with <math>a_{-m} \neq 0</math>. ===Proof 1: Removing Principal Part by Multiplication=== By multiplying with <math>(z-z_0)^m</math>, we get: :<math>g_m(z) := (z-z_0)^m \cdot f(z) = \sum_{k=0}^\infty a_{k-m} (z-z_0)^k.</math> The residue <math>a_{-1}</math> is then the coefficient of <math>(z-z_0)^{m-1}</math> in the power series of <math>g_m(z)</math>. ===Proof 2: Using (m-1)-fold Differentiation=== By differentiating <math>m-1</math> times, the first <math>m-1</math> terms from <math>n=0</math> to <math>n=m-2</math> vanish. The residue is then found as the coefficient of <math>(z-z_0)^0</math> in: :<math>g_m^{(m-1)}(z) = \sum_{k=m-1}^\infty \frac{k!}{(k-m+1)!} a_{k-m}(z-z_0)^{k-m+1}.</math> ===Proof 3: Limit Process to Compute Coefficient of <math>(z-z_0)^0</math>=== By shifting the index: :<math>g_m^{(m-1)}(z) = \sum_{k=-1}^\infty \frac{(m+k)!}{(k+1)!} a_k (z-z_0)^{k+1}.</math> Taking the limit <math>z \to z_0</math>, all terms with <math>k \geq 0</math> vanish, leaving: :<math>\lim_{z \to z_0} g_m^{(m-1)}(z) = \frac{(m-1)!}{0!} \cdot a_{-1} \cdot (z-z_0)^0 = (m-1)! \cdot a_{-1}.</math> Thus, the residue can be computed <math>z \to z_0</math> using: :<math>\mathrm{res}{z_0}(f) = a{-1} = \frac{1}{(m-1)!} \cdot \lim_{z \to z_0} g_m^{(m-1)}(z).</math> ==Exercises for Students== *Explain why, in the Laurent series expansion, all terms from the principal and outer parts, i.e., <math>n \in \mathbb{Z}</math> with <math>n \neq -1</math>, yield integrals that evaluate to zero: :<math>\int_{\partial D_r(z_0)} a_n \cdot (\xi - z_0)^n, d\xi = 0.</math> *Why can the order of integration and series expansion be interchanged? ::<math>\sum_{n=-\infty}^{+\infty} \int_{\partial D_r(z_0)} a_n (\xi-z_0)^n d\xi = \int_{\partial D_r(z_0)} \sum_{n=-\infty}^{+\infty} a_n (\xi-z_0)^n d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} f(\xi), d\xi = \mathrm{res}_{z_0}(f).</math> *Given the function <math>f:\mathbb{C}\setminus {i} \to \mathbb{C}</math> with <math>z \mapsto f(z) = \frac{e^{z-i}}{(z-i)^5}</math>, calculate the residue <math>\mathrm{res}_{z_0}(f)</math> at <math>z_0 := i</math>. ==See Also== *[[Residue|Residue]] *[[Complex_analysis/Examples_of_Laurent_series|Examples of Laurent Series]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Residuum|Kurs:Funktionentheorie/Residuum ]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum * Date: 12/26/2024<span type="translate" src="Kurs:Funktionentheorie/Residuum " srclang="de" date="12/26/2024" time="11:28" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Residuum ]] </noinclude> [[Category:Wiki2Reveal]] dka9bqwu6fehjngd5jt9gr5n44yohsu 2693294 2693293 2024-12-26T12:09:40Z Eshaa2024 2993595 /* Definition */ 2693294 wikitext text/x-wiki ==Definition== Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and let <math>f</math> be holomorphic except for isolated singularities <math>S \subset G</math>, i.e., <math>f\colon G \setminus S \to \mathbb{C}</math> is holomorphic. If <math>z_0 \in S \subset G</math> is an [[Complex Analysis/Isolated singularity|Isolated singularity]] of <math>f</math> with <math>D_r(z_0) \cap S = {z_0}</math>, the residue is defined as: :<math>\mathrm{res}{z_0}(f) := \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{|\xi-z_0|=r} f(\xi), d\xi</math>. == Relation Between Residue and Laurent Series == If <math>f</math> is represented around an isolated singularity <math>z_0 \in S \subset G</math> as a Laurent series, the residue can be computed as follows. With <math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n</math> as the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math>, it holds that: :<math>\mathrm{res}{z_0}(f) = \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} a_{-1}\cdot (\xi-z_0)^{-1}, d\xi = \frac{1}{2\pi i} a_{-1} \cdot \underbrace{\int_{\partial D_r(z_0)} (\xi-z_0)^{-1} , d\xi}{=2\pi i} = a{-1}</math>. It must be taken into account that the closed disk <math>\overline{D_r(z_0)}</math> contains only the singularity <math>z_0 \in S </math>, i.e<math>\overline{D_r(z_0)} \cap s = \{z_0\}</math>. Thus, one can read off the 'residue' <math>\mathrm{res}_{z_0}(f) = a_{-1}</math> from the Laurent expansion of around at the <math>f</math> um <math>z_0</math> an -1-ten coefficient of . ==Considerations== The closed disk <math>\overline{D_r(z_0)}</math> must contain only the singularity <math>z_0 \in S</math>, meaning<math>\overline{D_r(z_0)} \cap S = {z_0}</math>. Thus, the residue <math>\mathrm{res}{z_0}(f) = a{-1}</math> can be read off directly as the coefficient of <math>(z-z_0)^{-1}</math> in the Laurent series expansion of <math>f</math> around <math>z_0</math>. ==Etymology== The term "residue" (from Latin residuere – to remain) is used because in integration along the path <math>\gamma(t) := z_0 + r\cdot e^{it}</math> with <math>t \in [0, 2\pi]</math> around the circle centered at <math>z_0</math>, the following holds: <center><math> \begin{array}{rl} \displaystyle \int_{|w-z_0|=r} f(w)\, dw &= \displaystyle \sum_{n=-\infty}^{+\infty} a_n \int_{|w-z_0|=r} (w-z_0)^n \, dw\\ &= 2\pi i \cdot a_{-1}. \end{array} </math></center> Thus, the residue is what remains after integrating. ==Computation for Poles== If <math>z_0 \in U</math> is a pole of order <math>m</math> of <math>f</math>, the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math> has the form: :<math>f(z) = \sum_{k=-m}^\infty a_k (z-z_0)^k</math> with <math>a_{-m} \neq 0</math>. ===Proof 1: Removing Principal Part by Multiplication=== By multiplying with <math>(z-z_0)^m</math>, we get: :<math>g_m(z) := (z-z_0)^m \cdot f(z) = \sum_{k=0}^\infty a_{k-m} (z-z_0)^k.</math> The residue <math>a_{-1}</math> is then the coefficient of <math>(z-z_0)^{m-1}</math> in the power series of <math>g_m(z)</math>. ===Proof 2: Using (m-1)-fold Differentiation=== By differentiating <math>m-1</math> times, the first <math>m-1</math> terms from <math>n=0</math> to <math>n=m-2</math> vanish. The residue is then found as the coefficient of <math>(z-z_0)^0</math> in: :<math>g_m^{(m-1)}(z) = \sum_{k=m-1}^\infty \frac{k!}{(k-m+1)!} a_{k-m}(z-z_0)^{k-m+1}.</math> ===Proof 3: Limit Process to Compute Coefficient of <math>(z-z_0)^0</math>=== By shifting the index: :<math>g_m^{(m-1)}(z) = \sum_{k=-1}^\infty \frac{(m+k)!}{(k+1)!} a_k (z-z_0)^{k+1}.</math> Taking the limit <math>z \to z_0</math>, all terms with <math>k \geq 0</math> vanish, leaving: :<math>\lim_{z \to z_0} g_m^{(m-1)}(z) = \frac{(m-1)!}{0!} \cdot a_{-1} \cdot (z-z_0)^0 = (m-1)! \cdot a_{-1}.</math> Thus, the residue can be computed <math>z \to z_0</math> using: :<math>\mathrm{res}{z_0}(f) = a{-1} = \frac{1}{(m-1)!} \cdot \lim_{z \to z_0} g_m^{(m-1)}(z).</math> ==Exercises for Students== *Explain why, in the Laurent series expansion, all terms from the principal and outer parts, i.e., <math>n \in \mathbb{Z}</math> with <math>n \neq -1</math>, yield integrals that evaluate to zero: :<math>\int_{\partial D_r(z_0)} a_n \cdot (\xi - z_0)^n, d\xi = 0.</math> *Why can the order of integration and series expansion be interchanged? ::<math>\sum_{n=-\infty}^{+\infty} \int_{\partial D_r(z_0)} a_n (\xi-z_0)^n d\xi = \int_{\partial D_r(z_0)} \sum_{n=-\infty}^{+\infty} a_n (\xi-z_0)^n d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} f(\xi), d\xi = \mathrm{res}_{z_0}(f).</math> *Given the function <math>f:\mathbb{C}\setminus {i} \to \mathbb{C}</math> with <math>z \mapsto f(z) = \frac{e^{z-i}}{(z-i)^5}</math>, calculate the residue <math>\mathrm{res}_{z_0}(f)</math> at <math>z_0 := i</math>. ==See Also== *[[Residue|Residue]] *[[Complex_analysis/Examples_of_Laurent_series|Examples of Laurent Series]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Residuum|Kurs:Funktionentheorie/Residuum ]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum * Date: 12/26/2024<span type="translate" src="Kurs:Funktionentheorie/Residuum " srclang="de" date="12/26/2024" time="11:28" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Residuum ]] </noinclude> [[Category:Wiki2Reveal]] 2zy210qga87cks9bsl2hcx3f2gmgwna 2693295 2693294 2024-12-26T12:10:40Z Eshaa2024 2993595 /* Definition */ 2693295 wikitext text/x-wiki ==Definition== Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and let <math>f</math> be holomorphic except for isolated singularities <math>S \subset G</math>, i.e., <math>f\colon G \setminus S \to \mathbb{C}</math> is holomorphic. If <math>z_0 \in S \subset G</math> is an [[Complex_analysis/Isolated _singularity|isolated singularity]] of <math>f</math> with <math>D_r(z_0) \cap S = {z_0}</math>, the residue is defined as: :<math>\mathrm{res}{z_0}(f) := \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{|\xi-z_0|=r} f(\xi), d\xi</math>. == Relation Between Residue and Laurent Series == If <math>f</math> is represented around an isolated singularity <math>z_0 \in S \subset G</math> as a Laurent series, the residue can be computed as follows. With <math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n</math> as the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math>, it holds that: :<math>\mathrm{res}{z_0}(f) = \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} a_{-1}\cdot (\xi-z_0)^{-1}, d\xi = \frac{1}{2\pi i} a_{-1} \cdot \underbrace{\int_{\partial D_r(z_0)} (\xi-z_0)^{-1} , d\xi}{=2\pi i} = a{-1}</math>. It must be taken into account that the closed disk <math>\overline{D_r(z_0)}</math> contains only the singularity <math>z_0 \in S </math>, i.e<math>\overline{D_r(z_0)} \cap s = \{z_0\}</math>. Thus, one can read off the 'residue' <math>\mathrm{res}_{z_0}(f) = a_{-1}</math> from the Laurent expansion of around at the <math>f</math> um <math>z_0</math> an -1-ten coefficient of . ==Considerations== The closed disk <math>\overline{D_r(z_0)}</math> must contain only the singularity <math>z_0 \in S</math>, meaning<math>\overline{D_r(z_0)} \cap S = {z_0}</math>. Thus, the residue <math>\mathrm{res}{z_0}(f) = a{-1}</math> can be read off directly as the coefficient of <math>(z-z_0)^{-1}</math> in the Laurent series expansion of <math>f</math> around <math>z_0</math>. ==Etymology== The term "residue" (from Latin residuere – to remain) is used because in integration along the path <math>\gamma(t) := z_0 + r\cdot e^{it}</math> with <math>t \in [0, 2\pi]</math> around the circle centered at <math>z_0</math>, the following holds: <center><math> \begin{array}{rl} \displaystyle \int_{|w-z_0|=r} f(w)\, dw &= \displaystyle \sum_{n=-\infty}^{+\infty} a_n \int_{|w-z_0|=r} (w-z_0)^n \, dw\\ &= 2\pi i \cdot a_{-1}. \end{array} </math></center> Thus, the residue is what remains after integrating. ==Computation for Poles== If <math>z_0 \in U</math> is a pole of order <math>m</math> of <math>f</math>, the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math> has the form: :<math>f(z) = \sum_{k=-m}^\infty a_k (z-z_0)^k</math> with <math>a_{-m} \neq 0</math>. ===Proof 1: Removing Principal Part by Multiplication=== By multiplying with <math>(z-z_0)^m</math>, we get: :<math>g_m(z) := (z-z_0)^m \cdot f(z) = \sum_{k=0}^\infty a_{k-m} (z-z_0)^k.</math> The residue <math>a_{-1}</math> is then the coefficient of <math>(z-z_0)^{m-1}</math> in the power series of <math>g_m(z)</math>. ===Proof 2: Using (m-1)-fold Differentiation=== By differentiating <math>m-1</math> times, the first <math>m-1</math> terms from <math>n=0</math> to <math>n=m-2</math> vanish. The residue is then found as the coefficient of <math>(z-z_0)^0</math> in: :<math>g_m^{(m-1)}(z) = \sum_{k=m-1}^\infty \frac{k!}{(k-m+1)!} a_{k-m}(z-z_0)^{k-m+1}.</math> ===Proof 3: Limit Process to Compute Coefficient of <math>(z-z_0)^0</math>=== By shifting the index: :<math>g_m^{(m-1)}(z) = \sum_{k=-1}^\infty \frac{(m+k)!}{(k+1)!} a_k (z-z_0)^{k+1}.</math> Taking the limit <math>z \to z_0</math>, all terms with <math>k \geq 0</math> vanish, leaving: :<math>\lim_{z \to z_0} g_m^{(m-1)}(z) = \frac{(m-1)!}{0!} \cdot a_{-1} \cdot (z-z_0)^0 = (m-1)! \cdot a_{-1}.</math> Thus, the residue can be computed <math>z \to z_0</math> using: :<math>\mathrm{res}{z_0}(f) = a{-1} = \frac{1}{(m-1)!} \cdot \lim_{z \to z_0} g_m^{(m-1)}(z).</math> ==Exercises for Students== *Explain why, in the Laurent series expansion, all terms from the principal and outer parts, i.e., <math>n \in \mathbb{Z}</math> with <math>n \neq -1</math>, yield integrals that evaluate to zero: :<math>\int_{\partial D_r(z_0)} a_n \cdot (\xi - z_0)^n, d\xi = 0.</math> *Why can the order of integration and series expansion be interchanged? ::<math>\sum_{n=-\infty}^{+\infty} \int_{\partial D_r(z_0)} a_n (\xi-z_0)^n d\xi = \int_{\partial D_r(z_0)} \sum_{n=-\infty}^{+\infty} a_n (\xi-z_0)^n d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} f(\xi), d\xi = \mathrm{res}_{z_0}(f).</math> *Given the function <math>f:\mathbb{C}\setminus {i} \to \mathbb{C}</math> with <math>z \mapsto f(z) = \frac{e^{z-i}}{(z-i)^5}</math>, calculate the residue <math>\mathrm{res}_{z_0}(f)</math> at <math>z_0 := i</math>. ==See Also== *[[Residue|Residue]] *[[Complex_analysis/Examples_of_Laurent_series|Examples of Laurent Series]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Residuum|Kurs:Funktionentheorie/Residuum ]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum * Date: 12/26/2024<span type="translate" src="Kurs:Funktionentheorie/Residuum " srclang="de" date="12/26/2024" time="11:28" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Residuum ]] </noinclude> [[Category:Wiki2Reveal]] own8g3bl1knjepmyjfqyzz8b90byg7v 2693296 2693295 2024-12-26T12:12:59Z Eshaa2024 2993595 /* Definition */ 2693296 wikitext text/x-wiki ==Definition== Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and let <math>f</math> be holomorphic except for isolated singularities <math>S \subset G</math>, i.e., <math>f\colon G \setminus S \to \mathbb{C}</math> is holomorphic. If <math>z_0 \in S \subset G</math> is an [[Complex Analysis/Isolated singularity|Isolated singularity]] of <math>f</math> with <math>D_r(z_0) \cap S = {z_0}</math>, the residue is defined as: :<math>\mathrm{res}{z_0}(f) := \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{|\xi-z_0|=r} f(\xi), d\xi</math>. == Relation Between Residue and Laurent Series == If <math>f</math> is represented around an isolated singularity <math>z_0 \in S \subset G</math> as a Laurent series, the residue can be computed as follows. With <math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n</math> as the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math>, it holds that: :<math>\mathrm{res}{z_0}(f) = \frac{1}{2\pi i} \int{\partial D_r(z_0)} f(\xi), d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} a_{-1}\cdot (\xi-z_0)^{-1}, d\xi = \frac{1}{2\pi i} a_{-1} \cdot \underbrace{\int_{\partial D_r(z_0)} (\xi-z_0)^{-1} , d\xi}{=2\pi i} = a{-1}</math>. It must be taken into account that the closed disk <math>\overline{D_r(z_0)}</math> contains only the singularity <math>z_0 \in S </math>, i.e<math>\overline{D_r(z_0)} \cap s = \{z_0\}</math>. Thus, one can read off the 'residue' <math>\mathrm{res}_{z_0}(f) = a_{-1}</math> from the Laurent expansion of around at the <math>f</math> um <math>z_0</math> an -1-ten coefficient of . ==Considerations== The closed disk <math>\overline{D_r(z_0)}</math> must contain only the singularity <math>z_0 \in S</math>, meaning<math>\overline{D_r(z_0)} \cap S = {z_0}</math>. Thus, the residue <math>\mathrm{res}{z_0}(f) = a{-1}</math> can be read off directly as the coefficient of <math>(z-z_0)^{-1}</math> in the Laurent series expansion of <math>f</math> around <math>z_0</math>. ==Etymology== The term "residue" (from Latin residuere – to remain) is used because in integration along the path <math>\gamma(t) := z_0 + r\cdot e^{it}</math> with <math>t \in [0, 2\pi]</math> around the circle centered at <math>z_0</math>, the following holds: <center><math> \begin{array}{rl} \displaystyle \int_{|w-z_0|=r} f(w)\, dw &= \displaystyle \sum_{n=-\infty}^{+\infty} a_n \int_{|w-z_0|=r} (w-z_0)^n \, dw\\ &= 2\pi i \cdot a_{-1}. \end{array} </math></center> Thus, the residue is what remains after integrating. ==Computation for Poles== If <math>z_0 \in U</math> is a pole of order <math>m</math> of <math>f</math>, the [[Laurent Series|Laurent Series]] expansion of <math>f</math> around <math>z_0</math> has the form: :<math>f(z) = \sum_{k=-m}^\infty a_k (z-z_0)^k</math> with <math>a_{-m} \neq 0</math>. ===Proof 1: Removing Principal Part by Multiplication=== By multiplying with <math>(z-z_0)^m</math>, we get: :<math>g_m(z) := (z-z_0)^m \cdot f(z) = \sum_{k=0}^\infty a_{k-m} (z-z_0)^k.</math> The residue <math>a_{-1}</math> is then the coefficient of <math>(z-z_0)^{m-1}</math> in the power series of <math>g_m(z)</math>. ===Proof 2: Using (m-1)-fold Differentiation=== By differentiating <math>m-1</math> times, the first <math>m-1</math> terms from <math>n=0</math> to <math>n=m-2</math> vanish. The residue is then found as the coefficient of <math>(z-z_0)^0</math> in: :<math>g_m^{(m-1)}(z) = \sum_{k=m-1}^\infty \frac{k!}{(k-m+1)!} a_{k-m}(z-z_0)^{k-m+1}.</math> ===Proof 3: Limit Process to Compute Coefficient of <math>(z-z_0)^0</math>=== By shifting the index: :<math>g_m^{(m-1)}(z) = \sum_{k=-1}^\infty \frac{(m+k)!}{(k+1)!} a_k (z-z_0)^{k+1}.</math> Taking the limit <math>z \to z_0</math>, all terms with <math>k \geq 0</math> vanish, leaving: :<math>\lim_{z \to z_0} g_m^{(m-1)}(z) = \frac{(m-1)!}{0!} \cdot a_{-1} \cdot (z-z_0)^0 = (m-1)! \cdot a_{-1}.</math> Thus, the residue can be computed <math>z \to z_0</math> using: :<math>\mathrm{res}{z_0}(f) = a{-1} = \frac{1}{(m-1)!} \cdot \lim_{z \to z_0} g_m^{(m-1)}(z).</math> ==Exercises for Students== *Explain why, in the Laurent series expansion, all terms from the principal and outer parts, i.e., <math>n \in \mathbb{Z}</math> with <math>n \neq -1</math>, yield integrals that evaluate to zero: :<math>\int_{\partial D_r(z_0)} a_n \cdot (\xi - z_0)^n, d\xi = 0.</math> *Why can the order of integration and series expansion be interchanged? ::<math>\sum_{n=-\infty}^{+\infty} \int_{\partial D_r(z_0)} a_n (\xi-z_0)^n d\xi = \int_{\partial D_r(z_0)} \sum_{n=-\infty}^{+\infty} a_n (\xi-z_0)^n d\xi = \frac{1}{2\pi i} \int_{\partial D_r(z_0)} f(\xi), d\xi = \mathrm{res}_{z_0}(f).</math> *Given the function <math>f:\mathbb{C}\setminus {i} \to \mathbb{C}</math> with <math>z \mapsto f(z) = \frac{e^{z-i}}{(z-i)^5}</math>, calculate the residue <math>\mathrm{res}_{z_0}(f)</math> at <math>z_0 := i</math>. ==See Also== *[[Residue|Residue]] *[[Complex_analysis/Examples_of_Laurent_series|Examples of Laurent Series]] == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Residuum|Kurs:Funktionentheorie/Residuum ]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum * Date: 12/26/2024<span type="translate" src="Kurs:Funktionentheorie/Residuum " srclang="de" date="12/26/2024" time="11:28" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Residuum ]] </noinclude> [[Category:Wiki2Reveal]] 2zy210qga87cks9bsl2hcx3f2gmgwna 0 317507 2693328 2693278 2024-12-26T18:57:23Z Watchduck 137431 Watchduck moved page [[Boolf prop/3-ary/weight pair]] to [[Boolf prop/3-ary/burden]] 2693278 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> [[File:Set of 3-ary Boolean functions 1774342044350424423459706185431556916232666498796281775013496886931783680.svg|thumb|500px|(truth table) [[Boolf prop/3-ary/weight|weight]] 4]] [[File:Set of 3-ary Boolean functions 80069177398200828458670140847044886780.svg|thumb|500px|[[Boolf prop/3-ary/zhegalkin weight|Zhegalkin weight]] 4]] <div 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72, 80, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79229376457598622863384908800.svg|420px]] |- |class="size"| 12 |class="prop"| (3, 6) |class="block"| <span class="block-list">[11, 13, 14, 35, 41, 49, 50, 69, 73, 81, 84, 97]</span>[[File:Set_of_3-ary_Boolean_functions_158478095728312205693471975424.svg|420px]] |- |class="size"| 9 |class="prop"| (4, 2) |class="block"| <span class="block-list">[15, 51, 60, 85, 90, 102, 106, 108, 120]</span>[[File:Set_of_3-ary_Boolean_functions_1329638715856015485898113705029173248.svg|420px]] |- |class="size"| 23 |class="prop"| (4, 4) |class="block"| <span class="block-list small">[23, 27, 29, 30, 39, 45, 46, 53, 54, 57, 58, 71, 75, 78, 83, 86, 89, 92, 99, 101, 105, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103889676774519634310907416364974080.svg|420px]] |- |class="size"| 12 |class="prop"| (5, 4) |class="block"| <span class="block-list">[31, 47, 55, 59, 79, 87, 93, 110, 115, 117, 122, 124]</span>[[File:Set_of_3-ary_Boolean_functions_26793549874313211552840646019958964224.svg|420px]] |- |class="size"| 3 |class="prop"| (3, 2) |class="block"| <span class="block-list">[42, 76, 112]</span>[[File:Set_of_3-ary_Boolean_functions_5192296858610385492260808699150336.svg|420px]] |- |class="size"| 3 |class="prop"| (4, 6) |class="block"| <span class="block-list">[43, 77, 113]</span>[[File:Set_of_3-ary_Boolean_functions_10384593717220770984521617398300672.svg|420px]] |- |class="size"| 9 |class="prop"| (5, 6) |class="block"| <span class="block-list">[61, 62, 91, 94, 103, 107, 109, 118, 121]</span>[[File:Set_of_3-ary_Boolean_functions_2991584450387929320740109725133176832.svg|420px]] |- |class="size"| 3 |class="prop"| (6, 2) |class="block"| <span class="block-list">[63, 95, 119]</span>[[File:Set_of_3-ary_Boolean_functions_664614037506539202807444363766923264.svg|420px]] |- |class="size"| 3 |class="prop"| (6, 4) |class="block"| <span class="block-list">[111, 123, 125]</span>[[File:Set_of_3-ary_Boolean_functions_53171715979825902329966547659378393088.svg|420px]] |- |class="size"| 1 |class="prop"| (6, 6) |class="block"| <span class="block-list">[126]</span>[[File:Set_of_3-ary_Boolean_functions_85070591730234615865843651857942052864.svg|420px]] |- |class="size"| 1 |class="prop"| (7, 2) |class="block"| <span class="block-list">[127]</span>[[File:Set_of_3-ary_Boolean_functions_170141183460469231731687303715884105728.svg|420px]] |- |class="size"| 1 |class="prop"| (1, 1) |class="block"| <span class="block-list">[128]</span>[[File:Set_of_3-ary_Boolean_functions_340282366920938463463374607431768211456.svg|420px]] |- |class="size"| 1 |class="prop"| (2, 7) |class="block"| <span class="block-list">[129]</span>[[File:Set_of_3-ary_Boolean_functions_680564733841876926926749214863536422912.svg|420px]] |- |class="size"| 3 |class="prop"| (2, 3) |class="block"| <span class="block-list">[130, 132, 144]</span>[[File:Set_of_3-ary_Boolean_functions_22307550845869041910804985764796996870209536.svg|420px]] |- |class="size"| 9 |class="prop"| (3, 5) |class="block"| <span class="block-list">[131, 133, 134, 145, 146, 148, 152, 164, 194]</span>[[File:Set_of_3-ary_Boolean_functions_25108406964936948895192162057100570117740705509753615286272.svg|420px]] |- |class="size"| 20 |class="prop"| (4, 3) |class="block"| <span class="block-list small">[135, 147, 149, 150, 153, 154, 156, 165, 166, 172, 180, 184, 195, 198, 202, 210, 216, 226, 228, 232]</span>[[File:Set_of_3-ary_Boolean_functions_7441052244809532022255575855098061541171806476706749552323906890629120.svg|420px]] |- |class="size"| 3 |class="prop"| (2, 1) |class="block"| <span class="block-list">[136, 160, 192]</span>[[File:Set_of_3-ary_Boolean_functions_6277101736848182488278978273171597895442537966652629188608.svg|420px]] |- |class="size"| 3 |class="prop"| (3, 7) |class="block"| <span class="block-list">[137, 161, 193]</span>[[File:Set_of_3-ary_Boolean_functions_12554203473696364976557956546343195790885075933305258377216.svg|420px]] |- |class="size"| 9 |class="prop"| (3, 3) |class="block"| <span class="block-list">[138, 140, 162, 168, 176, 196, 200, 208, 224]</span>[[File:Set_of_3-ary_Boolean_functions_26960359750661738282316453158984117888600841619383213102677675212800.svg|420px]] |- |class="size"| 12 |class="prop"| (4, 5) |class="block"| <span class="block-list">[139, 141, 142, 163, 169, 177, 178, 197, 201, 209, 212, 225]</span>[[File:Set_of_3-ary_Boolean_functions_53927301519553144512686748242099354398634187143454433371679667257344.svg|420px]] |- |class="size"| 9 |class="prop"| (5, 3) |class="block"| <span class="block-list">[143, 179, 188, 213, 218, 230, 234, 236, 248]</span>[[File:Set_of_3-ary_Boolean_functions_452452609381202100815903215357421756553903823647912245591481221591322329088.svg|420px]] |- |class="size"| 23 |class="prop"| (5, 5) |class="block"| <span class="block-list small">[151, 155, 157, 158, 167, 173, 174, 181, 182, 185, 186, 199, 203, 206, 211, 214, 217, 220, 227, 229, 233, 242, 244]</span>[[File:Set_of_3-ary_Boolean_functions_35351825111484788975203103024835962725810281641519939081497323445399060480.svg|420px]] |- |class="size"| 12 |class="prop"| (6, 3) |class="block"| <span class="block-list">[159, 175, 183, 187, 207, 215, 221, 238, 243, 245, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9117372569445512904238451931420414304594892851168509083640346829510770950144.svg|420px]] |- |class="size"| 3 |class="prop"| (4, 1) |class="block"| <span class="block-list">[170, 204, 240]</span>[[File:Set_of_3-ary_Boolean_functions_1766847064804095338292937922828216753893183071784675679834258561581449216.svg|420px]] |- |class="size"| 3 |class="prop"| (5, 7) |class="block"| <span class="block-list">[171, 205, 241]</span>[[File:Set_of_3-ary_Boolean_functions_3533694129608190676585875845656433507786366143569351359668517123162898432.svg|420px]] |- |class="size"| 9 |class="prop"| (6, 5) |class="block"| <span class="block-list">[189, 190, 219, 222, 231, 235, 237, 246, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1017983437621879394163401061060446564310147155177051195411448556160016187392.svg|420px]] |- |class="size"| 3 |class="prop"| (7, 3) |class="block"| <span class="block-list">[191, 223, 247]</span>[[File:Set_of_3-ary_Boolean_functions_226156437771606530900532845121941495453627987222484645961029556549277712384.svg|420px]] |- |class="size"| 3 |class="prop"| (7, 5) |class="block"| <span class="block-list">[239, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_18093397366863044727097758056357857061331348845954775940644308304337072816128.svg|420px]] |- |class="size"| 1 |class="prop"| (7, 7) |class="block"| <span class="block-list">[254]</span>[[File:Set_of_3-ary_Boolean_functions_28948022309329048855892746252171976963317496166410141009864396001978282409984.svg|420px]] |- |class="size"| 1 |class="prop"| (8, 1) |class="block"| <span class="block-list">[255]</span>[[File:Set_of_3-ary_Boolean_functions_57896044618658097711785492504343953926634992332820282019728792003956564819968.svg|420px]] |} [[Category:Boolf prop/3-ary|weight pair]] rgsdjfp9h8sonebwo3d3i5pydf3p8ot 0 317509 2693333 2693284 2024-12-26T18:59:56Z Watchduck 137431 Watchduck moved page [[Boolf prop/3-ary/battalion]] to [[Boolf prop/3-ary/super chunky burden]] 2693284 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">7</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">8</span> + <span class="count">3</span>⋅<span class="size">24</span> + <span class="count">1</span>⋅<span class="size">48</span> + <span class="count">1</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">72</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| battalion !class="block"| block |- |class="size"| 8 |class="prop"| ((0, 0), (6, 6)) |class="block"| <span class="block-list">[0, 1, 126, 127, 128, 129, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844066927987146567678238756515930891228547375183942267580842783803978022915.svg|420px]] |- |class="size"| 24 |class="prop"| ((1, 4), (5, 4)) |class="block"| <span class="block-list small">[2, 3, 4, 5, 16, 17, 110, 111, 122, 123, 124, 125, 130, 131, 132, 133, 144, 145, 238, 239, 250, 251, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_27140096050294567090646637084536852514729318450027635179418706668985287639100.svg|420px]] |- |class="size"| 48 |class="prop"| ((2, 4), (4, 2)) |class="block"| <span class="block-list small">[6, 7, 18, 19, 20, 21, 24, 25, 36, 37, 60, 61, 66, 67, 90, 91, 102, 103, 106, 107, 108, 109, 120, 121, 134, 135, 146, 147, 148, 149, 152, 153, 164, 165, 188, 189, 194, 195, 218, 219, 230, 231, 234, 235, 236, 237, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1357357828104114268394922852559926565150399615546410821754293924513292484800.svg|420px]] |- |class="size"| 24 |class="prop"| ((1, 2), (5, 6)) |class="block"| <span class="block-list small">[8, 9, 32, 33, 62, 63, 64, 65, 94, 95, 118, 119, 136, 137, 160, 161, 190, 191, 192, 193, 222, 223, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339234656657409815182104478227459708018373721404780464128709633561468076800.svg|420px]] |- |class="size"| 72 |class="prop"| ((2, 2), (4, 4)) |class="block"| [[File:Set_of_3-ary_Boolean_functions_106010909902065063019574201811588137846492846153495048081510250921936829440.svg|420px]] |- |class="size"| 24 |class="prop"| ((3, 2), (3, 6)) |class="block"| <span class="block-list small">[14, 15, 42, 43, 50, 51, 76, 77, 84, 85, 112, 113, 142, 143, 170, 171, 178, 179, 204, 205, 212, 213, 240, 241]</span>[[File:Set_of_3-ary_Boolean_functions_5300541214158340703882657930010422655050990403499600699345936983898767360.svg|420px]] |- |class="size"| 56 |class="prop"| ((3, 4), (3, 4)) |class="block"| <span class="block-list small">[22, 23, 26, 27, 28, 29, 38, 39, 44, 45, 52, 53, 56, 57, 70, 71, 74, 75, 82, 83, 88, 89, 98, 99, 100, 101, 104, 105, 150, 151, 154, 155, 156, 157, 166, 167, 172, 173, 180, 181, 184, 185, 198, 199, 202, 203, 210, 211, 216, 217, 226, 227, 228, 229, 232, 233]</span>[[File:Set_of_3-ary_Boolean_functions_22323156734277945624977106139613641801666920800657794174809143267819520.svg|420px]] |} [[Category:Boolf prop/3-ary|battalion]] 9uwwmaruv2z20ozrrk07zpiirmwv9ah 2693336 2693333 2024-12-26T19:06:08Z Watchduck 137431 2693336 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">7</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">8</span> + <span class="count">3</span>⋅<span class="size">24</span> + <span class="count">1</span>⋅<span class="size">48</span> + <span class="count">1</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">72</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| super chunky burden !class="block"| block |- |class="size"| 8 |class="prop"| (0, 0),<br>(1, 1),<br>(1, 8),<br>(2, 7),<br>(6, 6),<br>(7, 2),<br>(7, 7),<br>(8, 1) |class="block"| <span class="block-list">[0, 1, 126, 127, 128, 129, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844066927987146567678238756515930891228547375183942267580842783803978022915.svg|420px]] |- |class="size"| 24 |class="prop"| (1, 4),<br>(2, 3),<br>(2, 4),<br>(3, 5),<br>(5, 4),<br>(6, 3),<br>(6, 4),<br>(7, 5) |class="block"| <span class="block-list small">[2, 3, 4, 5, 16, 17, 110, 111, 122, 123, 124, 125, 130, 131, 132, 133, 144, 145, 238, 239, 250, 251, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_27140096050294567090646637084536852514729318450027635179418706668985287639100.svg|420px]] |- |class="size"| 48 |class="prop"| (2, 4),<br>(3, 4),<br>(3, 5),<br>(4, 2),<br>(4, 3),<br>(5, 3),<br>(5, 6),<br>(6, 5) |class="block"| <span class="block-list small">[6, 7, 18, 19, 20, 21, 24, 25, 36, 37, 60, 61, 66, 67, 90, 91, 102, 103, 106, 107, 108, 109, 120, 121, 134, 135, 146, 147, 148, 149, 152, 153, 164, 165, 188, 189, 194, 195, 218, 219, 230, 231, 234, 235, 236, 237, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1357357828104114268394922852559926565150399615546410821754293924513292484800.svg|420px]] |- |class="size"| 24 |class="prop"| (1, 2),<br>(2, 1),<br>(2, 6),<br>(3, 7),<br>(5, 6),<br>(6, 2),<br>(6, 5),<br>(7, 3) |class="block"| <span class="block-list small">[8, 9, 32, 33, 62, 63, 64, 65, 94, 95, 118, 119, 136, 137, 160, 161, 190, 191, 192, 193, 222, 223, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339234656657409815182104478227459708018373721404780464128709633561468076800.svg|420px]] |- |class="size"| 72 |class="prop"| (2, 2),<br>(3, 3),<br>(3, 6),<br>(4, 4),<br>(4, 5),<br>(5, 4),<br>(5, 5),<br>(6, 3) |class="block"| [[File:Set_of_3-ary_Boolean_functions_106010909902065063019574201811588137846492846153495048081510250921936829440.svg|420px]] |- |class="size"| 24 |class="prop"| (3, 2),<br>(3, 6),<br>(4, 1),<br>(4, 2),<br>(4, 5),<br>(4, 6),<br>(5, 3),<br>(5, 7) |class="block"| <span class="block-list small">[14, 15, 42, 43, 50, 51, 76, 77, 84, 85, 112, 113, 142, 143, 170, 171, 178, 179, 204, 205, 212, 213, 240, 241]</span>[[File:Set_of_3-ary_Boolean_functions_5300541214158340703882657930010422655050990403499600699345936983898767360.svg|420px]] |- |class="size"| 56 |class="prop"| (3, 4),<br>(3, 4),<br>(4, 3),<br>(4, 3),<br>(4, 4),<br>(4, 4),<br>(5, 5),<br>(5, 5) |class="block"| <span class="block-list small">[22, 23, 26, 27, 28, 29, 38, 39, 44, 45, 52, 53, 56, 57, 70, 71, 74, 75, 82, 83, 88, 89, 98, 99, 100, 101, 104, 105, 150, 151, 154, 155, 156, 157, 166, 167, 172, 173, 180, 181, 184, 185, 198, 199, 202, 203, 210, 211, 216, 217, 226, 227, 228, 229, 232, 233]</span>[[File:Set_of_3-ary_Boolean_functions_22323156734277945624977106139613641801666920800657794174809143267819520.svg|420px]] |} [[Category:Boolf prop/3-ary|super chunky burden]] 4q3wld1sn64ofns044dy5qs13za7r0c 2693338 2693336 2024-12-26T19:08:44Z Watchduck 137431 2693338 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">7</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">8</span> + <span class="count">3</span>⋅<span class="size">24</span> + <span class="count">1</span>⋅<span class="size">48</span> + <span class="count">1</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">72</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| super chunky burden !class="block"| block |- |class="size"| 8 |class="prop"| (0, 0),<br>(1, 1),<br>(1, 8),<br>(2, 7),<br>(6, 6),<br>(7, 2),<br>(7, 7),<br>(8, 1)&nbsp; |class="block"| <span class="block-list">[0, 1, 126, 127, 128, 129, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844066927987146567678238756515930891228547375183942267580842783803978022915.svg|420px]] |- |class="size"| 24 |class="prop"| (1, 4),<br>(2, 3),<br>(2, 4),<br>(3, 5),<br>(5, 4),<br>(6, 3),<br>(6, 4),<br>(7, 5)&nbsp; |class="block"| <span class="block-list small">[2, 3, 4, 5, 16, 17, 110, 111, 122, 123, 124, 125, 130, 131, 132, 133, 144, 145, 238, 239, 250, 251, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_27140096050294567090646637084536852514729318450027635179418706668985287639100.svg|420px]] |- |class="size"| 48 |class="prop"| (2, 4),<br>(3, 4),<br>(3, 5),<br>(4, 2),<br>(4, 3),<br>(5, 3),<br>(5, 6),<br>(6, 5)&nbsp; |class="block"| <span class="block-list small">[6, 7, 18, 19, 20, 21, 24, 25, 36, 37, 60, 61, 66, 67, 90, 91, 102, 103, 106, 107, 108, 109, 120, 121, 134, 135, 146, 147, 148, 149, 152, 153, 164, 165, 188, 189, 194, 195, 218, 219, 230, 231, 234, 235, 236, 237, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1357357828104114268394922852559926565150399615546410821754293924513292484800.svg|420px]] |- |class="size"| 24 |class="prop"| (1, 2),<br>(2, 1),<br>(2, 6),<br>(3, 7),<br>(5, 6),<br>(6, 2),<br>(6, 5),<br>(7, 3)&nbsp; |class="block"| <span class="block-list small">[8, 9, 32, 33, 62, 63, 64, 65, 94, 95, 118, 119, 136, 137, 160, 161, 190, 191, 192, 193, 222, 223, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339234656657409815182104478227459708018373721404780464128709633561468076800.svg|420px]] |- |class="size"| 72 |class="prop"| (2, 2),<br>(3, 3),<br>(3, 6),<br>(4, 4),<br>(4, 5),<br>(5, 4),<br>(5, 5),<br>(6, 3)&nbsp; |class="block"| [[File:Set_of_3-ary_Boolean_functions_106010909902065063019574201811588137846492846153495048081510250921936829440.svg|420px]] |- |class="size"| 24 |class="prop"| (3, 2),<br>(3, 6),<br>(4, 1),<br>(4, 2),<br>(4, 5),<br>(4, 6),<br>(5, 3),<br>(5, 7)&nbsp; |class="block"| <span class="block-list small">[14, 15, 42, 43, 50, 51, 76, 77, 84, 85, 112, 113, 142, 143, 170, 171, 178, 179, 204, 205, 212, 213, 240, 241]</span>[[File:Set_of_3-ary_Boolean_functions_5300541214158340703882657930010422655050990403499600699345936983898767360.svg|420px]] |- |class="size"| 56 |class="prop"| (3, 4),<br>(3, 4),<br>(4, 3),<br>(4, 3),<br>(4, 4),<br>(4, 4),<br>(5, 5),<br>(5, 5)&nbsp; |class="block"| <span class="block-list small">[22, 23, 26, 27, 28, 29, 38, 39, 44, 45, 52, 53, 56, 57, 70, 71, 74, 75, 82, 83, 88, 89, 98, 99, 100, 101, 104, 105, 150, 151, 154, 155, 156, 157, 166, 167, 172, 173, 180, 181, 184, 185, 198, 199, 202, 203, 210, 211, 216, 217, 226, 227, 228, 229, 232, 233]</span>[[File:Set_of_3-ary_Boolean_functions_22323156734277945624977106139613641801666920800657794174809143267819520.svg|420px]] |} [[Category:Boolf prop/3-ary|super chunky burden]] 8fenijkhgabn2gyyjtah2cva0ipi6a9 0 317510 2693326 2693286 2024-12-26T18:56:18Z Watchduck 137431 Watchduck moved page [[Boolf prop/3-ary/half battalion]] to [[Boolf prop/3-ary/chunky burden]] 2693286 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">13</span></span> Integer partition: &nbsp; <span class="count">2</span>⋅<span class="size">4</span> + <span class="count">6</span>⋅<span class="size">12</span> + <span class="count">2</span>⋅<span class="size">24</span> + <span class="count">2</span>⋅<span class="size">36</span> + <span class="count">1</span>⋅<span class="size">56</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| half battalion !class="block"| block |- |class="size"| 4 |class="prop"| (0, 0) |class="block"| <span class="block-list">[0, 1, 128, 129]</span>[[File:Set_of_3-ary_Boolean_functions_1020847100762815390390123822295304634371.svg|420px]] |- |class="size"| 12 |class="prop"| (1, 4) |class="block"| <span class="block-list">[2, 3, 4, 5, 16, 17, 130, 131, 132, 133, 144, 145]</span>[[File:Set_of_3-ary_Boolean_functions_66922652537607125732414957294390990610825276.svg|420px]] |- |class="size"| 24 |class="prop"| (2, 4) |class="block"| <span class="block-list small">[6, 7, 18, 19, 20, 21, 24, 25, 36, 37, 66, 67, 134, 135, 146, 147, 148, 149, 152, 153, 164, 165, 194, 195]</span>[[File:Set_of_3-ary_Boolean_functions_75325220894810712840271410957050245523528888676370351915200.svg|420px]] |- |class="size"| 12 |class="prop"| (1, 2) |class="block"| <span class="block-list">[8, 9, 32, 33, 64, 65, 136, 137, 160, 161, 192, 193]</span>[[File:Set_of_3-ary_Boolean_functions_18831305210544547464836934819514793686382954132191901123328.svg|420px]] |- |class="size"| 36 |class="prop"| (2, 2) |class="block"| <span class="block-list small">[10, 11, 12, 13, 34, 35, 40, 41, 48, 49, 68, 69, 72, 73, 80, 81, 96, 97, 138, 139, 140, 141, 162, 163, 168, 169, 176, 177, 196, 197, 200, 201, 208, 209, 224, 225]</span>[[File:Set_of_3-ary_Boolean_functions_80881079251985214846949359476952353666040212987522435176623180364800.svg|420px]] |- |class="size"| 12 |class="prop"| (3, 6) |class="block"| <span class="block-list">[14, 15, 50, 51, 84, 85, 142, 143, 178, 179, 212, 213]</span>[[File:Set_of_3-ary_Boolean_functions_19746054689003844161525772393355864297569742503366378873056968704.svg|420px]] |- |class="size"| 56 |class="prop"| (3, 4) |class="block"| <span class="block-list small">[22, 23, 26, 27, 28, 29, 38, 39, 44, 45, 52, 53, 56, 57, 70, 71, 74, 75, 82, 83, 88, 89, 98, 99, 100, 101, 104, 105, 150, 151, 154, 155, 156, 157, 166, 167, 172, 173, 180, 181, 184, 185, 198, 199, 202, 203, 210, 211, 216, 217, 226, 227, 228, 229, 232, 233]</span>[[File:Set_of_3-ary_Boolean_functions_22323156734277945624977106139613641801666920800657794174809143267819520.svg|420px]] |- |class="size"| 36 |class="prop"| (4, 4) |class="block"| <span class="block-list small">[30, 31, 46, 47, 54, 55, 58, 59, 78, 79, 86, 87, 92, 93, 114, 115, 116, 117, 158, 159, 174, 175, 182, 183, 186, 187, 206, 207, 214, 215, 220, 221, 242, 243, 244, 245]</span>[[File:Set_of_3-ary_Boolean_functions_106010829020985811034359354862228660894139180113282060559075074298756464640.svg|420px]] |- |class="size"| 12 |class="prop"| (3, 2) |class="block"| <span class="block-list">[42, 43, 76, 77, 112, 113, 170, 171, 204, 205, 240, 241]</span>[[File:Set_of_3-ary_Boolean_functions_5300541194412286014878813768484650261695126105929858195979558110841798656.svg|420px]] |- |class="size"| 24 |class="prop"| (4, 2) |class="block"| <span class="block-list small">[60, 61, 90, 91, 102, 103, 106, 107, 108, 109, 120, 121, 188, 189, 218, 219, 230, 231, 234, 235, 236, 237, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1357357828104114193069701957749213724878988658496165298225405248142940569600.svg|420px]] |- |class="size"| 12 |class="prop"| (5, 6) |class="block"| <span class="block-list">[62, 63, 94, 95, 118, 119, 190, 191, 222, 223, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339234656657409796350799267682912243181438901889986777745755501369566953472.svg|420px]] |- |class="size"| 12 |class="prop"| (5, 4) |class="block"| <span class="block-list">[110, 111, 122, 123, 124, 125, 238, 239, 250, 251, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_27140096050294567090646637084536785592076780842901902764461412277994676813824.svg|420px]] |- |class="size"| 4 |class="prop"| (6, 6) |class="block"| <span class="block-list">[126, 127, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844066927987146567678238756515930890207700274421126877190718961508673388544.svg|420px]] |} [[Category:Boolf prop/3-ary|half battalion]] lw9k1eql8p7jv130d7kb1b8ezrc1cgd 2693332 2693326 2024-12-26T18:58:18Z Watchduck 137431 2693332 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">13</span></span> Integer partition: &nbsp; <span class="count">2</span>⋅<span class="size">4</span> + <span class="count">6</span>⋅<span class="size">12</span> + <span class="count">2</span>⋅<span class="size">24</span> + <span class="count">2</span>⋅<span class="size">36</span> + <span class="count">1</span>⋅<span class="size">56</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chunky burden !class="block"| block |- |class="size"| 4 |class="prop"| ((0, 0), (1, 1), (1, 8), (2, 7)) |class="block"| <span class="block-list">[0, 1, 128, 129]</span>[[File:Set_of_3-ary_Boolean_functions_1020847100762815390390123822295304634371.svg|420px]] |- |class="size"| 12 |class="prop"| ((1, 4), (2, 3), (2, 4), (3, 5)) |class="block"| <span class="block-list">[2, 3, 4, 5, 16, 17, 130, 131, 132, 133, 144, 145]</span>[[File:Set_of_3-ary_Boolean_functions_66922652537607125732414957294390990610825276.svg|420px]] |- |class="size"| 24 |class="prop"| ((2, 4), (3, 4), (3, 5), (4, 3)) |class="block"| <span class="block-list small">[6, 7, 18, 19, 20, 21, 24, 25, 36, 37, 66, 67, 134, 135, 146, 147, 148, 149, 152, 153, 164, 165, 194, 195]</span>[[File:Set_of_3-ary_Boolean_functions_75325220894810712840271410957050245523528888676370351915200.svg|420px]] |- |class="size"| 12 |class="prop"| ((1, 2), (2, 1), (2, 6), (3, 7)) |class="block"| <span class="block-list">[8, 9, 32, 33, 64, 65, 136, 137, 160, 161, 192, 193]</span>[[File:Set_of_3-ary_Boolean_functions_18831305210544547464836934819514793686382954132191901123328.svg|420px]] |- |class="size"| 36 |class="prop"| ((2, 2), (3, 3), (3, 6), (4, 5)) |class="block"| <span class="block-list small">[10, 11, 12, 13, 34, 35, 40, 41, 48, 49, 68, 69, 72, 73, 80, 81, 96, 97, 138, 139, 140, 141, 162, 163, 168, 169, 176, 177, 196, 197, 200, 201, 208, 209, 224, 225]</span>[[File:Set_of_3-ary_Boolean_functions_80881079251985214846949359476952353666040212987522435176623180364800.svg|420px]] |- |class="size"| 12 |class="prop"| ((3, 6), (4, 2), (4, 5), (5, 3)) |class="block"| <span class="block-list">[14, 15, 50, 51, 84, 85, 142, 143, 178, 179, 212, 213]</span>[[File:Set_of_3-ary_Boolean_functions_19746054689003844161525772393355864297569742503366378873056968704.svg|420px]] |- |class="size"| 56 |class="prop"| ((3, 4), (4, 3), (4, 4), (5, 5)) |class="block"| <span class="block-list small">[22, 23, 26, 27, 28, 29, 38, 39, 44, 45, 52, 53, 56, 57, 70, 71, 74, 75, 82, 83, 88, 89, 98, 99, 100, 101, 104, 105, 150, 151, 154, 155, 156, 157, 166, 167, 172, 173, 180, 181, 184, 185, 198, 199, 202, 203, 210, 211, 216, 217, 226, 227, 228, 229, 232, 233]</span>[[File:Set_of_3-ary_Boolean_functions_22323156734277945624977106139613641801666920800657794174809143267819520.svg|420px]] |- |class="size"| 36 |class="prop"| ((4, 4), (5, 4), (5, 5), (6, 3)) |class="block"| <span class="block-list small">[30, 31, 46, 47, 54, 55, 58, 59, 78, 79, 86, 87, 92, 93, 114, 115, 116, 117, 158, 159, 174, 175, 182, 183, 186, 187, 206, 207, 214, 215, 220, 221, 242, 243, 244, 245]</span>[[File:Set_of_3-ary_Boolean_functions_106010829020985811034359354862228660894139180113282060559075074298756464640.svg|420px]] |- |class="size"| 12 |class="prop"| ((3, 2), (4, 1), (4, 6), (5, 7)) |class="block"| <span class="block-list">[42, 43, 76, 77, 112, 113, 170, 171, 204, 205, 240, 241]</span>[[File:Set_of_3-ary_Boolean_functions_5300541194412286014878813768484650261695126105929858195979558110841798656.svg|420px]] |- |class="size"| 24 |class="prop"| ((4, 2), (5, 3), (5, 6), (6, 5)) |class="block"| <span class="block-list small">[60, 61, 90, 91, 102, 103, 106, 107, 108, 109, 120, 121, 188, 189, 218, 219, 230, 231, 234, 235, 236, 237, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1357357828104114193069701957749213724878988658496165298225405248142940569600.svg|420px]] |- |class="size"| 12 |class="prop"| ((5, 6), (6, 2), (6, 5), (7, 3)) |class="block"| <span class="block-list">[62, 63, 94, 95, 118, 119, 190, 191, 222, 223, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339234656657409796350799267682912243181438901889986777745755501369566953472.svg|420px]] |- |class="size"| 12 |class="prop"| ((5, 4), (6, 3), (6, 4), (7, 5)) |class="block"| <span class="block-list">[110, 111, 122, 123, 124, 125, 238, 239, 250, 251, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_27140096050294567090646637084536785592076780842901902764461412277994676813824.svg|420px]] |- |class="size"| 4 |class="prop"| ((6, 6), (7, 2), (7, 7), (8, 1)) |class="block"| <span class="block-list">[126, 127, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844066927987146567678238756515930890207700274421126877190718961508673388544.svg|420px]] |} [[Category:Boolf prop/3-ary|chunky burden]] hilyvgjp4xehaei8j3uh99yte8nwotb 2693335 2693332 2024-12-26T19:05:35Z Watchduck 137431 2693335 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">13</span></span> Integer partition: &nbsp; <span class="count">2</span>⋅<span class="size">4</span> + <span class="count">6</span>⋅<span class="size">12</span> + <span class="count">2</span>⋅<span class="size">24</span> + <span class="count">2</span>⋅<span class="size">36</span> + <span class="count">1</span>⋅<span class="size">56</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chunky burden !class="block"| block |- |class="size"| 4 |class="prop"| (0, 0),<br>(1, 1),<br>(1, 8),<br>(2, 7) |class="block"| <span class="block-list">[0, 1, 128, 129]</span>[[File:Set_of_3-ary_Boolean_functions_1020847100762815390390123822295304634371.svg|420px]] |- |class="size"| 12 |class="prop"| (1, 4),<br>(2, 3),<br>(2, 4),<br>(3, 5) |class="block"| <span class="block-list">[2, 3, 4, 5, 16, 17, 130, 131, 132, 133, 144, 145]</span>[[File:Set_of_3-ary_Boolean_functions_66922652537607125732414957294390990610825276.svg|420px]] |- |class="size"| 24 |class="prop"| (2, 4),<br>(3, 4),<br>(3, 5),<br>(4, 3) |class="block"| <span class="block-list small">[6, 7, 18, 19, 20, 21, 24, 25, 36, 37, 66, 67, 134, 135, 146, 147, 148, 149, 152, 153, 164, 165, 194, 195]</span>[[File:Set_of_3-ary_Boolean_functions_75325220894810712840271410957050245523528888676370351915200.svg|420px]] |- |class="size"| 12 |class="prop"| (1, 2),<br>(2, 1),<br>(2, 6),<br>(3, 7) |class="block"| <span class="block-list">[8, 9, 32, 33, 64, 65, 136, 137, 160, 161, 192, 193]</span>[[File:Set_of_3-ary_Boolean_functions_18831305210544547464836934819514793686382954132191901123328.svg|420px]] |- |class="size"| 36 |class="prop"| (2, 2),<br>(3, 3),<br>(3, 6),<br>(4, 5) |class="block"| <span class="block-list small">[10, 11, 12, 13, 34, 35, 40, 41, 48, 49, 68, 69, 72, 73, 80, 81, 96, 97, 138, 139, 140, 141, 162, 163, 168, 169, 176, 177, 196, 197, 200, 201, 208, 209, 224, 225]</span>[[File:Set_of_3-ary_Boolean_functions_80881079251985214846949359476952353666040212987522435176623180364800.svg|420px]] |- |class="size"| 12 |class="prop"| (3, 6),<br>(4, 2),<br>(4, 5),<br>(5, 3) |class="block"| <span class="block-list">[14, 15, 50, 51, 84, 85, 142, 143, 178, 179, 212, 213]</span>[[File:Set_of_3-ary_Boolean_functions_19746054689003844161525772393355864297569742503366378873056968704.svg|420px]] |- |class="size"| 56 |class="prop"| (3, 4),<br>(4, 3),<br>(4, 4),<br>(5, 5) |class="block"| <span class="block-list small">[22, 23, 26, 27, 28, 29, 38, 39, 44, 45, 52, 53, 56, 57, 70, 71, 74, 75, 82, 83, 88, 89, 98, 99, 100, 101, 104, 105, 150, 151, 154, 155, 156, 157, 166, 167, 172, 173, 180, 181, 184, 185, 198, 199, 202, 203, 210, 211, 216, 217, 226, 227, 228, 229, 232, 233]</span>[[File:Set_of_3-ary_Boolean_functions_22323156734277945624977106139613641801666920800657794174809143267819520.svg|420px]] |- |class="size"| 36 |class="prop"| (4, 4),<br>(5, 4),<br>(5, 5),<br>(6, 3) |class="block"| <span class="block-list small">[30, 31, 46, 47, 54, 55, 58, 59, 78, 79, 86, 87, 92, 93, 114, 115, 116, 117, 158, 159, 174, 175, 182, 183, 186, 187, 206, 207, 214, 215, 220, 221, 242, 243, 244, 245]</span>[[File:Set_of_3-ary_Boolean_functions_106010829020985811034359354862228660894139180113282060559075074298756464640.svg|420px]] |- |class="size"| 12 |class="prop"| (3, 2),<br>(4, 1),<br>(4, 6),<br>(5, 7) |class="block"| <span class="block-list">[42, 43, 76, 77, 112, 113, 170, 171, 204, 205, 240, 241]</span>[[File:Set_of_3-ary_Boolean_functions_5300541194412286014878813768484650261695126105929858195979558110841798656.svg|420px]] |- |class="size"| 24 |class="prop"| (4, 2),<br>(5, 3),<br>(5, 6),<br>(6, 5) |class="block"| <span class="block-list small">[60, 61, 90, 91, 102, 103, 106, 107, 108, 109, 120, 121, 188, 189, 218, 219, 230, 231, 234, 235, 236, 237, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1357357828104114193069701957749213724878988658496165298225405248142940569600.svg|420px]] |- |class="size"| 12 |class="prop"| (5, 6),<br>(6, 2),<br>(6, 5),<br>(7, 3) |class="block"| <span class="block-list">[62, 63, 94, 95, 118, 119, 190, 191, 222, 223, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339234656657409796350799267682912243181438901889986777745755501369566953472.svg|420px]] |- |class="size"| 12 |class="prop"| (5, 4),<br>(6, 3),<br>(6, 4),<br>(7, 5) |class="block"| <span class="block-list">[110, 111, 122, 123, 124, 125, 238, 239, 250, 251, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_27140096050294567090646637084536785592076780842901902764461412277994676813824.svg|420px]] |- |class="size"| 4 |class="prop"| (6, 6),<br>(7, 2),<br>(7, 7),<br>(8, 1) |class="block"| <span class="block-list">[126, 127, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844066927987146567678238756515930890207700274421126877190718961508673388544.svg|420px]] |} [[Category:Boolf prop/3-ary|chunky burden]] er0rajv9niyd1aph8gzd2209nz4chy6 2693337 2693335 2024-12-26T19:08:09Z Watchduck 137431 2693337 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">13</span></span> Integer partition: &nbsp; <span class="count">2</span>⋅<span class="size">4</span> + <span class="count">6</span>⋅<span class="size">12</span> + <span class="count">2</span>⋅<span class="size">24</span> + <span class="count">2</span>⋅<span class="size">36</span> + <span class="count">1</span>⋅<span class="size">56</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chunky burden !class="block"| block |- |class="size"| 4 |class="prop"| (0, 0),<br>(1, 1),<br>(1, 8),<br>(2, 7)&nbsp; |class="block"| <span class="block-list">[0, 1, 128, 129]</span>[[File:Set_of_3-ary_Boolean_functions_1020847100762815390390123822295304634371.svg|420px]] |- |class="size"| 12 |class="prop"| (1, 4),<br>(2, 3),<br>(2, 4),<br>(3, 5)&nbsp; |class="block"| <span class="block-list">[2, 3, 4, 5, 16, 17, 130, 131, 132, 133, 144, 145]</span>[[File:Set_of_3-ary_Boolean_functions_66922652537607125732414957294390990610825276.svg|420px]] |- |class="size"| 24 |class="prop"| (2, 4),<br>(3, 4),<br>(3, 5),<br>(4, 3)&nbsp; |class="block"| <span class="block-list small">[6, 7, 18, 19, 20, 21, 24, 25, 36, 37, 66, 67, 134, 135, 146, 147, 148, 149, 152, 153, 164, 165, 194, 195]</span>[[File:Set_of_3-ary_Boolean_functions_75325220894810712840271410957050245523528888676370351915200.svg|420px]] |- |class="size"| 12 |class="prop"| (1, 2),<br>(2, 1),<br>(2, 6),<br>(3, 7)&nbsp; |class="block"| <span class="block-list">[8, 9, 32, 33, 64, 65, 136, 137, 160, 161, 192, 193]</span>[[File:Set_of_3-ary_Boolean_functions_18831305210544547464836934819514793686382954132191901123328.svg|420px]] |- |class="size"| 36 |class="prop"| (2, 2),<br>(3, 3),<br>(3, 6),<br>(4, 5)&nbsp; |class="block"| <span class="block-list small">[10, 11, 12, 13, 34, 35, 40, 41, 48, 49, 68, 69, 72, 73, 80, 81, 96, 97, 138, 139, 140, 141, 162, 163, 168, 169, 176, 177, 196, 197, 200, 201, 208, 209, 224, 225]</span>[[File:Set_of_3-ary_Boolean_functions_80881079251985214846949359476952353666040212987522435176623180364800.svg|420px]] |- |class="size"| 12 |class="prop"| (3, 6),<br>(4, 2),<br>(4, 5),<br>(5, 3)&nbsp; |class="block"| <span class="block-list">[14, 15, 50, 51, 84, 85, 142, 143, 178, 179, 212, 213]</span>[[File:Set_of_3-ary_Boolean_functions_19746054689003844161525772393355864297569742503366378873056968704.svg|420px]] |- |class="size"| 56 |class="prop"| (3, 4),<br>(4, 3),<br>(4, 4),<br>(5, 5)&nbsp; |class="block"| <span class="block-list small">[22, 23, 26, 27, 28, 29, 38, 39, 44, 45, 52, 53, 56, 57, 70, 71, 74, 75, 82, 83, 88, 89, 98, 99, 100, 101, 104, 105, 150, 151, 154, 155, 156, 157, 166, 167, 172, 173, 180, 181, 184, 185, 198, 199, 202, 203, 210, 211, 216, 217, 226, 227, 228, 229, 232, 233]</span>[[File:Set_of_3-ary_Boolean_functions_22323156734277945624977106139613641801666920800657794174809143267819520.svg|420px]] |- |class="size"| 36 |class="prop"| (4, 4),<br>(5, 4),<br>(5, 5),<br>(6, 3)&nbsp; |class="block"| <span class="block-list small">[30, 31, 46, 47, 54, 55, 58, 59, 78, 79, 86, 87, 92, 93, 114, 115, 116, 117, 158, 159, 174, 175, 182, 183, 186, 187, 206, 207, 214, 215, 220, 221, 242, 243, 244, 245]</span>[[File:Set_of_3-ary_Boolean_functions_106010829020985811034359354862228660894139180113282060559075074298756464640.svg|420px]] |- |class="size"| 12 |class="prop"| (3, 2),<br>(4, 1),<br>(4, 6),<br>(5, 7)&nbsp; |class="block"| <span class="block-list">[42, 43, 76, 77, 112, 113, 170, 171, 204, 205, 240, 241]</span>[[File:Set_of_3-ary_Boolean_functions_5300541194412286014878813768484650261695126105929858195979558110841798656.svg|420px]] |- |class="size"| 24 |class="prop"| (4, 2),<br>(5, 3),<br>(5, 6),<br>(6, 5)&nbsp; |class="block"| <span class="block-list small">[60, 61, 90, 91, 102, 103, 106, 107, 108, 109, 120, 121, 188, 189, 218, 219, 230, 231, 234, 235, 236, 237, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1357357828104114193069701957749213724878988658496165298225405248142940569600.svg|420px]] |- |class="size"| 12 |class="prop"| (5, 6),<br>(6, 2),<br>(6, 5),<br>(7, 3)&nbsp; |class="block"| <span class="block-list">[62, 63, 94, 95, 118, 119, 190, 191, 222, 223, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339234656657409796350799267682912243181438901889986777745755501369566953472.svg|420px]] |- |class="size"| 12 |class="prop"| (5, 4),<br>(6, 3),<br>(6, 4),<br>(7, 5)&nbsp; |class="block"| <span class="block-list">[110, 111, 122, 123, 124, 125, 238, 239, 250, 251, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_27140096050294567090646637084536785592076780842901902764461412277994676813824.svg|420px]] |- |class="size"| 4 |class="prop"| (6, 6),<br>(7, 2),<br>(7, 7),<br>(8, 1)&nbsp; |class="block"| <span class="block-list">[126, 127, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844066927987146567678238756515930890207700274421126877190718961508673388544.svg|420px]] |} [[Category:Boolf prop/3-ary|chunky burden]] etfpcp3n88909fxdvzqwhpuo3bobna3 0 317512 2693297 2024-12-26T12:13:36Z Eshaa2024 2993595 New resource with "== Definition == Let <math>G \subseteq \mathbb C</math> be a domain and <math>z_0 \in G</math>. If <math>f \colon G\setminus {z_0} \to \mathbb C</math> is a [[w:en:Holomorphic function|holomorphic]] function, then <math>z_0</math> is called an ''isolated singularity'' of <math>f</math>. == Classification == Depending on the behavior of <math>f</math> in the neighborhood of <math>z_0</math>, one distinguishes three different types of isolated singularities of <math>f</ma..." 2693297 wikitext text/x-wiki == Definition == Let <math>G \subseteq \mathbb C</math> be a domain and <math>z_0 \in G</math>. If <math>f \colon G\setminus {z_0} \to \mathbb C</math> is a [[w:en:Holomorphic function|holomorphic]] function, then <math>z_0</math> is called an ''isolated singularity'' of <math>f</math>. == Classification == Depending on the behavior of <math>f</math> in the neighborhood of <math>z_0</math>, one distinguishes three different types of isolated singularities of <math>f</math>. === Removable Singularities === If <math>f</math> can be holomorphically extended to the entire domain <math>G</math>, then we say that <math>z_0</math> is a ''removable singularity''. According to the [[Riemann Removability Theorem|Riemann Removability Theorem]], this is the case if <math>f</math> is bounded in a neighborhood of <math>z_0</math>. === Poles === If <math>z_0</math> is not a removable singularity, but there exists an <math>n \ge 1</math> such that <math>(\cdot - z_0)^n \cdot f</math> has a removable singularity at <math>z_0</math>, then we say that <math>f</math> has a ''pole'' at <math>z_0</math>. The smallest such <math>n</math> is called the ''order'' of the pole. === Essential Singularities === If <math>z_0</math> is neither removable nor a pole, then <math>z_0</math> is called an ''essential singularity'' of <math>f</math>. == Examples == *Since <math>\lim_{z\to 0} \frac{\sin z}z = 1</math>, the function <math>f_1(z) = \frac{\sin z}z</math> has a removable singularity at <math>z_0 = 0</math>. *The function <math>f_2(z) = \frac 1{\sin z}</math> does not have <math>z_0 = 0</math> a removable singularity at, since<math>f_2</math> is unbounded at <math>0</math>, but <math>f_2</math> has a first-order pole at <math>0</math>, because <math>f_2(z) \cdot (z - 0)^1 = f_2(z)z = \frac{z}{\sin z}</math> and <math>\lim_{z \to 0} \frac{z}{\sin z} = 1</math>, which has a removable singularity at 0 . *The function <math>f_3(z) = \sin \frac 1z</math> has an essential singularity at <math>z_0 = 0</math>, since for every <math>n \ge 1</math>, the function <math>f_3(z)z^n = z^n \sin \frac 1z</math> is unbounded in any neighborhood of <math>0</math>. To see this, consider<math>\sin z^{-1} = \frac{e^{iz^{-1}} - e^{-iz^{-1}}}{2i}</math>.For <math>z = it</math> with <math>t \in \mathbb R</math> is also <math>f_3(it)(it)^n = (it)^n \frac{e^{t^{-1}} - e^{-t^{-1}}}{2i}</math>,which diverges as <math>t \to 0^+</math> . == Laurent Expansions == The type of isolated singularity can also be inferred from the [[Complex Analysis/Laurent Expansion|Laurent Expansion]] of <math>f</math> around <math>z_0</math>. Let<center><math> f(z) = \sum_{n = -\infty}^\infty a_n (z-z_0)^n </math></center> be the [[Laurent Series]] of <math>f</math> around <math>z_0</math>. We define <center><math> o_z(f) = \sup\{n \in \mathbb Z | \forall k < n : a_k = 0\}</math>.</center> Then, <math>f</math> has the following singularities:If <math>o_z(f) \ge 0</math>, i.e., all negative coefficients vanish, the [[Laurent Series#Main part and remainder|main part]] of the series is zero, and the singularity is removable. If <math>-\infty < o_z(f) < 0</math>, i.e., only finitely many negative coefficients are nonzero, there is a pole of order <math>-o_z(f)</math>. If <math>o_z(f) = -\infty</math>, i.e., infinitely many negative coefficients are nonzero, the singularity is essential. === Examples === Let us consider our three examples again: It is <math>f_1(z) = \frac{\sin z}z = \sum_{k=0}^\infty (-1)^n\frac{z^{2n}}{(2n+1)!}</math>, so <math>o_0(f_1) = 0</math>, a removable singularity. It is <center><math>f_2(z) = \frac 1{\sin z} = \frac 1z + \frac z6 + \frac 7{360}z^3 + \ldots </math></center> so <math>o_0(f_2) = -1</math>, a pole of first order. It is <math>f_3(z) = \sin z^{-1} = \sum_{n=-\infty}^0 \frac{(-1)^n}{(-2n+1)!}z^{2n-1}</math>, so <math>o_0(f_3) = -\infty</math>, an essential singularity. == Page information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/isolierte Singularität Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/isolierte Singularität |Kurs:Funktionentheorie/isolierte Singularität]] - URL:https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/isolierte Singularität * Date: 11/20/2024 <span type="translate" src="Kurs:Funktionentheorie/isolierte Singularität" srclang="de" date="12/17/2024" time="11:42" status="inprogress"></span> <noinclude>[[de:Kurs:Funktionentheorie/isolierte Singularität]]</noinclude> [[Category:Wiki2Reveal]] htfdpfg2vky4yvcjcx8hpdmh9itiyw7 2693301 2693297 2024-12-26T12:47:21Z Eshaa2024 2993595 2693301 wikitext text/x-wiki == Definition == Let <math>G \subseteq \mathbb C</math> be a domain and <math>z_0 \in G</math>. If <math>f \colon G\setminus {z_0} \to \mathbb C</math> is a [[w:en:Holomorphic function|holomorphic]] function, then <math>z_0</math> is called an ''isolated singularity'' of <math>f</math>. == Classification == Depending on the behavior of <math>f</math> in the neighborhood of <math>z_0</math>, one distinguishes three different types of isolated singularities of <math>f</math>. === Removable Singularities === If <math>f</math> can be holomorphically extended to the entire domain <math>G</math>, then we say that <math>z_0</math> is a ''removable singularity''. According to the [[w:en:Riemann Removability Theorem|Riemann Removability Theorem]], this is the case if <math>f</math> is bounded in a neighborhood of <math>z_0</math>. === Poles === If <math>z_0</math> is not a removable singularity, but there exists an <math>n \ge 1</math> such that <math>(\cdot - z_0)^n \cdot f</math> has a removable singularity at <math>z_0</math>, then we say that <math>f</math> has a ''pole'' at <math>z_0</math>. The smallest such <math>n</math> is called the ''order'' of the pole. === Essential Singularities === If <math>z_0</math> is neither removable nor a pole, then <math>z_0</math> is called an ''essential singularity'' of <math>f</math>. == Examples == *Since <math>\lim_{z\to 0} \frac{\sin z}z = 1</math>, the function <math>f_1(z) = \frac{\sin z}z</math> has a removable singularity at <math>z_0 = 0</math>. *The function <math>f_2(z) = \frac 1{\sin z}</math> does not have <math>z_0 = 0</math> a removable singularity at, since<math>f_2</math> is unbounded at <math>0</math>, but <math>f_2</math> has a first-order pole at <math>0</math>, because <math>f_2(z) \cdot (z - 0)^1 = f_2(z)z = \frac{z}{\sin z}</math> and <math>\lim_{z \to 0} \frac{z}{\sin z} = 1</math>, which has a removable singularity at 0 . *The function <math>f_3(z) = \sin \frac 1z</math> has an essential singularity at <math>z_0 = 0</math>, since for every <math>n \ge 1</math>, the function <math>f_3(z)z^n = z^n \sin \frac 1z</math> is unbounded in any neighborhood of <math>0</math>. To see this, consider<math>\sin z^{-1} = \frac{e^{iz^{-1}} - e^{-iz^{-1}}}{2i}</math>.For <math>z = it</math> with <math>t \in \mathbb R</math> is also <math>f_3(it)(it)^n = (it)^n \frac{e^{t^{-1}} - e^{-t^{-1}}}{2i}</math>,which diverges as <math>t \to 0^+</math> . == Laurent Expansions == The type of isolated singularity can also be inferred from the [[w:en:Complex Analysis/Laurent Expansion|Laurent Expansion]] of <math>f</math> around <math>z_0</math>. Let<center><math> f(z) = \sum_{n = -\infty}^\infty a_n (z-z_0)^n </math></center> be the [[Laurent Series]] of <math>f</math> around <math>z_0</math>. We define <center><math> o_z(f) = \sup\{n \in \mathbb Z | \forall k < n : a_k = 0\}</math>.</center> Then, <math>f</math> has the following singularities:If <math>o_z(f) \ge 0</math>, i.e., all negative coefficients vanish, the [[Laurent Series#Main part and remainder|main part]] of the series is zero, and the singularity is removable. If <math>-\infty < o_z(f) < 0</math>, i.e., only finitely many negative coefficients are nonzero, there is a pole of order <math>-o_z(f)</math>. If <math>o_z(f) = -\infty</math>, i.e., infinitely many negative coefficients are nonzero, the singularity is essential. === Examples === Let us consider our three examples again: It is <math>f_1(z) = \frac{\sin z}z = \sum_{k=0}^\infty (-1)^n\frac{z^{2n}}{(2n+1)!}</math>, so <math>o_0(f_1) = 0</math>, a removable singularity. It is <center><math>f_2(z) = \frac 1{\sin z} = \frac 1z + \frac z6 + \frac 7{360}z^3 + \ldots </math></center> so <math>o_0(f_2) = -1</math>, a pole of first order. It is <math>f_3(z) = \sin z^{-1} = \sum_{n=-\infty}^0 \frac{(-1)^n}{(-2n+1)!}z^{2n-1}</math>, so <math>o_0(f_3) = -\infty</math>, an essential singularity. == Page information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/isolierte Singularität Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/isolierte Singularität |Kurs:Funktionentheorie/isolierte Singularität]] - URL:https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/isolierte Singularität * Date: 11/20/2024 <span type="translate" src="Kurs:Funktionentheorie/isolierte Singularität" srclang="de" date="12/17/2024" time="11:42" status="inprogress"></span> <noinclude>[[de:Kurs:Funktionentheorie/isolierte Singularität]]</noinclude> [[Category:Wiki2Reveal]] q3p203z4mcavwbrinw4ljcsnnmnjeb9 2693302 2693301 2024-12-26T12:48:02Z Eshaa2024 2993595 /* Laurent Expansions */ 2693302 wikitext text/x-wiki == Definition == Let <math>G \subseteq \mathbb C</math> be a domain and <math>z_0 \in G</math>. If <math>f \colon G\setminus {z_0} \to \mathbb C</math> is a [[w:en:Holomorphic function|holomorphic]] function, then <math>z_0</math> is called an ''isolated singularity'' of <math>f</math>. == Classification == Depending on the behavior of <math>f</math> in the neighborhood of <math>z_0</math>, one distinguishes three different types of isolated singularities of <math>f</math>. === Removable Singularities === If <math>f</math> can be holomorphically extended to the entire domain <math>G</math>, then we say that <math>z_0</math> is a ''removable singularity''. According to the [[w:en:Riemann Removability Theorem|Riemann Removability Theorem]], this is the case if <math>f</math> is bounded in a neighborhood of <math>z_0</math>. === Poles === If <math>z_0</math> is not a removable singularity, but there exists an <math>n \ge 1</math> such that <math>(\cdot - z_0)^n \cdot f</math> has a removable singularity at <math>z_0</math>, then we say that <math>f</math> has a ''pole'' at <math>z_0</math>. The smallest such <math>n</math> is called the ''order'' of the pole. === Essential Singularities === If <math>z_0</math> is neither removable nor a pole, then <math>z_0</math> is called an ''essential singularity'' of <math>f</math>. == Examples == *Since <math>\lim_{z\to 0} \frac{\sin z}z = 1</math>, the function <math>f_1(z) = \frac{\sin z}z</math> has a removable singularity at <math>z_0 = 0</math>. *The function <math>f_2(z) = \frac 1{\sin z}</math> does not have <math>z_0 = 0</math> a removable singularity at, since<math>f_2</math> is unbounded at <math>0</math>, but <math>f_2</math> has a first-order pole at <math>0</math>, because <math>f_2(z) \cdot (z - 0)^1 = f_2(z)z = \frac{z}{\sin z}</math> and <math>\lim_{z \to 0} \frac{z}{\sin z} = 1</math>, which has a removable singularity at 0 . *The function <math>f_3(z) = \sin \frac 1z</math> has an essential singularity at <math>z_0 = 0</math>, since for every <math>n \ge 1</math>, the function <math>f_3(z)z^n = z^n \sin \frac 1z</math> is unbounded in any neighborhood of <math>0</math>. To see this, consider<math>\sin z^{-1} = \frac{e^{iz^{-1}} - e^{-iz^{-1}}}{2i}</math>.For <math>z = it</math> with <math>t \in \mathbb R</math> is also <math>f_3(it)(it)^n = (it)^n \frac{e^{t^{-1}} - e^{-t^{-1}}}{2i}</math>,which diverges as <math>t \to 0^+</math> . == Laurent Expansions == The type of isolated singularity can also be inferred from the [[Complex Analysis/Laurent Expansion|Laurent Expansion]] of <math>f</math> around <math>z_0</math>. Let<center><math> f(z) = \sum_{n = -\infty}^\infty a_n (z-z_0)^n </math></center> be the [[Laurent Series]] of <math>f</math> around <math>z_0</math>. We define <center><math> o_z(f) = \sup\{n \in \mathbb Z | \forall k < n : a_k = 0\}</math>.</center> Then, <math>f</math> has the following singularities:If <math>o_z(f) \ge 0</math>, i.e., all negative coefficients vanish, the [[Laurent Series#Main part and remainder|main part]] of the series is zero, and the singularity is removable. If <math>-\infty < o_z(f) < 0</math>, i.e., only finitely many negative coefficients are nonzero, there is a pole of order <math>-o_z(f)</math>. If <math>o_z(f) = -\infty</math>, i.e., infinitely many negative coefficients are nonzero, the singularity is essential. === Examples === Let us consider our three examples again: It is <math>f_1(z) = \frac{\sin z}z = \sum_{k=0}^\infty (-1)^n\frac{z^{2n}}{(2n+1)!}</math>, so <math>o_0(f_1) = 0</math>, a removable singularity. It is <center><math>f_2(z) = \frac 1{\sin z} = \frac 1z + \frac z6 + \frac 7{360}z^3 + \ldots </math></center> so <math>o_0(f_2) = -1</math>, a pole of first order. It is <math>f_3(z) = \sin z^{-1} = \sum_{n=-\infty}^0 \frac{(-1)^n}{(-2n+1)!}z^{2n-1}</math>, so <math>o_0(f_3) = -\infty</math>, an essential singularity. == Page information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/isolierte Singularität Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/isolierte Singularität |Kurs:Funktionentheorie/isolierte Singularität]] - URL:https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/isolierte Singularität * Date: 11/20/2024 <span type="translate" src="Kurs:Funktionentheorie/isolierte Singularität" srclang="de" date="12/17/2024" time="11:42" status="inprogress"></span> <noinclude>[[de:Kurs:Funktionentheorie/isolierte Singularität]]</noinclude> [[Category:Wiki2Reveal]] 31wk4oh49gn8jv0rhv0ppdqouqmmiag 2693303 2693302 2024-12-26T12:50:55Z Eshaa2024 2993595 2693303 wikitext text/x-wiki == Definition == Let <math>G \subseteq \mathbb C</math> be a domain and <math>z_0 \in G</math>. If <math>f \colon G\setminus {z_0} \to \mathbb C</math> is a [[w:en:Holomorphic function|holomorphic]] function, then <math>z_0</math> is called an ''isolated singularity'' of <math>f</math>. == Classification == Depending on the behavior of <math>f</math> in the neighborhood of <math>z_0</math>, one distinguishes three different types of isolated singularities of <math>f</math>. === Removable Singularities === If <math>f</math> can be holomorphically extended to the entire domain <math>G</math>, then we say that <math>z_0</math> is a ''removable singularity''. According to the [[Riemann Removability Theorem|Riemann Removability Theorem]], this is the case if <math>f</math> is bounded in a neighborhood of <math>z_0</math>. === Poles === If <math>z_0</math> is not a removable singularity, but there exists an <math>n \ge 1</math> such that <math>(\cdot - z_0)^n \cdot f</math> has a removable singularity at <math>z_0</math>, then we say that <math>f</math> has a ''pole'' at <math>z_0</math>. The smallest such <math>n</math> is called the ''order'' of the pole. === Essential Singularities === If <math>z_0</math> is neither removable nor a pole, then <math>z_0</math> is called an ''essential singularity'' of <math>f</math>. == Examples == *Since <math>\lim_{z\to 0} \frac{\sin z}z = 1</math>, the function <math>f_1(z) = \frac{\sin z}z</math> has a removable singularity at <math>z_0 = 0</math>. *The function <math>f_2(z) = \frac 1{\sin z}</math> does not have <math>z_0 = 0</math> a removable singularity at, since<math>f_2</math> is unbounded at <math>0</math>, but <math>f_2</math> has a first-order pole at <math>0</math>, because <math>f_2(z) \cdot (z - 0)^1 = f_2(z)z = \frac{z}{\sin z}</math> and <math>\lim_{z \to 0} \frac{z}{\sin z} = 1</math>, which has a removable singularity at 0 . *The function <math>f_3(z) = \sin \frac 1z</math> has an essential singularity at <math>z_0 = 0</math>, since for every <math>n \ge 1</math>, the function <math>f_3(z)z^n = z^n \sin \frac 1z</math> is unbounded in any neighborhood of <math>0</math>. To see this, consider<math>\sin z^{-1} = \frac{e^{iz^{-1}} - e^{-iz^{-1}}}{2i}</math>.For <math>z = it</math> with <math>t \in \mathbb R</math> is also <math>f_3(it)(it)^n = (it)^n \frac{e^{t^{-1}} - e^{-t^{-1}}}{2i}</math>,which diverges as <math>t \to 0^+</math> . == Laurent Expansions == The type of isolated singularity can also be inferred from the [[Complex Analysis/Laurent Expansion|Laurent Expansion]] of <math>f</math> around <math>z_0</math>. Let<center><math> f(z) = \sum_{n = -\infty}^\infty a_n (z-z_0)^n </math></center> be the [[Laurent Series]] of <math>f</math> around <math>z_0</math>. We define <center><math> o_z(f) = \sup\{n \in \mathbb Z | \forall k < n : a_k = 0\}</math>.</center> Then, <math>f</math> has the following singularities:If <math>o_z(f) \ge 0</math>, i.e., all negative coefficients vanish, the [[Laurent Series#Main part and remainder|main part]] of the series is zero, and the singularity is removable. If <math>-\infty < o_z(f) < 0</math>, i.e., only finitely many negative coefficients are nonzero, there is a pole of order <math>-o_z(f)</math>. If <math>o_z(f) = -\infty</math>, i.e., infinitely many negative coefficients are nonzero, the singularity is essential. === Examples === Let us consider our three examples again: It is <math>f_1(z) = \frac{\sin z}z = \sum_{k=0}^\infty (-1)^n\frac{z^{2n}}{(2n+1)!}</math>, so <math>o_0(f_1) = 0</math>, a removable singularity. It is <center><math>f_2(z) = \frac 1{\sin z} = \frac 1z + \frac z6 + \frac 7{360}z^3 + \ldots </math></center> so <math>o_0(f_2) = -1</math>, a pole of first order. It is <math>f_3(z) = \sin z^{-1} = \sum_{n=-\infty}^0 \frac{(-1)^n}{(-2n+1)!}z^{2n-1}</math>, so <math>o_0(f_3) = -\infty</math>, an essential singularity. == Page information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/isolierte Singularität Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/isolierte Singularität |Kurs:Funktionentheorie/isolierte Singularität]] - URL:https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/isolierte Singularität * Date: 11/20/2024 <span type="translate" src="Kurs:Funktionentheorie/isolierte Singularität" srclang="de" date="12/17/2024" time="11:42" status="inprogress"></span> <noinclude>[[de:Kurs:Funktionentheorie/isolierte Singularität]]</noinclude> [[Category:Wiki2Reveal]] 3xtd9icl7lakl2cv0vtf50m9t2cwaid 0 317513 2693298 2024-12-26T12:37:29Z Eshaa2024 2993595 New resource with "== Laurent Expansion Around a Point == Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and <math>f \colon G \setminus {z_0} \to \mathbb{C}</math> a [[w:en:Holomorphic function|holomorphic]] function. A Laurent expansion of <math>f</math> around <math>z_0</math> is a representation of <math>f</math> as a [[Laurent Series]]: <center><math>f(z) = \sum_{n=-\infty}^\infty a_n (z - z_0)^n</math></center>where <math>a_n \in \mathbb{C}</math>, and..." 2693298 wikitext text/x-wiki == Laurent Expansion Around a Point == Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and <math>f \colon G \setminus {z_0} \to \mathbb{C}</math> a [[w:en:Holomorphic function|holomorphic]] function. A Laurent expansion of <math>f</math> around <math>z_0</math> is a representation of <math>f</math> as a [[Laurent Series]]: <center><math>f(z) = \sum_{n=-\infty}^\infty a_n (z - z_0)^n</math></center>where <math>a_n \in \mathbb{C}</math>, and the series converges on an annular region around <math>z_0</math> (i.e., excluding the point <math>z_0</math>). == Laurent Expansion on an Annulus == A slightly more general form of the expansion above is the following: Let <math>0 \leq r_1 < r_2</math> be two radii (the expansion around a point corresponds to <math>r_1 = 0</math>), and let <math>A_{r_1, r_2} := {z \in \mathbb{C}: r_1 < |z - z_0| < r_2}</math> be an annular region around <math>z_0</math>, and let <math>f \colon A_{r_1, r_2} \to \mathbb{C}</math> be a holomorphic function. Then the [[Laurent Series]] <center><math>f(z) = \sum_{n=-\infty}^\infty a_n (z - z_0)^n</math></center>with <math>a_n \in \mathbb{C}</math> is a Laurent expansion of <math>f</math> on <math>A_{r_1, r_2}</math>, provided the series converges for all <math>z \in A_{r_1, r_2}</math>. === Existence === Every holomorphic function on <math>A_{r_1, r_2}</math> has a Laurent expansion around <math>z_0</math>, and the coefficients <math>a_n</math> in the expansion are given by: <center><math>a_n = \frac{1}{2\pi i} \int_{|z - z_0| = r} \frac{f(z)}{(z - z_0)^{n+1}} \, dz</math></center>for a radius <math>r</math> with <math>r_1 < r < r_2</math>. === Uniqueness === The coefficients are uniquely determined by: <center><math>a_n = \frac{1}{2\pi i} \int_{|z - z_0| = r} \frac{f(z)}{(z - z_0)^{n+1}} \, dz</math></center> === Proof of Existence and Uniqueness of the Laurent Representation=== Uniqueness follows from the Identity Theorem for [[Laurent Series]]. To prove existence, choose a radius <math>r</math> such that <math>r_1 < r < r_2</math> and choose <math>R_1, R_2</math> so that <math>r_1 < R_1 < r < R_2 < r_2</math>. Let <math>z \in A_{R_1, R_2}</math> be arbitrary. "Cut" the annular region <math>A_{R_1, R_2}</math> at two points using radii <math>D_1</math> and <math>D_2</math> such that the cycle <math>\partial K_{R_2} - \partial K_{R_1}</math> is represented as the sum of two closed curves <math>C_1</math> and <math>C_2</math> in <math>A</math> that are null-homotopic. Choose <math>D_1</math> and <math>D_2</math> so that <math>z</math> is encircled by <math>C_1</math>. By the [[Complex Analysis/Cauchy Integral Theorem|Cauchy Integral Theorem]], we have: <center><math>f(z) = \frac{1}{2\pi i} \int_{C_1} \frac{f(w)}{w - z} \, dw</math></center>and <center><math>0 = \frac{1}{2\pi i} \int_{C_2} \frac{f(w)}{w - z} \, dw</math></center>since <math>C_2</math> does not encircle <math>z</math>. Thus, because <math>C_1 + C_2 = \partial K_{R_2} - \partial K_{R_1}</math>, we have: <center><math>f(z) = \frac{1}{2\pi i} \int_{|w - z_0| = R_2} \frac{f(w)}{w - z} \, dw - \frac{1}{2\pi i} \int_{|w - z_0| = R_1} \frac{f(w)}{w - z} \, dw</math></center>For <math>|w - z_0| = R_2</math>, we have: <center><math> \begin{array}{rl} \displaystyle \frac{1}{w - z} &= \displaystyle \frac{1}{(w - z_0) - (z - z_0)} \\ &= \displaystyle \frac{1}{w - z_0} \cdot \frac{1}{1 - \frac{z - z_0}{w - z_0}} \\ &= \displaystyle \frac{1}{w - z_0} \sum_{n=0}^\infty \frac{(z - z_0)^n}{(w - z_0)^n} \end{array} </math></center>The series converges absolutely because <math>|z - z_0| < |w - z_0|</math>, and we obtain: <center><math> \begin{array}{rl} \frac{1}{2\pi i} \int_{|w - z_0| = R_2} \frac{f(w)}{w - z} \, dw &= \frac{1}{2\pi i} \int_{|w - z_0| = R_2} \frac{1}{w - z_0} \cdot \frac{f(w)(z - z_0)^n}{(w - z_0)^n} \, dw \\ &= \frac{1}{2\pi i} \sum_{n=0}^\infty \int_{|w - z_0| = R_2} \frac{f(w)}{(w - z_0)^{n+1}} \, dw \cdot (z - z_0)^n \\ &= \frac{1}{2\pi i} \sum_{n=0}^\infty \int_{|w - z_0| = r} \frac{f(w)}{(w - z_0)^{n+1}} \, dw \cdot (z - z_0)^n \end{array} </math></center>Now, consider the integral over the inner circle, which is analogous to the above for <math>|w - z_0| = R_1</math>: <center><math> \begin{array}{rl} \displaystyle \frac{1}{w - z} &= \displaystyle \frac{1}{(w - z_0) - (z - z_0)} \\ &= \displaystyle \frac{-1}{z - z_0} \cdot \frac{1}{1 - \frac{w - z_0}{z - z_0}} \\ &= \displaystyle \frac{-1}{z - z_0} \sum_{n=0}^\infty \frac{(w - z_0)^n}{(z - z_0)^n} \end{array} </math></center>Thus, due to <math>R_1 = |w - z_0| < |z - z_0|</math>, the series converges, and we obtain: <center><math> \begin{array}{rl} -\frac{1}{2\pi i} \int_{|w - z_0| = R_1} \frac{f(w)}{w - z} \, dw &= \frac{1}{2\pi i} \int_{|w - z_0| = R_1} \frac{-1}{z - z_0} \cdot \frac{f(w)(w - z_0)^n}{(z - z_0)^n} \, dw \\ &= \frac{1}{2\pi i} \sum_{n=0}^\infty \int_{|w - z_0| = R_1} \frac{f(w)}{(w - z_0)^{-n}} \, dw \cdot (z - z_0)^{-n-1} \\ &= \frac{1}{2\pi i} \sum_{n=0}^\infty \int_{|w - z_0| = r} \frac{f(w)}{(w - z_0)^{-n}} \, dw \cdot (z - z_0)^{-n-1} \end{array} </math></center>Thus, it follows that for <math>z \in A_{R_1, R_2}</math>: <center><math>f(z) = \frac{1}{2\pi i} \sum_{n=-\infty}^\infty \int_{|w - z_0| = r} \frac{f(w)}{(w - z_0)^{n+1}} \, dw \cdot (z - z_0)^n</math></center>which proves the existence of the claimed Laurent expansion. == See Also == *[[Complex Analysis/Examples of Laurent Series Expansion|Examples of Laurent Series Expansion]] *[[Laurent Series]] == Page information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Laurententwicklung Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Laurententwicklung |Kurs:Funktionentheorie/Laurententwicklung]] - URL:https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Laurententwicklung * Date: 11/26/2024 <span type="translate" src="Kurs:Funktionentheorie/Laurententwicklung " srclang="de" date="12/26/2024" time="01:39" status="inprogress"></span> <noinclude>[[de:Kurs:Funktionentheorie/Laurententwicklung]]</noinclude> [[Category:Wiki2Reveal]] carynhhuarp8uo9u5zblghexxr558bu 2693300 2693298 2024-12-26T12:43:37Z Eshaa2024 2993595 Undo all revisions. Resource is empty, but not [[Wikiversity:Deletions|deleted]]. 2693300 wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 2693304 2693300 2024-12-26T13:01:02Z Eshaa2024 2993595 2693304 wikitext text/x-wiki ==Statement== Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and <math>f \colon G \setminus {z_0} \to \mathbb{C}</math> be holomorphic. Then <math>f</math> can be holomorphically extended to <math>z_0</math> if and only if there exists a neighborhood <math>U \subseteq G</math> of <math>z_0</math> such that <math>f</math> is bounded on <math>U \setminus {z_0}</math>. ==Proof== Let <math>r > 0</math> be chosen such that <math>\bar{B}_r(z_0) \subseteq U</math>, and let <math>M</math> be an upper bound for <math>f</math> on <math>U</math>. We consider the [[Laurent Series]] of <math>f</math> around <math>z_0</math>. It is <center><math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n, \qquad a_n = \frac{1}{2\pi i} \int_{|w-z_0| = r} \frac{f(w)}{(w-z_0)^{n+1}}\, dw </math></center>Estimating <math>a_n</math> gives the so-called Cauchy estimates, namely <center><math> \begin{array}{rl} |a_n| &= \displaystyle \left|\frac{1}{2\pi i} \int_{|w-z_0| = r} \frac{f(w)}{(w-z_0)^{n+1}}\, dw\right|\\ &\le \displaystyle \frac{1}{2\pi} \int_{|w-z_0| = r} \frac{|f(w)|}{|w-z_0|^{n+1}}\, |dw|\\ &\le \displaystyle \frac{1}{2\pi} \int_{|w-z_0| = r} \frac{M}{r^{n+1}}\, |dw|\\ &= \displaystyle \frac{M}{r^n} \\ \end{array} </math></center>For <math>n < 0</math>, it follows that <center><math> |a_n| \le \frac{M}{r^n} = Mr^{-n} \to 0, \quad r \to 0 </math></center>Thus, <math>a_n = 0</math> for all <math>n < 0</math>, meaning we have <math>f(z) = \sum_{n=0}^\infty a_n(z-z_0)^n</math>, and <math>f(z_0) := a_0</math> is a holomorphic extension of <math>f</math> to <math>z_0</math>. == Page information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Riemannscher Hebbarkeitssatz Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Riemannscher Hebbarkeitssatz|Riemannscher Hebbarkeitssatz]] - URL:https://de.wikiversity.org/wiki/Riemannscher Hebbarkeitssatz * Date: 11/26/2024 <span type="translate" src="Riemannscher Hebbarkeitssatz" srclang="de" date="12/26/2024" time="02:00" status="inprogress"></span> <noinclude>[[de:Riemannscher Hebbarkeitssatz]]</noinclude> [[Category:Wiki2Reveal]] 0a5qk901gfrdeeyuo5nk1z3eacxlgte 0 317514 2693299 2024-12-26T12:43:01Z Eshaa2024 2993595 New resource with "== Laurent Expansion Around a Point == Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and <math>f \colon G \setminus {z_0} \to \mathbb{C}</math> a [[w:en:Holomorphic function|holomorphic]] function. A Laurent expansion of <math>f</math> around <math>z_0</math> is a representation of <math>f</math> as a [[Laurent Series]]: <center><math>f(z) = \sum_{n=-\infty}^\infty a_n (z - z_0)^n</math></center>where <math>a_n \in \mathbb{C}</math>, and..." 2693299 wikitext text/x-wiki == Laurent Expansion Around a Point == Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and <math>f \colon G \setminus {z_0} \to \mathbb{C}</math> a [[w:en:Holomorphic function|holomorphic]] function. A Laurent expansion of <math>f</math> around <math>z_0</math> is a representation of <math>f</math> as a [[Laurent Series]]: <center><math>f(z) = \sum_{n=-\infty}^\infty a_n (z - z_0)^n</math></center>where <math>a_n \in \mathbb{C}</math>, and the series converges on an annular region around <math>z_0</math> (i.e., excluding the point <math>z_0</math>). == Laurent Expansion on an Annulus == A slightly more general form of the expansion above is the following: Let <math>0 \leq r_1 < r_2</math> be two radii (the expansion around a point corresponds to <math>r_1 = 0</math>), and let <math>A_{r_1, r_2} := {z \in \mathbb{C}: r_1 < |z - z_0| < r_2}</math> be an annular region around <math>z_0</math>, and let <math>f \colon A_{r_1, r_2} \to \mathbb{C}</math> be a holomorphic function. Then the [[Laurent Series]] <center><math>f(z) = \sum_{n=-\infty}^\infty a_n (z - z_0)^n</math></center>with <math>a_n \in \mathbb{C}</math> is a Laurent expansion of <math>f</math> on <math>A_{r_1, r_2}</math>, provided the series converges for all <math>z \in A_{r_1, r_2}</math>. === Existence === Every holomorphic function on <math>A_{r_1, r_2}</math> has a Laurent expansion around <math>z_0</math>, and the coefficients <math>a_n</math> in the expansion are given by: <center><math>a_n = \frac{1}{2\pi i} \int_{|z - z_0| = r} \frac{f(z)}{(z - z_0)^{n+1}} \, dz</math></center>for a radius <math>r</math> with <math>r_1 < r < r_2</math>. === Uniqueness === The coefficients are uniquely determined by: <center><math>a_n = \frac{1}{2\pi i} \int_{|z - z_0| = r} \frac{f(z)}{(z - z_0)^{n+1}} \, dz</math></center> === Proof of Existence and Uniqueness of the Laurent Representation=== Uniqueness follows from the Identity Theorem for [[Laurent Series]]. To prove existence, choose a radius <math>r</math> such that <math>r_1 < r < r_2</math> and choose <math>R_1, R_2</math> so that <math>r_1 < R_1 < r < R_2 < r_2</math>. Let <math>z \in A_{R_1, R_2}</math> be arbitrary. "Cut" the annular region <math>A_{R_1, R_2}</math> at two points using radii <math>D_1</math> and <math>D_2</math> such that the cycle <math>\partial K_{R_2} - \partial K_{R_1}</math> is represented as the sum of two closed curves <math>C_1</math> and <math>C_2</math> in <math>A</math> that are null-homotopic. Choose <math>D_1</math> and <math>D_2</math> so that <math>z</math> is encircled by <math>C_1</math>. By the [[Complex Analysis/Cauchy Integral Theorem|Cauchy Integral Theorem]], we have: <center><math>f(z) = \frac{1}{2\pi i} \int_{C_1} \frac{f(w)}{w - z} \, dw</math></center>and <center><math>0 = \frac{1}{2\pi i} \int_{C_2} \frac{f(w)}{w - z} \, dw</math></center>since <math>C_2</math> does not encircle <math>z</math>. Thus, because <math>C_1 + C_2 = \partial K_{R_2} - \partial K_{R_1}</math>, we have: <center><math>f(z) = \frac{1}{2\pi i} \int_{|w - z_0| = R_2} \frac{f(w)}{w - z} \, dw - \frac{1}{2\pi i} \int_{|w - z_0| = R_1} \frac{f(w)}{w - z} \, dw</math></center>For <math>|w - z_0| = R_2</math>, we have: <center><math> \begin{array}{rl} \displaystyle \frac{1}{w - z} &= \displaystyle \frac{1}{(w - z_0) - (z - z_0)} \\ &= \displaystyle \frac{1}{w - z_0} \cdot \frac{1}{1 - \frac{z - z_0}{w - z_0}} \\ &= \displaystyle \frac{1}{w - z_0} \sum_{n=0}^\infty \frac{(z - z_0)^n}{(w - z_0)^n} \end{array} </math></center>The series converges absolutely because <math>|z - z_0| < |w - z_0|</math>, and we obtain: <center><math> \begin{array}{rl} \frac{1}{2\pi i} \int_{|w - z_0| = R_2} \frac{f(w)}{w - z} \, dw &= \frac{1}{2\pi i} \int_{|w - z_0| = R_2} \frac{1}{w - z_0} \cdot \frac{f(w)(z - z_0)^n}{(w - z_0)^n} \, dw \\ &= \frac{1}{2\pi i} \sum_{n=0}^\infty \int_{|w - z_0| = R_2} \frac{f(w)}{(w - z_0)^{n+1}} \, dw \cdot (z - z_0)^n \\ &= \frac{1}{2\pi i} \sum_{n=0}^\infty \int_{|w - z_0| = r} \frac{f(w)}{(w - z_0)^{n+1}} \, dw \cdot (z - z_0)^n \end{array} </math></center>Now, consider the integral over the inner circle, which is analogous to the above for <math>|w - z_0| = R_1</math>: <center><math> \begin{array}{rl} \displaystyle \frac{1}{w - z} &= \displaystyle \frac{1}{(w - z_0) - (z - z_0)} \\ &= \displaystyle \frac{-1}{z - z_0} \cdot \frac{1}{1 - \frac{w - z_0}{z - z_0}} \\ &= \displaystyle \frac{-1}{z - z_0} \sum_{n=0}^\infty \frac{(w - z_0)^n}{(z - z_0)^n} \end{array} </math></center>Thus, due to <math>R_1 = |w - z_0| < |z - z_0|</math>, the series converges, and we obtain: <center><math> \begin{array}{rl} -\frac{1}{2\pi i} \int_{|w - z_0| = R_1} \frac{f(w)}{w - z} \, dw &= \frac{1}{2\pi i} \int_{|w - z_0| = R_1} \frac{-1}{z - z_0} \cdot \frac{f(w)(w - z_0)^n}{(z - z_0)^n} \, dw \\ &= \frac{1}{2\pi i} \sum_{n=0}^\infty \int_{|w - z_0| = R_1} \frac{f(w)}{(w - z_0)^{-n}} \, dw \cdot (z - z_0)^{-n-1} \\ &= \frac{1}{2\pi i} \sum_{n=0}^\infty \int_{|w - z_0| = r} \frac{f(w)}{(w - z_0)^{-n}} \, dw \cdot (z - z_0)^{-n-1} \end{array} </math></center>Thus, it follows that for <math>z \in A_{R_1, R_2}</math>: <center><math>f(z) = \frac{1}{2\pi i} \sum_{n=-\infty}^\infty \int_{|w - z_0| = r} \frac{f(w)}{(w - z_0)^{n+1}} \, dw \cdot (z - z_0)^n</math></center>which proves the existence of the claimed Laurent expansion. == See Also == *[[Complex Analysis/Examples of Laurent Series Expansion|Examples of Laurent Series Expansion]] *[[Laurent Series]] == Page information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Laurententwicklung Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Laurententwicklung |Kurs:Funktionentheorie/Laurententwicklung]] - URL:https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Laurententwicklung * Date: 11/26/2024 <span type="translate" src="Kurs:Funktionentheorie/Laurententwicklung " srclang="de" date="12/26/2024" time="01:39" status="inprogress"></span> <noinclude>[[de:Kurs:Funktionentheorie/Laurententwicklung]]</noinclude> [[Category:Wiki2Reveal]] carynhhuarp8uo9u5zblghexxr558bu 0 317515 2693307 2024-12-26T14:49:41Z Eyoungstrom 1933979 start page with instructions for cloning package list from ChatGPT 2693307 wikitext text/x-wiki =RStudio Setup= We suggest R Studio as the platform for working with R (rather than Base R, or JASP until it develops a robust way of exporting code for reproducibility). Steps: # Download R # Download R Studio (personal installation) # ''(Optional)'' Download R tools Next we can get a list of packages that are often used in our workflows. A few approaches are commonly used to replicate (or “clone”) an R package environment across different machines. Below are two popular strategies: one using base R commands, and another using renv for stricter reproducibility. '''1. Using Base R Commands''' Exporting the Package List On the source machine, get a list of all user-installed packages (excluding base and recommended packages) and save it to a file: r Copy code # List of all installed packages installed <- installed.packages() # We usually want to exclude base and recommended packages user_installed <- installed[is.na(installed[, "Priority"]), "Package"] # Save package list to an RDS file saveRDS(user_installed, "user_installed_pkgs.rds") Copy user_installed_pkgs.rds to the target machine. Installing on the New Machine On the target machine, read the package list and install: r Copy code # Load the list of packages pkgs_to_install <- readRDS("user_installed_pkgs.rds") # Install all packages in that list install.packages(pkgs_to_install) This approach is straightforward and very flexible. You can edit the list or exclude certain packages if needed before installing. However, you may lose fine-grained control over package versions or dependencies if CRAN has been updated in the meantime. 2. Using the renv Package for Reproducibility If your goal is to exactly replicate package versions (including dependencies), renv is often considered the most robust approach. It takes a “snapshot” of your project’s environment (with package versions) so you can restore it on another machine. This is especially important for reproducible research and collaboration. On the source machine (in your R project folder), initialize renv: r Copy code install.packages("renv") library(renv) renv::init() This will create a renv.lock file that records all of the package names and exact versions used in the project. Copy the entire project folder (including the renv folder and renv.lock file) to the target machine. On the target machine, open the R project folder and run: r Copy code renv::restore() This will install all packages (and their dependencies) in the same versions that were recorded in renv.lock. Because renv locks both package names and exact versions, this is the preferred way to ensure strict reproducibility—much more so than simply exporting and installing whatever is current on CRAN. Which Method Should You Use? Simple duplication of current packages: Use the base R approach (export a list, then install on the new machine). Exact duplication of package versions: Use renv for a fully reproducible environment—especially critical for research, production code, or collaboration. In most modern workflows, renv is recommended because it solves version mismatch headaches and ensures your code runs exactly the same way on different systems or at different points in time. na2i4mv9lg3e3mx74pyd273vikln5ye 1 317516 2693308 2024-12-26T14:52:26Z Eyoungstrom 1933979 /* Tips for working with R and Studio */ new section 2693308 wikitext text/x-wiki == Tips for working with R and Studio == Here's the link back to the ChatGPT: https://chatgpt.com/share/e/676d6d2b-c8bc-800d-bc17-0022cfc03243 [[User:Eyoungstrom|Eyoungstrom]] ([[User talk:Eyoungstrom|discuss]] • [[Special:Contributions/Eyoungstrom|contribs]]) 14:52, 26 December 2024 (UTC) pxtsw72grh45n16i3fqzlkq6yqioazy 0 317517 2693309 2024-12-26T15:07:22Z Eshaa2024 2993595 New resource with "== Introduction == In the following lesson, we first make an identification of the complex numbers <math>\mathbb{C}</math> with the two-dimensional <math>\mathbb{R}</math>-vector space <math>\mathbb{R}^2</math>, then we consider the classical real partial derivatives and the Jacobian matrix, and investigate the relationship between complex differentiability and partial derivatives of component functions of a map from <math>\mathbb{R}^2</math> to <math>\mathbb{R}^2</math>..." 2693309 wikitext text/x-wiki == Introduction == In the following lesson, we first make an identification of the complex numbers <math>\mathbb{C}</math> with the two-dimensional <math>\mathbb{R}</math>-vector space <math>\mathbb{R}^2</math>, then we consider the classical real partial derivatives and the Jacobian matrix, and investigate the relationship between complex differentiability and partial derivatives of component functions of a map from <math>\mathbb{R}^2</math> to <math>\mathbb{R}^2</math>. After that, the Cauchy-Riemann differential equations are proven based on these preliminary considerations. == Identification of Complex Numbers with <math>\mathbb{R}^2</math> == Let <math>R: \mathbb{C} \rightarrow \mathbb{R}^2, \ x+iy \mapsto R(x+iy) = \begin{pmatrix} x \\ y \end{pmatrix}</math>. Since the mapping <math>R</math> is bijective, the inverse mapping : <math> R^{-1}: \mathbb{R}^2 \rightarrow \mathbb{C}, \ \begin{pmatrix} x \\ y \end{pmatrix} \mapsto R^{-1}\begin{pmatrix} x \\ y \end{pmatrix} = x+iy </math> maps vectors from <math>\mathbb{R}^2</math> one-to-one back to a complex number. [[File:Cauchy Riemann DGL audio0.ogg|Identification of <math>\mathbb{C}</math> with <math>\mathbb{R}^2</math>]] === Real and Imaginary Part Functions === Now, if we decompose a function <math>f:U \rightarrow \mathbb{C}</math> with <math>f\left(x+iy\right)=u\left(x,y\right) + i,v\left(x,y\right)</math> into its real and imaginary parts with real functions <math>u: U_R \rightarrow \mathbb{R}</math>, <math>v: U_R \rightarrow \mathbb{R}</math> where <math>U_R \subset \mathbb{R}^2</math> and <math>U = {x+iy\in\mathbb{C} \ | \ (x,y) \in U_R}</math>, then the total derivative of the function <math>f_R: U_R\rightarrow \mathbb{R}^2, (x,y)\mapsto \begin{pmatrix} u\left(x,y\right) \ v\left(x,y\right) \end{pmatrix} </math> has the following [[w:en:Jacobian matrix and determinant|Jacobian matrix]] as its representation: <math>\begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}.</math> [[File:Cauchy Riemann DGL audio1.ogg]] == Task == For the complex-valued function <math>f:\mathbb{C}\rightarrow \mathbb{C}, \ z \mapsto f(z)=z^3</math>, give the mappings <math>u,v</math> with <math>f\left(x+iy\right)=u\left(x,y\right) + i\,v\left(x,y\right) </math> explicitly. [[File:Cauchy Riemann DGL Task audio2.ogg|Task]] === Evaluation of the Jacobian Matrix at a Point === The evaluation of the Jacobian matrix at a point <math>(x_o,y_o)\in \mathbb{R}^2</math> gives the total derivative at the point <math>x_o + iy_o \in \mathbb{C}</math> :<math> \begin{pmatrix} \frac{\partial u}{\partial x}(x_o,y_o) & \frac{\partial u}{\partial y}(x_o,y_o) \ \frac{\partial v}{\partial x}(x_o,y_o) & \frac{\partial v}{\partial y}(x_o,y_o)\end{pmatrix} </math> [[File:Cauchy Riemann DGL Evaluation Jacobian audio3.ogg|Evaluation of partial derivatives at a point]] === Cauchy-Riemann Differential Equations === A function <math>f</math> is complex differentiable at <math>z_o:=x_o+iy_o</math> if and only if it is real differentiable and the Cauchy-Riemann differential equations hold for <math>u, v</math> with <math>u: U_R \rightarrow \mathbb{R}</math>, <math>v: U_R \rightarrow \mathbb{R}</math> where <math>U_R \subset \mathbb{R}^2</math>: :<math>\frac{\partial u}{\partial x}(x_o,y_o) = \frac{\partial v}{\partial y}(x_o,y_o)</math> :<math>\displaystyle\frac{\partial u}{\partial y}(x_o,y_o) = -\frac{\partial v}{\partial x} (x_o,y_o) </math> are satisfied. [[File:Cauchy Riemann DGL audio4.ogg|Cauchy-Riemann Differential Equations]] == Relationship Between the Partial Derivatives == In the following explanations, the definition of differentiability in <math>\mathbb{C}</math> to properties of the partial derivatives in the Jacobian matrix. === Part 1 === If the following limit exists for <math>f:G\rightarrow \mathbb{C}</math> at <math>z_o\in G</math> with <math>G\subset \mathbb{C}</math> open: :<math>f'(z_o) = \lim_{z\rightarrow z_o} \frac{f(z)-f(z_o)}{z-z_o}</math>, then for any sequences <math>(z_n){n\in\mathbb{N}}</math> in the domain <math>G\subset \mathbb{C}</math> with <math>\lim{n\rightarrow \infty} z_n = z_o</math>, we also have: :<math>f'(z_o) = \lim_{n\rightarrow \infty} \frac{f(z_n)-f(z_o)}{z_n - z_o}</math> [[File:Cauchy Riemann DGL audio5.ogg|Sequence difference quotient]] === Part 2 === Now consider only the sequences for the two following limit processes with <math>h\in \mathbb{R}</math>: :<math>f'(z_o) = \lim_{h\rightarrow 0} \frac{f(z_o+h)-f(z_o)}{(z_o+h)-z_o} = \lim_{h\rightarrow 0} \frac{f(z_o+h)-f(z_o)}{h}</math>, :<math>f'(z_o) = \lim_{ih\rightarrow 0} \frac{f(z_o+ih)-f(z_o)}{(z_o+ih)-z_o} = \lim_{ih\rightarrow 0} \frac{f(z_o+ih)-f(z_o)}{ih}</math>, [[File:Cauchy Riemann DGL audio6.ogg|Real and imaginary sequences]] === Part 3: Limit Process for Real Part === By inserting the component functions for the real and imaginary parts <math>u,v</math>, we get with <math>h\in \mathbb{R}</math>: :<math>f'(z_o) = \lim_{h\rightarrow 0} \frac{f(z_o+h)-f(z_o)}{h} = </math> ::<math> = \lim_{h\rightarrow 0} \frac{u(x_o+h,y_o)-u(x_o,y_o)}{h} + i \lim_{h\rightarrow 0} \frac{v(x_o+h,y_o)-v(x_o,y_o)}{h}</math> ::<math> = \frac{\partial u}{\partial x}(x_o,y_o) + i \frac{\partial v}{\partial x}(x_o,y_o)</math> [[File:Cauchy Riemann DGL audio7.ogg|Limit process in the direction of the real part]] === Part 4: Limit Process for Imaginary Part === Applying this to the second equation, we get with <math>h\in \mathbb{R}</math>: :<math>f'(z_o) = \lim_{ih\rightarrow 0} \frac{f(z_o+ih)-f(z_o)}{ih} </math> ::<math> = \lim_{h\rightarrow 0} \frac{u(x_o,y_o+h)-u(x_o,y_o)}{ih} + i \lim_{h\rightarrow 0} \frac{v(x_o,y_o+h)-v(x_o,y_o)}{ih} </math> ::<math>= -i \lim_{h\rightarrow 0} \frac{u(x_o,y_o+h)-u(x_o,y_o)}{h} + \lim_{h\rightarrow 0} \frac{v(x_o,y_o+h)-v(x_o,y_o)}{h} </math>, ::<math>= -i \frac{\partial u}{\partial y}(x_o,y_o) + \frac{\partial v}{\partial y}(x_o,y_o)</math> [[File:Cauchy Riemann Differential Equations Audio8.ogg|Limit Process in the Direction of the Imaginary Part]] === Remark on Part 4 === In the first summand, the fraction is extended by <math>i</math> , and in the second summand <math>i</math>, the is canceled so that the denominator becomes real-valued and <math>h</math> corresponds. === Part 5: Comparison of Real and Imaginary Parts === By equating the terms from (3) and (4) and comparing the real and imaginary parts, we obtain the Cauchy-Riemann differential equations. *Real part: <math>\frac{\partial u}{\partial x}(x_o,y_o) = \frac{\partial v}{\partial y}(x_o,y_o)</math> *Imaginary part: <math>\displaystyle\frac{\partial u}{\partial y}(x_o,y_o) = -\frac{\partial v}{\partial x} (x_o,y_o) </math> [[File: Cauchy Riemann Differential Equations Audio9.ogg | Comparison of Real and Imaginary Parts of the Derivatives]] === Part 6: Partial Derivative in the Direction of the Real Part === The partial derivatives in <math>\mathbb{R}^2</math> of the Cauchy-Riemann differential equations can also be expressed in <math>\mathbb{C}</math> with <math>f := \mathfrak{Re}(f) + i \mathfrak{Im}(f)</math>, <math>\mathfrak{Re}(f): \mathbb{C} \rightarrow \mathbb{R}</math>, <math>\mathfrak{Im}(f): \mathbb{C} \rightarrow \mathbb{R}</math>, and <math>h \in \mathbb{R}</math>. :<math>\frac{\partial f}{\partial x}(z_o) = \lim_{h\rightarrow 0} \frac{f(z_o+h)-f(z_o)}{h}\in \mathbb{C}</math>, :<math>\frac{\partial \mathfrak{Re}(f)}{\partial x}(z_o) = \lim_{h\rightarrow 0} \frac{\mathfrak{Re}(f)(z_o+h)-\mathfrak{Re}(f)(z_o)}{h}\in \mathbb{R}</math>, :<math>\frac{\partial \mathfrak{Im}(f)}{\partial x}(z_o) = \lim_{h\rightarrow 0} \frac{\mathfrak{Im}(f)(z_o+h)-\mathfrak{Im}(f)(z_o)}{h} \in \mathbb{R}</math>. [[File:Cauchy Riemann DGL audio10.ogg|Partial Derivative in the Direction of the Real Part]] === Part 7: Partial Derivative in the Direction of the Imaginary Part === The partial derivatives in <math>\mathbb{R}^2</math> of the Cauchy-Riemann differential equations can also be expressed in <math>\mathbb{C}</math> with <math>f := \mathfrak{Re}(f) + i \mathfrak{Im}(f)</math>, <math>\mathfrak{Re}(f): \mathbb{C} \rightarrow \mathbb{R}</math>, <math>\mathfrak{Im}(f): \mathbb{C} \rightarrow \mathbb{R}</math>, and <math>h \in \mathbb{R}</math>. :<math>\frac{\partial f}{\partial y}(z_o) = \lim_{h\rightarrow 0} \frac{f(z_o+ih)-f(z_o)}{h}\in \mathbb{C}</math>, :<math>\frac{\partial \mathfrak{Re}(f)}{\partial y}(z_o) = \lim_{h\rightarrow 0} \frac{\mathfrak{Re}(f)(z_o+ih)-\mathfrak{Re}(f)(z_o)}{h}\in \mathbb{R}</math>, :<math>\frac{\partial \mathfrak{Im}(f)}{\partial y}(z_o) = \lim_{h\rightarrow 0} \frac{\mathfrak{Im}(f)(z_o+ih)-\mathfrak{Im}(f)(z_o)}{h} \in \mathbb{R}</math>. [[File:Cauchy Riemann DGL audio10.ogg|Partial Derivative in the Direction of the Imaginary Part]] === Part 8: Cauchy-Riemann DGL with Functions in <math>\mathbb{C}</math> === The partial derivatives of the Cauchy-Riemann differential equations can also be expressed in <math>\mathbb{C}</math> with <math>f := f_x + i f_y</math>, <math>f_x := \mathfrak{Re}(f)</math>, <math>f_y := \mathfrak{Im}(f)</math>: Real part: <math>\frac{\partial \mathfrak{Re}(f)}{\partial x}(z_o) = \frac{\partial \mathfrak{Im}(f)}{\partial y}(z_o)</math> Imaginary part: <math>\displaystyle\frac{\partial \mathfrak{Re}(f)}{\partial y}(z_o) = -\frac{\partial \mathfrak{Im}(f)}{\partial x}(z_o)</math> == Theorem - Cauchy-Riemann DGL == Let <math display="inline">G \subseteq \mathbb{C}</math> be an open subset. The function <math display="inline">f = u + i v</math> is complex differentiable at a point <math display="inline">z = x+iy \in G</math>. Then, the partial derivatives of <math display="inline">u</math> and <math display="inline">v</math> exist at <math display="inline">\left( x,y \right) \in \mathbb{R}^2</math>, and the following Cauchy-Riemann differential equations hold: <math display="block">\dfrac{\partial u}{\partial x} \left( x,y \right) = \dfrac{\partial v}{\partial y} \left( x,y \right)</math> <math display="block">\dfrac{\partial u}{\partial y} \left( x,y \right) = - \dfrac{\partial v}{\partial x} \left( x,y \right)</math> === Remark on CR-DGL === In this case, the derivative of <math display="inline">f</math> at the point <math display="inline">z \in \mathbb{C}</math> can be represented in two ways using the component functions <math display="inline">u</math> and <math display="inline">v</math>: <math display="block">f'\left( z \right) = \dfrac{\partial u}{\partial x} \left( x,y \right) - i \dfrac{\partial u}{\partial y} \left( x,y \right) = \dfrac{\partial v}{\partial y} \left( x,y \right) + i \dfrac{\partial v}{\partial x} \left( x,y \right)</math> The proof of the Cauchy-Riemann differential equations uses a comparison of the real and imaginary parts to derive the above equations. == Proof == The proof considers two directional derivatives: *(DG1) the derivative in the direction of the real part and *(DG2) the derivative in the direction of the imaginary part. Since these coincide for complex differentiability, the Cauchy-Riemann differential equations are obtained by setting them equal and comparing the real and imaginary parts. === Step 1 - Derivative in the Direction of the Real Part === In the first step, let <math>h \in \mathbb{C}</math> converge to 0 in the direction of the real part. To achieve this, choose <math display="inline"> h := h_1 + i \cdot 0 </math> with <math display="inline"> h_1 \in \mathbb{R} </math>. The decomposition of the function <math display="inline"> f = u + i v </math> into its real part <math>u</math> and imaginary part <math>v</math> then yields (DG1). === Step 2 - Calculation of the Derivative - Real Part === <math display="block"> \begin{array}{rcl} f'\left( z \right) & = & \lim\limits_{h \to 0} \dfrac{f\left( z+h \right) - f\left( z \right)}{h} \\ & = &\lim\limits_{h_1 \to 0} \dfrac{u\left( x+h_1,y \right) + iv\left(x+h_1,y \right) -u\left( x,y \right) -iv\left( x,y \right) }{h_1} \\ & = &\lim\limits_{h_1 \to 0} \dfrac{u\left( x+h_1 ,y \right) - u\left( x,y \right) }{h_1} +i\dfrac{v\left( x+h_1 ,y \right) - v\left( x,y \right) }{h_1} \\ & = & \dfrac{\partial u}{\partial x} \left( x,y \right) + i\dfrac{\partial v}{\partial x} \left( x,y \right) \end{array} </math> === Step 3 - Derivative in the Direction of the Imaginary Part === Similarly, the partial derivative for the imaginary part can be considered with <math display="inline"> h := 0 + i \cdot h_2 </math> and <math display="inline"> h_2 \in \mathbb{R} </math>. This yields equation (DG2). === Step 4 - Calculation of the Derivative - Imaginary Part === <math display="inline"> \begin{array}{rcl} f'\left( z \right) & = & \lim\limits_{h \to 0} \dfrac{f\left( z+h \right) - f\left( z \right)}{h}\\ & = &\lim\limits_{h_2 \to 0} \dfrac{u\left( x,y+h_2 \right) + iv\left(x,y+h_2 \right) -u\left( x,y \right) -iv\left( x,y \right) }{i\cdot h_2}\\ & = &\lim\limits_{l \to 0} \dfrac{1}{i}\dfrac{u\left( x,y+h_2 \right) - u\left( x,y \right) }{h_2} +\dfrac{v\left( x,y+h_2 \right) - v\left( x,y \right) }{h_2}\\ & = & \dfrac{\partial v}{\partial y} \left( x,y \right) - i\dfrac{\partial u}{\partial y} \left( x,y \right) \end{array} </math> === Step 5 - Equating the Derivatives === By equating the two derivatives, one can compare the real and imaginary parts of the two derivatives (DG1) and (DG2): <math display="block"> f'\left( z \right) = \dfrac{\partial u}{\partial x} \left( x,y \right) + i \dfrac{\partial v}{\partial x} \left( x,y \right) = \dfrac{\partial v}{\partial y} \left( x,y \right) - i \dfrac{\partial u}{\partial y} \left( x,y \right) </math> === Step 6 - Comparison of Real and Imaginary Parts === Two complex numbers are equal if and only if their real and imaginary parts are equal. This results in the Cauchy-Riemann differential equations. The two representation formulas follow from the above equation and the Cauchy-Riemann equations. == See also == *[[Complex Analysis]] *[[w:en:Wirtinger calculus|Wirtinger derivatives or Wirtinger calculus]] *[[Complex Analysis/Maximum Principle|Maximum principle]] *[[Complex_number#Real_and_imaginary_part_functions|Real and imaginary part functions]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Cauchy-Riemann-Differential%20equation https://en.wikiversity.org/wiki/Complex%20Analysis/Cauchy-Riemann-Differential%20equation] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Cauchy-Riemann-Differential%20equation This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Cauchy-Riemann-Differentialgleichungen Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Cauchy-Riemann-Differentialgleichungen/Cauchy-Riemann-Differentialgleichungen]] - URL: https://de.wikiversity.org/wiki/Cauchy-Riemann-Differentialgleichungen * Date: 12/26/2024 <span type="translate" src="Cauchy-Riemann-Differentialgleichungen" srclang="de" date="12/26/2024" time="04:10" status="inprogress"></span> <noinclude> [[de:Cauchy-Riemann-Differentialgleichungen]] </noinclude> [[Category:Wiki2Reveal]] 6u0rtl2pqt51e0601ft9xlb8a5iyf9k 0 317518 2693311 2024-12-26T15:44:02Z Eshaa2024 2993595 New resource with "== Introduction == The maximum principle is a statement about holomorphic functions from the [[Complex Analysis|Complex Analysis]]. The magnitude <math>|f|</math> of a holomorphic function <math>f: G \to \mathbb{C}</math> cannot attain any strict local maxima within the domain of definition. Specifically, it asserts the following statement. == Statement == Let <math>U \subseteq \mathbb{C}</math> be a domain, and let <math>f: U \to \mathbb{C}</math> be w:en:Holomorphic..." 2693311 wikitext text/x-wiki == Introduction == The maximum principle is a statement about holomorphic functions from the [[Complex Analysis|Complex Analysis]]. The magnitude <math>|f|</math> of a holomorphic function <math>f: G \to \mathbb{C}</math> cannot attain any strict local maxima within the domain of definition. Specifically, it asserts the following statement. == Statement == Let <math>U \subseteq \mathbb{C}</math> be a domain, and let <math>f: U \to \mathbb{C}</math> be [[w:en:Holomorphic function|holomorphic]]. If <math>|f|</math> has a local maximum in <math>U</math>, then <math>f</math> is constant. If <math>U</math> is bounded and <math>f</math> can be continuously extended to <math>\bar{U}</math>, then <math>f</math> attains its maximum on <math>\partial U</math>. To prove this, we require a lemma that locally implies the conclusion. === Lemma === Let <math>G \subseteq \mathbb{C}</math> be open, and <math>f: G \to \mathbb{C}</math> be [[w:en:Holomorphic function|holomorphic]]. Let <math>z_0 \in G</math> be a local maximum point of <math>|f|</math>. Then <math>f</math> is constant in a neighborhood of <math>z_0</math>. Proof of Lemma 1 Let <math>r > 0</math> be chosen such that <math>|f(z)| \leq |f(z_0)|</math> for all <math>z \in \bar{D}r(z_0) \subseteq G</math>. The [[w:en:Cauchy Integral Formula#For disks|Cauchy integral formula]] gives, for all <math>\varepsilon \leq r</math>: :<math>f(z_0) = \frac{1}{2\pi i} \int{\partial D_\varepsilon(z_0)} \frac{f(z)}{z-z_0} , dz</math> This allows us to establish the following estimation: ==Proof of Lemma 2== We derive the following estimation: :<math> \begin{array}{rl} |f(z_0)| &= \frac 1{2\pi} \left| \int_{\partial D_\varepsilon(z_0)} \frac{f(z)}{z-z_0}\, dz \right| \\ &= \frac 1{2\pi} \left| \int_0^{2\pi} \frac {f(z_0 + \varepsilon e^{it})}{\varepsilon \cdot e^{it}} \varepsilon \cdot i \cdot e^{it}\, dt\right|\\ &\le \frac 1{2\pi} \int_0^{2\pi} |f(z_0 + \varepsilon \cdot e^{it})|\, dt\\ &\le \sup_{t \in[0,2\pi]} |f(z_0 + \varepsilon \cdot e^{it})|\\ &\le |f(z_0)| \end{array} </math> ==Proof of Lemma 3== It follows that the inequality <math>\leq</math> must be an equality chain, implying :<math> \begin{array}{rl} |f(z_0)| = & \int_0 ^{2 \pi}|f(z_0)| \cdot \frac 1 {2 \pi} dt = \frac 1{2\pi} \int_0^{2\pi} |f(z_0 + \varepsilon \cdot e^{it})| \, dt\\ \Rightarrow & 0 = \int_0 ^{2 \pi} (| f(z)| - |f(z_0)|)dt \\ \Rightarrow & |f(z)| = |f(z_0)| \end{array} </math>. === Proof of Lemma 4 === Thus, we establish the constancy of <math>|f|</math> using the property: :<math>|f(z)| = |f(z_0)|</math> for all <math>z \in \bar D_r(z_0)</math>, i.e., <math>|f|</math> is constant on <math>D_r(z_0)</math>. === Proof of Lemma 5 === If <math>|f|=\sqrt{\mathfrak{Re}(f)^2+\mathfrak{Im}(f)^2}</math> is constant on <math>D_r(z_0)</math>, then <math>\mathfrak{Re}(f)^2+\mathfrak{Im}(f)^2 = c</math> must also be constant, where <math>c \in \mathbb{R}</math> is a constant. === Proof of Lemma 6 === Since <math>f</math> is holomorphic on <math>D_r(z_0)</math>, the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]'apply: :<math> u := \mathfrak{Re}(f), \ v := \mathfrak{Im}(f), \ \text{and} \ f(z) := u(x,y) + i \cdot v(x,y)</math>, and the following holds: :<math>\frac{\partial u(x,y)^2 + v(x,y)^2}{\partial x} = 0 \ \text{and} \ \frac{\partial u(x,y)^2 + v(x,y)^2}{\partial y} = 0</math>. === Proof of Lemma 7 === Let <math>u_x = \frac{\partial u}{\partial x}</math> and <math>u_y = \frac{\partial u}{\partial y}</math>. Applying the chain rule to the partial derivatives, we obtain: :<math>0 = 2 u u_x + 2 v v_x</math> and <math>0 = 2 u u_y + 2 v v_y</math>. Using the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]', replace the partial derivatives of <math>v</math> with those of <math>u</math>: :<math>u_x = v_y</math> and <math>u_y = -v_x</math>, leading to: :<math>0 = 2 \cdot (u u_x - v u_y)</math> and <math>0 = 2 \cdot (u u_y + v u_x)</math>. === Proof of Lemma 8 === Squaring the above equations yields: :<math>0 = (u u_x - v u_y)^2 = u^2 u_x^2 - 2 u u_x \cdot v u_y + v^2 u_y^2</math>, :<math>0 = (u u_y + v u_x)^2 = u^2 u_y^2 + 2 u u_y \cdot v u_x + v^2 u_x^2</math>. Adding these equations gives: :<math>0 = u^2 u_x^2 + v^2 u_y^2 + u^2 u_y^2 + v^2 u_x^2</math>. === Proof of Lemma 9 === Factoring out <math>u^2</math> and <math>v^2</math>: :<math>0 = u^2 (u_x^2 + u_y^2) + v^2 (u_x^2 + u_y^2) = (u^2 + v^2) \cdot (u_x^2 + u_y^2)</math>. Thus, :<math>0 = u^2 + v^2</math> or <math>0 = u_x^2 + u_y^2</math>. === Proof of Lemma 10 === With <math>u^2 = -v^2</math>, it follows that <math>u = v = 0</math> since <math>u(x,y)</math> and <math>v(x,y)</math> are real-valued, implying <math>f = u + iv = 0</math>. If <math>u_x^2 = -u_y^2</math>, then <math>u_x^2 = u_y^2 = 0</math>, and <math>u_x = u_y = 0</math>. By the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]], <math>f' = 0</math>. Thus, <math>f</math> is constant on <math>D_r(z_0)</math>. == Proof == Let <math>z_0 \in G</math> be a local maximum point of <math>|f|</math> in the domain <math>G</math>. Define <math>V := {z \in G : f(z) = f(z_0)}</math> as the set of all <math>z \in G</math> mapped to <math>w := f(z_0) \in \mathbb{C}</math> (level set). === Proof 1: V is closed === Since <math>f</math> is continuous, preimages of open sets are open, and preimages of closed sets are closed (in the relative topology of <math>G</math>). Thus, <math>V = f^{-1}({w})</math> is closed in <math>G</math>. === Proof 2: V is open === Using the lemma, <math>V</math> can also be represented as a union of open disks, and unions of open sets are open. === Proof 3: Connectivity === Thus, <math>V = G</math> due to the connectivity of <math>U</math>, i.e., <math>f</math> is constant. === Proof 4: G is bounded === If <math>G</math> is bounded, then <math>\bar G</math> is compact. Therefore, the continuous function <math>f</math> attains its maximum on <math>\bar G</math>, say at <math>z_0 \in \bar G</math>. If <math>z_0 \in G</math>, then <math>f</math> is constant on <math>G</math> (by the lemma) and hence on <math>\bar G</math>, so <math>f</math> also attains its maximum on <math>\partial G</math>. Otherwise, <math>z_0 \in \partial G</math>, completing the proof. == See Also == *[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]' *[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Maximum%20Principle https://en.wikiversity.org/wiki/Complex%20Analysis/Maximum%20Principle] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Maximum%20Principle This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/MaximumprinzipWikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Kurven|Kurs:Funktionentheorie/Maximumprinzip]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Maximumprinzip * Date: 12/26/2024 <span type="translate" src="Kurs:Funktionentheorie/Maximumprinzip " srclang="de" date="12/17/2024" time="04:45" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Maximumprinzip]] </noinclude> [[Category:Wiki2Reveal]] aa85v2xb9fqxvux62joea8g3rrv722g 2693312 2693311 2024-12-26T15:47:08Z Eshaa2024 2993595 /* Lemma */ 2693312 wikitext text/x-wiki == Introduction == The maximum principle is a statement about holomorphic functions from the [[Complex Analysis|Complex Analysis]]. The magnitude <math>|f|</math> of a holomorphic function <math>f: G \to \mathbb{C}</math> cannot attain any strict local maxima within the domain of definition. Specifically, it asserts the following statement. == Statement == Let <math>U \subseteq \mathbb{C}</math> be a domain, and let <math>f: U \to \mathbb{C}</math> be [[w:en:Holomorphic function|holomorphic]]. If <math>|f|</math> has a local maximum in <math>U</math>, then <math>f</math> is constant. If <math>U</math> is bounded and <math>f</math> can be continuously extended to <math>\bar{U}</math>, then <math>f</math> attains its maximum on <math>\partial U</math>. To prove this, we require a lemma that locally implies the conclusion. === Lemma === Let <math>G \subseteq \mathbb{C}</math> be open, and <math>f: G \to \mathbb{C}</math> be [[w:en:Holomorphic function|holomorphic]]. Let <math>z_0 \in G</math> be a local maximum point of <math>|f|</math>. Then <math>f</math> is constant in a neighborhood of <math>z_0</math>. ==Proof of Lemma 1== Let <math>r > 0</math> be chosen such that <math>|f(z)| \leq |f(z_0)|</math> for all <math>z \in \bar{D}r(z_0) \subseteq G</math>. The [[Cauchy's integral formula|Cauchy's integral formula]]'gives, for all <math>\varepsilon \leq r</math>: :<math>f(z_0) = \frac{1}{2\pi i} \int{\partial D_\varepsilon(z_0)} \frac{f(z)}{z-z_0} , dz</math> This allows us to establish the following estimation: ==Proof of Lemma 2== We derive the following estimation: :<math> \begin{array}{rl} |f(z_0)| &= \frac 1{2\pi} \left| \int_{\partial D_\varepsilon(z_0)} \frac{f(z)}{z-z_0}\, dz \right| \\ &= \frac 1{2\pi} \left| \int_0^{2\pi} \frac {f(z_0 + \varepsilon e^{it})}{\varepsilon \cdot e^{it}} \varepsilon \cdot i \cdot e^{it}\, dt\right|\\ &\le \frac 1{2\pi} \int_0^{2\pi} |f(z_0 + \varepsilon \cdot e^{it})|\, dt\\ &\le \sup_{t \in[0,2\pi]} |f(z_0 + \varepsilon \cdot e^{it})|\\ &\le |f(z_0)| \end{array} </math> ==Proof of Lemma 3== It follows that the inequality <math>\leq</math> must be an equality chain, implying :<math> \begin{array}{rl} |f(z_0)| = & \int_0 ^{2 \pi}|f(z_0)| \cdot \frac 1 {2 \pi} dt = \frac 1{2\pi} \int_0^{2\pi} |f(z_0 + \varepsilon \cdot e^{it})| \, dt\\ \Rightarrow & 0 = \int_0 ^{2 \pi} (| f(z)| - |f(z_0)|)dt \\ \Rightarrow & |f(z)| = |f(z_0)| \end{array} </math>. === Proof of Lemma 4 === Thus, we establish the constancy of <math>|f|</math> using the property: :<math>|f(z)| = |f(z_0)|</math> for all <math>z \in \bar D_r(z_0)</math>, i.e., <math>|f|</math> is constant on <math>D_r(z_0)</math>. === Proof of Lemma 5 === If <math>|f|=\sqrt{\mathfrak{Re}(f)^2+\mathfrak{Im}(f)^2}</math> is constant on <math>D_r(z_0)</math>, then <math>\mathfrak{Re}(f)^2+\mathfrak{Im}(f)^2 = c</math> must also be constant, where <math>c \in \mathbb{R}</math> is a constant. === Proof of Lemma 6 === Since <math>f</math> is holomorphic on <math>D_r(z_0)</math>, the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]'apply: :<math> u := \mathfrak{Re}(f), \ v := \mathfrak{Im}(f), \ \text{and} \ f(z) := u(x,y) + i \cdot v(x,y)</math>, and the following holds: :<math>\frac{\partial u(x,y)^2 + v(x,y)^2}{\partial x} = 0 \ \text{and} \ \frac{\partial u(x,y)^2 + v(x,y)^2}{\partial y} = 0</math>. === Proof of Lemma 7 === Let <math>u_x = \frac{\partial u}{\partial x}</math> and <math>u_y = \frac{\partial u}{\partial y}</math>. Applying the chain rule to the partial derivatives, we obtain: :<math>0 = 2 u u_x + 2 v v_x</math> and <math>0 = 2 u u_y + 2 v v_y</math>. Using the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]', replace the partial derivatives of <math>v</math> with those of <math>u</math>: :<math>u_x = v_y</math> and <math>u_y = -v_x</math>, leading to: :<math>0 = 2 \cdot (u u_x - v u_y)</math> and <math>0 = 2 \cdot (u u_y + v u_x)</math>. === Proof of Lemma 8 === Squaring the above equations yields: :<math>0 = (u u_x - v u_y)^2 = u^2 u_x^2 - 2 u u_x \cdot v u_y + v^2 u_y^2</math>, :<math>0 = (u u_y + v u_x)^2 = u^2 u_y^2 + 2 u u_y \cdot v u_x + v^2 u_x^2</math>. Adding these equations gives: :<math>0 = u^2 u_x^2 + v^2 u_y^2 + u^2 u_y^2 + v^2 u_x^2</math>. === Proof of Lemma 9 === Factoring out <math>u^2</math> and <math>v^2</math>: :<math>0 = u^2 (u_x^2 + u_y^2) + v^2 (u_x^2 + u_y^2) = (u^2 + v^2) \cdot (u_x^2 + u_y^2)</math>. Thus, :<math>0 = u^2 + v^2</math> or <math>0 = u_x^2 + u_y^2</math>. === Proof of Lemma 10 === With <math>u^2 = -v^2</math>, it follows that <math>u = v = 0</math> since <math>u(x,y)</math> and <math>v(x,y)</math> are real-valued, implying <math>f = u + iv = 0</math>. If <math>u_x^2 = -u_y^2</math>, then <math>u_x^2 = u_y^2 = 0</math>, and <math>u_x = u_y = 0</math>. By the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]], <math>f' = 0</math>. Thus, <math>f</math> is constant on <math>D_r(z_0)</math>. == Proof == Let <math>z_0 \in G</math> be a local maximum point of <math>|f|</math> in the domain <math>G</math>. Define <math>V := {z \in G : f(z) = f(z_0)}</math> as the set of all <math>z \in G</math> mapped to <math>w := f(z_0) \in \mathbb{C}</math> (level set). === Proof 1: V is closed === Since <math>f</math> is continuous, preimages of open sets are open, and preimages of closed sets are closed (in the relative topology of <math>G</math>). Thus, <math>V = f^{-1}({w})</math> is closed in <math>G</math>. === Proof 2: V is open === Using the lemma, <math>V</math> can also be represented as a union of open disks, and unions of open sets are open. === Proof 3: Connectivity === Thus, <math>V = G</math> due to the connectivity of <math>U</math>, i.e., <math>f</math> is constant. === Proof 4: G is bounded === If <math>G</math> is bounded, then <math>\bar G</math> is compact. Therefore, the continuous function <math>f</math> attains its maximum on <math>\bar G</math>, say at <math>z_0 \in \bar G</math>. If <math>z_0 \in G</math>, then <math>f</math> is constant on <math>G</math> (by the lemma) and hence on <math>\bar G</math>, so <math>f</math> also attains its maximum on <math>\partial G</math>. Otherwise, <math>z_0 \in \partial G</math>, completing the proof. == See Also == *[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]' *[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Maximum%20Principle https://en.wikiversity.org/wiki/Complex%20Analysis/Maximum%20Principle] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Maximum%20Principle This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/MaximumprinzipWikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Kurven|Kurs:Funktionentheorie/Maximumprinzip]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Maximumprinzip * Date: 12/26/2024 <span type="translate" src="Kurs:Funktionentheorie/Maximumprinzip " srclang="de" date="12/17/2024" time="04:45" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Maximumprinzip]] </noinclude> [[Category:Wiki2Reveal]] pyhezzz8ykug5jjg3r65p3x7rxtof5k 2693313 2693312 2024-12-26T15:50:44Z Eshaa2024 2993595 2693313 wikitext text/x-wiki == Introduction == The maximum principle is a statement about holomorphic functions from the [[Complex Analysis|Complex Analysis]]. The magnitude <math>|f|</math> of a holomorphic function <math>f: G \to \mathbb{C}</math> cannot attain any strict local maxima within the domain of definition. Specifically, it asserts the following statement. == Statement == Let <math>U \subseteq \mathbb{C}</math> be a domain, and let <math>f: U \to \mathbb{C}</math> be [[w:en:Holomorphic function|holomorphic]]. If <math>|f|</math> has a local maximum in <math>U</math>, then <math>f</math> is constant. If <math>U</math> is bounded and <math>f</math> can be continuously extended to <math>\bar{U}</math>, then <math>f</math> attains its maximum on <math>\partial U</math>. To prove this, we require a lemma that locally implies the conclusion. === Lemma === Let <math>G \subseteq \mathbb{C}</math> be open, and <math>f: G \to \mathbb{C}</math> be [[w:en:Holomorphic function|holomorphic]]. Let <math>z_0 \in G</math> be a local maximum point of <math>|f|</math>. Then <math>f</math> is constant in a neighborhood of <math>z_0</math>. ==Proof of Lemma 1== Let <math>r > 0</math> be chosen such that <math>|f(z)| \leq |f(z_0)|</math> for all <math>z \in \bar{D}r(z_0) \subseteq G</math>. The [[Cauchy's integral formula|Cauchy's integral formula]]'gives, for all <math>\varepsilon \leq r</math>: :<math>f(z_0) = \frac{1}{2\pi i} \int{\partial D_\varepsilon(z_0)} \frac{f(z)}{z-z_0} , dz</math> This allows us to establish the following estimation: ==Proof of Lemma 2== We derive the following estimation: :<math> \begin{array}{rl} |f(z_0)| &= \frac 1{2\pi} \left| \int_{\partial D_\varepsilon(z_0)} \frac{f(z)}{z-z_0}\, dz \right| \\ &= \frac 1{2\pi} \left| \int_0^{2\pi} \frac {f(z_0 + \varepsilon e^{it})}{\varepsilon \cdot e^{it}} \varepsilon \cdot i \cdot e^{it}\, dt\right|\\ &\le \frac 1{2\pi} \int_0^{2\pi} |f(z_0 + \varepsilon \cdot e^{it})|\, dt\\ &\le \sup_{t \in[0,2\pi]} |f(z_0 + \varepsilon \cdot e^{it})|\\ &\le |f(z_0)| \end{array} </math> ==Proof of Lemma 3== It follows that the inequality <math>\leq</math> must be an equality chain, implying :<math> \begin{array}{rl} |f(z_0)| = & \int_0 ^{2 \pi}|f(z_0)| \cdot \frac 1 {2 \pi} dt = \frac 1{2\pi} \int_0^{2\pi} |f(z_0 + \varepsilon \cdot e^{it})| \, dt\\ \Rightarrow & 0 = \int_0 ^{2 \pi} (| f(z)| - |f(z_0)|)dt \\ \Rightarrow & |f(z)| = |f(z_0)| \end{array} </math>. === Proof of Lemma 4 === Thus, we establish the constancy of <math>|f|</math> using the property: :<math>|f(z)| = |f(z_0)|</math> for all <math>z \in \bar D_r(z_0)</math>, i.e., <math>|f|</math> is constant on <math>D_r(z_0)</math>. === Proof of Lemma 5 === If <math>|f|=\sqrt{\mathfrak{Re}(f)^2+\mathfrak{Im}(f)^2}</math> is constant on <math>D_r(z_0)</math>, then <math>\mathfrak{Re}(f)^2+\mathfrak{Im}(f)^2 = c</math> must also be constant, where <math>c \in \mathbb{R}</math> is a constant. === Proof of Lemma 6 === Since <math>f</math> is holomorphic on <math>D_r(z_0)</math>, the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]'apply: :<math> u := \mathfrak{Re}(f), \ v := \mathfrak{Im}(f), \ \text{and} \ f(z) := u(x,y) + i \cdot v(x,y)</math>, and the following holds: :<math>\frac{\partial u(x,y)^2 + v(x,y)^2}{\partial x} = 0 \ \text{and} \ \frac{\partial u(x,y)^2 + v(x,y)^2}{\partial y} = 0</math>. === Proof of Lemma 7 === Let <math>u_x = \frac{\partial u}{\partial x}</math> and <math>u_y = \frac{\partial u}{\partial y}</math>. Applying the chain rule to the partial derivatives, we obtain: :<math>0 = 2 u u_x + 2 v v_x</math> and <math>0 = 2 u u_y + 2 v v_y</math>. Using the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]', replace the partial derivatives of <math>v</math> with those of <math>u</math>: :<math>u_x = v_y</math> and <math>u_y = -v_x</math>, leading to: :<math>0 = 2 \cdot (u u_x - v u_y)</math> and <math>0 = 2 \cdot (u u_y + v u_x)</math>. === Proof of Lemma 8 === Squaring the above equations yields: :<math>0 = (u u_x - v u_y)^2 = u^2 u_x^2 - 2 u u_x \cdot v u_y + v^2 u_y^2</math>, :<math>0 = (u u_y + v u_x)^2 = u^2 u_y^2 + 2 u u_y \cdot v u_x + v^2 u_x^2</math>. Adding these equations gives: :<math>0 = u^2 u_x^2 + v^2 u_y^2 + u^2 u_y^2 + v^2 u_x^2</math>. === Proof of Lemma 9 === Factoring out <math>u^2</math> and <math>v^2</math>: :<math>0 = u^2 (u_x^2 + u_y^2) + v^2 (u_x^2 + u_y^2) = (u^2 + v^2) \cdot (u_x^2 + u_y^2)</math>. Thus, :<math>0 = u^2 + v^2</math> or <math>0 = u_x^2 + u_y^2</math>. === Proof of Lemma 10 === With <math>u^2 = -v^2</math>, it follows that <math>u = v = 0</math> since <math>u(x,y)</math> and <math>v(x,y)</math> are real-valued, implying <math>f = u + iv = 0</math>. If <math>u_x^2 = -u_y^2</math>, then <math>u_x^2 = u_y^2 = 0</math>, and <math>u_x = u_y = 0</math>. By the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]], <math>f' = 0</math>. Thus, <math>f</math> is constant on <math>D_r(z_0)</math>. == Proof == Let <math>z_0 \in G</math> be a local maximum point of <math>|f|</math> in the domain <math>G</math>. Define <math>V := {z \in G : f(z) = f(z_0)}</math> as the set of all <math>z \in G</math> mapped to <math>w := f(z_0) \in \mathbb{C}</math> (level set). === Proof 1: V is closed === Since <math>f</math> is continuous, preimages of open sets are open, and preimages of closed sets are closed (in the relative topology of <math>G</math>). Thus, <math>V = f^{-1}({w})</math> is closed in <math>G</math>. === Proof 2: V is open === Using the lemma, <math>V</math> can also be represented as a union of open disks, and unions of open sets are open. === Proof 3: Connectivity === Thus, <math>V = G</math> due to the connectivity of <math>U</math>, i.e., <math>f</math> is constant. === Proof 4: G is bounded === If <math>G</math> is bounded, then <math>\bar G</math> is compact. Therefore, the continuous function <math>f</math> attains its maximum on <math>\bar G</math>, say at <math>z_0 \in \bar G</math>. If <math>z_0 \in G</math>, then <math>f</math> is constant on <math>G</math> (by the lemma) and hence on <math>\bar G</math>, so <math>f</math> also attains its maximum on <math>\partial G</math>. Otherwise, <math>z_0 \in \partial G</math>, completing the proof. == See Also == *[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]' *[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Maximum%20Principle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Maximum%20Principle https://en.wikiversity.org/wiki/Complex%20Analysis/Maximum%20Principle] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Maximum%20Principle This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/MaximumprinzipWikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Maximumprinzip|Kurs:Funktionentheorie/Maximumprinzip]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Maximumprinzip * Date: 12/26/2024 <span type="translate" src="Kurs:Funktionentheorie/Maximumprinzip " 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Acronym !! Full name |- | A || Address |- | ACL || Access Control List |- | AH || Authentication Header |- | AP || Access Point |- | API || Application Programming Interface |- | APIPA || Automatic Private Internet Protocol Addressing |- | ARP || Address Resolution Protocol |- | AUP || Acceptable Use Policy |- | BGP || Border Gateway Protocol |- | BNC || Bayonet Neill–Concelman |- | BSSID || Basic Service Set Identifier |- | BYOD || Bring Your Own Device |- | CAM || Content-addressable Memory |- | CDN || Content Delivery Network |- | CDP || Cisco Discovery Protocol |- | CIA || Confidentiality, Integrity, and Availability |- | CIDR || Classless Inter-domain Routing |- | CLI || Command-line Interface |- | CNAME || Canonical Name |- | CPU || Central Processing Unit |- | CRC || Cyclic Redundancy Check |- | DAC || Direct Attach Copper |- | DAS || Direct-attached Storage |- | DCI || Data Center Interconnect |- | DDoS || Distributed Denial-of-service |- | DHCP || Dynamic Host Configuration Protocol |- | DLP || Data Loss Prevention |- | DNS || Domain Name System |- | DNSSEC || Domain Name System Security Extensions |- | DoH || DNS over Hypertext Transfer Protocol Secure |- | DoS || Denial-of-service |- | DoT || DNS over Transport Layer Security |- | DR || Disaster Recovery |- | EAPoL || Extensible Authentication Protocol over LAN |- | EIGRP || Enhanced Interior Gateway Routing Protocol |- | EOL || End-of-life |- | EOS || End-of-support |- | ESP || Encapsulating Security Payload |- | ESSID || Extended Service Set Identifier |- | EULA || End User License Agreement |- | FC || Fibre Channel |- | FHRP || First Hop Redundancy Protocol |- | FTP || File Transfer Protocol |- | GDPR || General Data Protection Regulation |- | GRE || Generic Routing Encapsulation |- | GUI || Graphical User Interface |- | HTTP || Hypertext Transfer Protocol |- | HTTPS || Hypertext Transfer Protocol Secure |- | IaaS || Infrastructure as a Service |- | IaC || Infrastructure as Code |- | IAM || Identity and Access Management |- | ICMP || Internet Control Message Protocol |- | ICS || Industrial Control System |- | IDF || Intermediate Distribution Frame |- | IDS || Intrusion Detection System |- | IoT || Internet of Things |- | IIoT || Industrial Internet of Things |- | IKE || Internet Key Exchange |- | IP || Internet Protocol |- | IPAM || Internet Protocol Address Management |- | IPS || Intrusion Prevention System |- | IPSec || Internet Protocol Security |- | IS-IS || Intermediate System to Intermediate System |- | LACP || Link Aggregation Control Protocol |- | LAN || Local Area Network |- | LC || Local Connector |- | LDAP || Lightweight Directory Access Protocol |- | LDAPS || Lightweight Directory Access Protocol over SSL |- | LLDP || Link Layer Discovery Protocol |- | MAC || Media Access Control |- | MDF || Main Distribution Frame |- | MDIX || Medium Dependent Interface Crossover |- | MFA || Multifactor Authentication |- | MIB || Management Information Base |- | MPO || Multifiber Push On |- | MTBF || Mean Time Between Failure |- | MTTR || Mean Time To Repair |- | MTU || Maximum Transmission Unit |- | MX || Mail Exchange |- | NAC || Network Access Control |- | NAS || Network-attached Storage |- | NAT || Network Address Translation |- | NFV || Network Functions Virtualization |- | NIC || Network Interface Cards |- | NS || Name Server |- | NTP || Network Time Protocol |- | NTS || Network Time Security |- | OS || Operating System |- | OSPF || Open Shortest Path First |- | OSI || Open Systems Interconnection |- | OT || Operational Technology |- | PaaS || Platform as a Service |- | PAT || Port Address Translation |- | PCI || DSS Payment Card Industry Data Security Standards |- | PDU || Power Distribution Unit |- | PKI || Public Key Infrastructure |- | PoE || Power over Ethernet |- | PSK || Pre-shared Key |- | PTP || Precision Time Protocol |- | PTR || Pointer |- | QoS || Quality of Service |- | QSFP || Quad Small Form-factor Pluggable |- | RADIUS || Remote Authentication Dial-in User Service |- | RDP || Remote Desktop Protocol |- | RFID || Radio Frequency Identifier |- | RIP || Routing Information Protocol |- | RJ || Registered Jack |- | RPO || Recovery Point Objective |- | RSTP || Rapid Spanning Tree Protocol |- | RTO || Recovery Time Objective |- | RX || Receiver |- | SaaS || Software as a Service |- | SAML || Security Assertion Markup Language |- | SAN || Storage Area Network |- | SASE || Secure Access Service Edge |- | SC || Subscriber Connector |- | SCADA || Supervisory Control and Data Acquisition |- | SDN || Software-defined Network |- | SD-WAN || Software-defined Wide Area Network |- | SFP || Small Form-factor Pluggable |- | SFTP || Secure File Transfer Protocol |- | SIP || Session Initiation Protocol |- | SIEM || Security Information and Event Management |- | SLA || Service-level Agreement |- | SLAAC || Stateless Address Autoconfiguration |- | SMB || Server Message Block |- | SMTP || Simple Mail Transfer Protocol |- | SMTPS || Simple Mail Transfer Protocol Secure |- | SNMP || Simple Network Management Protocol |- | SOA || Start of Authority |- | SQL || Structured Query Language |- | SSE || Security Service Edge |- | SSH || Secure Shell |- | SSID || Service Set Identifier |- | SSL || Secure Socket Layer |- | SSO || Single Sign-on |- | ST || Straight Tip |- | STP || Shielded Twisted Pair |- | SVI || Switch Virtual Interface |- | TACAS+ || Terminal Access Controller Access Control System Plus |- | TCP || Transmission Control Protocol |- | TFTP || Trivial File Transfer Protocol |- | TTL || Time to Live |- | TX || Transmitter |- | TXT || Text |- | UDP || User Datagram Protocol |- | UPS || Uninterruptible Power Supply |- | URL || Uniform Resource Locator |- | USB || Universal Serial Bus |- | UTM || Unified Threat Management |- | UTP || Unshielded Twisted Pair |- | VIP || Virtual IP |- | VLAN || Virtual Local Area Network |- | VLSM || Variable Length Subnet Mask |- | VoIP || Voice over IP |- | VPC || Virtual Private Cloud |- | VPN || Virtual Private Network |- | WAN || Wide Area Network |- | WPA || Wi-Fi Protected Access |- | WPS || Wi-Fi Protected Setup |- | VXLAN || Virtual Extensible LAN |- | ZTA || Zero Trust Architecture |} hwkoy4w2po6nybfdcmw2q6m9389wvsh 0 317522 2693327 2024-12-26T18:56:18Z Watchduck 137431 Watchduck moved page [[Boolf prop/3-ary/half battalion]] to [[Boolf prop/3-ary/chunky burden]] 2693327 wikitext text/x-wiki #REDIRECT [[Boolf prop/3-ary/chunky burden]] e6l6bwrnuxd57me8w3dqtnxoz1t8goa 0 317523 2693329 2024-12-26T18:57:23Z Watchduck 137431 Watchduck moved page [[Boolf prop/3-ary/weight pair]] to [[Boolf prop/3-ary/burden]] 2693329 wikitext text/x-wiki #REDIRECT [[Boolf prop/3-ary/burden]] 9nukm2bg8j0nw73tcwwj526c6k5o80y 0 317524 2693334 2024-12-26T18:59:56Z Watchduck 137431 Watchduck moved page [[Boolf prop/3-ary/battalion]] to [[Boolf prop/3-ary/super chunky burden]] 2693334 wikitext text/x-wiki #REDIRECT [[Boolf prop/3-ary/super chunky burden]] pxnesjelf0chg712ubux1lsduvkm1ef 0 317525 2693341 2024-12-26T19:12:26Z Watchduck 137431 Watchduck moved page [[Boolf prop/3-ary/greater twin mentor]] to [[Boolf prop/3-ary/super great twin mentor]] 2693341 wikitext text/x-wiki #REDIRECT [[Boolf prop/3-ary/super great twin mentor]] 409u2d1imhhkpvz6degrclvbobpqqaw 0 317526 2693347 2024-12-26T19:16:52Z Watchduck 137431 Watchduck moved page [[Boolf prop/3-ary/greater guild]] to [[Boolf prop/3-ary/super great guild]] 2693347 wikitext text/x-wiki #REDIRECT [[Boolf prop/3-ary/super great guild]] ej97ny31khs2aolihw2e6xgnw64we3l 3 317527 2693353 2024-12-26T19:28:58Z Tule-hog 2984180 vandal1 2693353 wikitext text/x-wiki ==Reverted edits December 2024== {{uw-vandal1}} [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:28, 26 December 2024 (UTC) kfnuralcoee8oq1xwqbbrq4fd1llxia 0 317528 2693356 2024-12-26T19:35:19Z Tule-hog 2984180 Tule-hog moved page [[Network+/Architecture/Routing/Introduction]] to [[Network+/Old guides/Routing]]: alter parent 2693356 wikitext text/x-wiki #REDIRECT [[Network+/Old guides/Routing]] jq0ek2v8xxyyjc3folg5rmanhny92q9 2693357 2693356 2024-12-26T19:36:32Z Tule-hog 2984180 nominate speedy 2693357 wikitext text/x-wiki {{speedy|C10}} #REDIRECT [[Network+/Old guides/Routing]] cb4u2666xti31dsh7pnx9p71pgjugq4 0 317529 2693358 2024-12-26T19:37:14Z Tule-hog 2984180 mk pg 2693358 wikitext text/x-wiki This page includes links to pages relevant to material covered on the exam. * [[:w:Virtualization|Virtualization]] ==Cloud concepts== * [[:w:Cloud computing|Cloud computing]] qsq33j91r0iqjgspsw7c2ljhe1diyb6 2693363 2693358 2024-12-26T19:56:30Z Tule-hog 2984180 add item 2693363 wikitext text/x-wiki This page includes links to pages relevant to material covered on the exam. * [[:w:Virtualization|Virtualization]] ==Networking== * [[:w:IP address|IP address]] ==Cloud concepts== * [[:w:Cloud computing|Cloud computing]] ak8j9lm2r0jvcdxvfwcfdxm3zj2um1t 2693367 2693363 2024-12-26T19:59:34Z Tule-hog 2984180 Undo revision [[Special:Diff/2693363|2693363]] by [[Special:Contributions/Tule-hog|Tule-hog]] ([[User talk:Tule-hog|talk]]) - added to [[Network+/Objectives/Networking Concepts|obj]] 2693367 wikitext text/x-wiki This page includes links to pages relevant to material covered on the exam. * [[:w:Virtualization|Virtualization]] ==Cloud concepts== * [[:w:Cloud computing|Cloud computing]] qsq33j91r0iqjgspsw7c2ljhe1diyb6 2693375 2693367 2024-12-26T20:19:44Z Tule-hog 2984180 add item 2693375 wikitext text/x-wiki This page includes links to pages relevant to material covered on the exam. * [[:w:Virtualization|Virtualization]] * [[:w:Videoconferencing|Videoconferencing]] ==Cloud concepts== * [[:w:Cloud computing|Cloud computing]] 8f4055piasoirjfsnmgk07vkzpvyrdm 2693380 2693375 2024-12-26T20:40:11Z Tule-hog 2984180 add {[[template:bookcat|bookcat]] 2693380 wikitext text/x-wiki This page includes links to pages relevant to material covered on the exam. * [[:w:Virtualization|Virtualization]] * [[:w:Videoconferencing|Videoconferencing]] ==Cloud concepts== * [[:w:Cloud computing|Cloud computing]] {{BookCat}} 2z8lv7a5h9zu0owg4as86n9yf8gz09t 2693414 2693380 2024-12-26T23:04:53Z Tule-hog 2984180 /* Software based firewalls */ adapt from [[Network+/Security/Firewalls]] 2693414 wikitext text/x-wiki This page includes links to pages relevant to material covered on the exam. * [[:w:Virtualization|Virtualization]] * [[:w:Videoconferencing|Videoconferencing]] ===Software based firewalls=== * [[:w:Netfilter|Netfilter]] ([[:w:iptables|iptables]]) * [[:w:ipfirewall|ipfirewall]] * [[:w:PF (firewall)|PF (firewall)]] ==Cloud concepts== * [[:w:Cloud computing|Cloud computing]] {{BookCat}} 1d1ftymfp6glhy6lou6pucurrl531qx 0 317530 2693387 2024-12-26T21:36:10Z Eshaa2024 2993595 New resource with "==Statement== Let <math>U \subseteq\mathbb C</math> be a domain, and let <math>f \colon U \to \mathbb C</math> be a [[w:en:Holomorphicfunction|holomorphic]], non-constant function. Then, <math>f(U)</math> is a domain. ==Proof== According to the theorem of domain preservation, one must show that <math>f(U)</math> is a domain, i.e., the set <math>f(U)</math> *is connected, and *is open. The proof is divided into these two parts. === Proof 1: Connectedness === We show tha..." 2693387 wikitext text/x-wiki ==Statement== Let <math>U \subseteq\mathbb C</math> be a domain, and let <math>f \colon U \to \mathbb C</math> be a [[w:en:Holomorphicfunction|holomorphic]], non-constant function. Then, <math>f(U)</math> is a domain. ==Proof== According to the theorem of domain preservation, one must show that <math>f(U)</math> is a domain, i.e., the set <math>f(U)</math> *is connected, and *is open. The proof is divided into these two parts. === Proof 1: Connectedness === We show that if <math>f</math> is continuous and <math>U</math> is connected, then <math>f(U)</math> is also connected. === Proof 2: Connectedness === Let <math>w_1, w_2 \in f(U)</math> be arbitrarily chosen. Then, there exist <math>z_1,z_2 \in U</math> such that <math>f(z_1)=w_1</math> and <math>f(z_2)=w_2</math>. Since <math>U</math> is connected, there exists a path <math>\gamma:[a,b]\to U</math> such that <math>\gamma(a)= z_1</math> and <math>\gamma(b)= z_2</math>. === Proof 3: Connectedness === Because <math>f</math> is continuous and <math>\gamma:[a,b]\to U</math> is a continuous path in <math>U</math>, the composition <math>\gamma_f := f \circ \gamma</math> is a continuous path in <math>f(U)</math>, for which: : <math>\gamma_f(a) = f(\gamma (a)) = f(z_1)=w_1</math> and <math>\gamma_f(b) = f(\gamma (b)) = f(z_2)=w_2</math>. === Proof 4: Openness === It remains to show that <math>f(U)</math> is open. Let <math>w_0 \in f(U)</math> and <math>z_0 \in U</math> such that <math>f(z_0) = w_0</math>. Now, consider the set of <math>w_0</math>-preimages: : <math> S(f,w_0) := { z \in U \ | \ f(z) = w_0 } </math> === Proof 5: Openness - Identity Theorem === According to the [[Complex Analysis/Identity Theorem|Identity Theorem]], the set <math>S(f,w_0) := { z \in U \ | \ f(z) = w_0 } </math> cannot have accumulation points in <math>f(U)</math>. If <math>S(f,w_0) \subseteq f(U)</math> had accumulation points in <math>f(U)</math>, the holomorphic function <math>f \colon U \to \mathbb C</math> would be constant with <math>f(z) = w_0</math> for all <math>z \in U</math>. === Proof 6: Openness - Neighborhoods === If the set <math>S(f,w_0)</math> of <math>w_0</math>-preimages of <math>f</math> has no accumulation points, one can choose a neighborhood <math>V \subseteq U</math> of <math>z_0</math> where <math>z_0</math> is the only <math>w_0</math>-preimage. Let <math>r > 0</math> be such that <math>\bar D_r(z_0) \subseteq V</math>. === Proof 7: Openness === We then define the smallest lower bound for the distance of <math>f(z)</math> to <math>w_0</math>, where <math>z</math> lies on the boundary of the disk <math>D_r(z_0)</math>: : <math> \varepsilon := \inf_{z \in \partial D_r(z_0)} |f(z) - w_0| > 0 </math> Here, <math>\varepsilon > 0</math>, because <math>f</math> is continuous and attains a minimum on the compact set <math>\partial D_r(z_0)</math>. Since <math>\bar D_r(z_0) \subseteq V</math>, no <math>w_0</math>-preimages can lie on the boundary. === Proof 8: Openness - Maximum Principle === We show that <math>D_{\frac{\varepsilon}{3}}(w_0) \subseteq f(U)</math>. Let <math>|w - w_0| < \frac{\varepsilon}{3}</math>. We prove by contradiction that this arbitrary <math>w \in D_{\frac{\varepsilon}{3}}(w_0)</math> is in the image of <math>f</math>. === Proof 9: Openness - Maximum Principle === Assume <math>f(z) \neq w</math> for all <math>z \in \bar D_r(z_0)</math>. Then, <math>|g(z)|</math> with <math>g(z):=f(z) - w</math> attains a nonzero minimum on <math>\overline{D_r(z_0)}</math>. Since <math>f</math> is not constant, this minimum must lie on <math>\partial D_r(z_0)</math> (otherwise <math>h(z):=\frac{1}{f(z)-w}</math> would be constant by the [[Maximum Principle|Maximum Principle]]. If <math>h</math> were constant, <math>f</math> would also have to be constant—a contradiction to the assumption). === Proof 9: Openness === Since <math>w_0 \in f(U)</math> was chosen arbitrarily, and for every <math>w_0 \in f(U)</math>, there exists a <math>\frac{\epsilon}{3}</math>-neighborhood <math>D_{\frac{\epsilon}{3}}(w_0) \subseteq f(U)</math>, we obtain <math>f(U) = \bigcup_{w_0 \in f(U)} D_{\frac{\epsilon}{3}}(w_0)</math> as an [[Norms, metrics, topology|Norms, metrics, topology]], and thus <math>f(U)</math> is open. ==See also== *[[Norms, metrics, topology|Norms, metrics, topology]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Openness%20theorem/theorem%20of%20territorial%20loyalty&author=Openness%20theorem&language=en&audioslide=yes&shorttitle=theorem%20of%20territorial%20loyalty&coursetitle=Openness%20theorem Wiki2Reveal slides]''' === Wiki2Reveal === The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Openness%20theorem/theorem%20of%20territorial%20loyalty&author=Openness%20theorem&language=en&audioslide=yes&shorttitle=theorem%20of%20territorial%20loyalty&coursetitle=Openness%20theorem Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Openness theorem/theorem of territorial loyalty https://en.wikiversity.org/wiki/Complex%20Analysis/Openness theorem/theorem of territorial loyalty] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Openness theorem/theorem of territorial loyalty This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Satz von der Gebietstreue Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Satz von der Gebietstreue|Kurs:Funktionentheorie/Satz von der Gebietstreue]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Satz von der Gebietstreue * Date: 12/26/2024 <span type="translate" src="Kurs:Funktionentheorie/Satz von der Gebietstreue" srclang="de" date="12/26/2024" time="10:32" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Satz von der Gebietstreue]] </noinclude> [[Category:Wiki2Reveal]] riem3nola6m7couil5t1lj8hnmx5xkw 0 317531 2693392 2024-12-26T22:06:32Z Eshaa2024 2993595 New resource with "==Statement== Let <math>U \subseteq\mathbb C</math> be a domain, and let <math>f \colon U \to \mathbb C</math> be a [[w:en:Holomorphic function|holomorphic]], non-constant function. Then, <math>f(U)</math> is a domain. ==Proof== According to the theorem of domain preservation, one must show that <math>f(U)</math> is a domain, i.e., the set <math>f(U)</math> *is connected, and *is open. The proof is divided into these two parts. === Proof 1: Connectedness === We show t..." 2693392 wikitext text/x-wiki ==Statement== Let <math>U \subseteq\mathbb C</math> be a domain, and let <math>f \colon U \to \mathbb C</math> be a [[w:en:Holomorphic function|holomorphic]], non-constant function. Then, <math>f(U)</math> is a domain. ==Proof== According to the theorem of domain preservation, one must show that <math>f(U)</math> is a domain, i.e., the set <math>f(U)</math> *is connected, and *is open. The proof is divided into these two parts. === Proof 1: Connectedness === We show that if <math>f</math> is continuous and <math>U</math> is connected, then <math>f(U)</math> is also connected. === Proof 2: Connectedness === Let <math>w_1, w_2 \in f(U)</math> be arbitrarily chosen. Then, there exist <math>z_1,z_2 \in U</math> such that <math>f(z_1)=w_1</math> and <math>f(z_2)=w_2</math>. Since <math>U</math> is connected, there exists a path <math>\gamma:[a,b]\to U</math> such that <math>\gamma(a)= z_1</math> and <math>\gamma(b)= z_2</math>. === Proof 3: Connectedness === Because <math>f</math> is continuous and <math>\gamma:[a,b]\to U</math> is a continuous path in <math>U</math>, the composition <math>\gamma_f := f \circ \gamma</math> is a continuous path in <math>f(U)</math>, for which: : <math>\gamma_f(a) = f(\gamma (a)) = f(z_1)=w_1</math> and <math>\gamma_f(b) = f(\gamma (b)) = f(z_2)=w_2</math>. === Proof 4: Openness === It remains to show that <math>f(U)</math> is open. Let <math>w_0 \in f(U)</math> and <math>z_0 \in U</math> such that <math>f(z_0) = w_0</math>. Now, consider the set of <math>w_0</math>-preimages: : <math> S(f,w_0) := { z \in U \ | \ f(z) = w_0 } </math> === Proof 5: Openness - Identity Theorem === According to the [[Complex Analysis/Identity Theorem|Identity Theorem]], the set <math>S(f,w_0) := { z \in U \ | \ f(z) = w_0 } </math> cannot have accumulation points in <math>f(U)</math>. If <math>S(f,w_0) \subseteq f(U)</math> had accumulation points in <math>f(U)</math>, the holomorphic function <math>f \colon U \to \mathbb C</math> would be constant with <math>f(z) = w_0</math> for all <math>z \in U</math>. === Proof 6: Openness - Neighborhoods === If the set <math>S(f,w_0)</math> of <math>w_0</math>-preimages of <math>f</math> has no accumulation points, one can choose a neighborhood <math>V \subseteq U</math> of <math>z_0</math> where <math>z_0</math> is the only <math>w_0</math>-preimage. Let <math>r > 0</math> be such that <math>\bar D_r(z_0) \subseteq V</math>. === Proof 7: Openness === We then define the smallest lower bound for the distance of <math>f(z)</math> to <math>w_0</math>, where <math>z</math> lies on the boundary of the disk <math>D_r(z_0)</math>: : <math> \varepsilon := \inf_{z \in \partial D_r(z_0)} |f(z) - w_0| > 0 </math> Here, <math>\varepsilon > 0</math>, because <math>f</math> is continuous and attains a minimum on the compact set <math>\partial D_r(z_0)</math>. Since <math>\bar D_r(z_0) \subseteq V</math>, no <math>w_0</math>-preimages can lie on the boundary. === Proof 8: Openness - Maximum Principle === We show that <math>D_{\frac{\varepsilon}{3}}(w_0) \subseteq f(U)</math>. Let <math>|w - w_0| < \frac{\varepsilon}{3}</math>. We prove by contradiction that this arbitrary <math>w \in D_{\frac{\varepsilon}{3}}(w_0)</math> is in the image of <math>f</math>. === Proof 9: Openness - Maximum Principle === Assume <math>f(z) \neq w</math> for all <math>z \in \bar D_r(z_0)</math>. Then, <math>|g(z)|</math> with <math>g(z):=f(z) - w</math> attains a nonzero minimum on <math>\overline{D_r(z_0)}</math>. Since <math>f</math> is not constant, this minimum must lie on <math>\partial D_r(z_0)</math> (otherwise <math>h(z):=\frac{1}{f(z)-w}</math> would be constant by the [[Complex Analysis/Maximum Principle|Maximum Principle]]. If <math>h</math> were constant, <math>f</math> would also have to be constant—a contradiction to the assumption). === Proof 9: Openness === Since <math>w_0 \in f(U)</math> was chosen arbitrarily, and for every <math>w_0 \in f(U)</math>, there exists a <math>\frac{\epsilon}{3}</math>-neighborhood <math>D_{\frac{\epsilon}{3}}(w_0) \subseteq f(U)</math>, we obtain <math>f(U) = \bigcup_{w_0 \in f(U)} D_{\frac{\epsilon}{3}}(w_0)</math> as an [[Norms, metrics, topology|Norms, metrics, topology]], and thus <math>f(U)</math> is open. ==See also== * [[Norms, metrics, topology|Norms, metrics, topology]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Openness%20theorem%20theorem%20of%20territorial%20loyalty&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Openness%20theorem%20theorem%20of%20territorial%20loyalty&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty https://en.wikiversity.org/wiki/Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Satz von der Gebietstreue Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Satz von der Gebietstreue|Kurs:Funktionentheorie/Satz von der Gebietstreue]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Satz von der Gebietstreue * Date: 12/26/2024 <span type="translate" src="Kurs:Funktionentheorie/Satz von der Gebietstreue" srclang="de" date="12/26/2024" time="11:06" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Satz von der Gebietstreue]] </noinclude> [[Category:Wiki2Reveal]] 604u8j3qkdam3d5aayfglaugsfq8v1a 2693404 2693392 2024-12-26T22:41:04Z Eshaa2024 2993595 /* Proof 9: Openness - Maximum Principle */ 2693404 wikitext text/x-wiki ==Statement== Let <math>U \subseteq\mathbb C</math> be a domain, and let <math>f \colon U \to \mathbb C</math> be a [[w:en:Holomorphic function|holomorphic]], non-constant function. Then, <math>f(U)</math> is a domain. ==Proof== According to the theorem of domain preservation, one must show that <math>f(U)</math> is a domain, i.e., the set <math>f(U)</math> *is connected, and *is open. The proof is divided into these two parts. === Proof 1: Connectedness === We show that if <math>f</math> is continuous and <math>U</math> is connected, then <math>f(U)</math> is also connected. === Proof 2: Connectedness === Let <math>w_1, w_2 \in f(U)</math> be arbitrarily chosen. Then, there exist <math>z_1,z_2 \in U</math> such that <math>f(z_1)=w_1</math> and <math>f(z_2)=w_2</math>. Since <math>U</math> is connected, there exists a path <math>\gamma:[a,b]\to U</math> such that <math>\gamma(a)= z_1</math> and <math>\gamma(b)= z_2</math>. === Proof 3: Connectedness === Because <math>f</math> is continuous and <math>\gamma:[a,b]\to U</math> is a continuous path in <math>U</math>, the composition <math>\gamma_f := f \circ \gamma</math> is a continuous path in <math>f(U)</math>, for which: : <math>\gamma_f(a) = f(\gamma (a)) = f(z_1)=w_1</math> and <math>\gamma_f(b) = f(\gamma (b)) = f(z_2)=w_2</math>. === Proof 4: Openness === It remains to show that <math>f(U)</math> is open. Let <math>w_0 \in f(U)</math> and <math>z_0 \in U</math> such that <math>f(z_0) = w_0</math>. Now, consider the set of <math>w_0</math>-preimages: : <math> S(f,w_0) := { z \in U \ | \ f(z) = w_0 } </math> === Proof 5: Openness - Identity Theorem === According to the [[Complex Analysis/Identity Theorem|Identity Theorem]], the set <math>S(f,w_0) := { z \in U \ | \ f(z) = w_0 } </math> cannot have accumulation points in <math>f(U)</math>. If <math>S(f,w_0) \subseteq f(U)</math> had accumulation points in <math>f(U)</math>, the holomorphic function <math>f \colon U \to \mathbb C</math> would be constant with <math>f(z) = w_0</math> for all <math>z \in U</math>. === Proof 6: Openness - Neighborhoods === If the set <math>S(f,w_0)</math> of <math>w_0</math>-preimages of <math>f</math> has no accumulation points, one can choose a neighborhood <math>V \subseteq U</math> of <math>z_0</math> where <math>z_0</math> is the only <math>w_0</math>-preimage. Let <math>r > 0</math> be such that <math>\bar D_r(z_0) \subseteq V</math>. === Proof 7: Openness === We then define the smallest lower bound for the distance of <math>f(z)</math> to <math>w_0</math>, where <math>z</math> lies on the boundary of the disk <math>D_r(z_0)</math>: : <math> \varepsilon := \inf_{z \in \partial D_r(z_0)} |f(z) - w_0| > 0 </math> Here, <math>\varepsilon > 0</math>, because <math>f</math> is continuous and attains a minimum on the compact set <math>\partial D_r(z_0)</math>. Since <math>\bar D_r(z_0) \subseteq V</math>, no <math>w_0</math>-preimages can lie on the boundary. === Proof 8: Openness - Maximum Principle === We show that <math>D_{\frac{\varepsilon}{3}}(w_0) \subseteq f(U)</math>. Let <math>|w - w_0| < \frac{\varepsilon}{3}</math>. We prove by contradiction that this arbitrary <math>w \in D_{\frac{\varepsilon}{3}}(w_0)</math> is in the image of <math>f</math>. === Proof 9: Openness - Maximum Principle === Assume <math>f(z) \neq w</math> for all <math>z \in \bar D_r(z_0)</math>. Then, <math>|g(z)|</math> with <math>g(z):=f(z) - w</math> attains a nonzero minimum on <math>\overline{D_r(z_0)}</math>. Since <math>f</math> is not constant, this minimum must lie on <math>\partial D_r(z_0)</math> (otherwise <math>h(z):=\frac{1}{f(z)-w}</math> would be constant by the [[Complex Analysics/Maximum Principle|Maximum Principle]]. If <math>h</math> were constant, <math>f</math> would also have to be constant—a contradiction to the assumption). === Proof 9: Openness === Since <math>w_0 \in f(U)</math> was chosen arbitrarily, and for every <math>w_0 \in f(U)</math>, there exists a <math>\frac{\epsilon}{3}</math>-neighborhood <math>D_{\frac{\epsilon}{3}}(w_0) \subseteq f(U)</math>, we obtain <math>f(U) = \bigcup_{w_0 \in f(U)} D_{\frac{\epsilon}{3}}(w_0)</math> as an [[Norms, metrics, topology|Norms, metrics, topology]], and thus <math>f(U)</math> is open. ==See also== * [[Norms, metrics, topology|Norms, metrics, topology]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Openness%20theorem%20theorem%20of%20territorial%20loyalty&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Openness%20theorem%20theorem%20of%20territorial%20loyalty&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty https://en.wikiversity.org/wiki/Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Satz von der Gebietstreue Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Satz von der Gebietstreue|Kurs:Funktionentheorie/Satz von der Gebietstreue]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Satz von der Gebietstreue * Date: 12/26/2024 <span type="translate" src="Kurs:Funktionentheorie/Satz von der Gebietstreue" srclang="de" date="12/26/2024" time="11:06" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Satz von der Gebietstreue]] </noinclude> [[Category:Wiki2Reveal]] 5fqwjw3d1cuzlelrknh15yj188yyyrl 10 317532 2693397 2024-12-26T22:22:32Z Watchduck 137431 New resource with "table#super-families {text-align: center;} table#super-families td.super-clan-size {color: gray; 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The magnitude <math>|f|</math> of a holomorphic function <math>f: G \to \mathbb{C}</math> cannot attain any strict local maxima within the domain of definition. Specifically, it asserts the following statement. == Statement == Let <math>U \subseteq \mathbb{C}</math> be a domain, and let <math>f: U \to \mathbb{C}</math> be w:en:Holomorphic..." 2693399 wikitext text/x-wiki == Introduction == The maximum principle is a statement about holomorphic functions from the [[Complex Analysis|Complex Analysis]]. The magnitude <math>|f|</math> of a holomorphic function <math>f: G \to \mathbb{C}</math> cannot attain any strict local maxima within the domain of definition. Specifically, it asserts the following statement. == Statement == Let <math>U \subseteq \mathbb{C}</math> be a domain, and let <math>f: U \to \mathbb{C}</math> be [[w:en:Holomorphic function|holomorphic]]. If <math>|f|</math> has a local maximum in <math>U</math>, then <math>f</math> is constant. If <math>U</math> is bounded and <math>f</math> can be continuously extended to <math>\bar{U}</math>, then <math>f</math> attains its maximum on <math>\partial U</math>. To prove this, we require a lemma that locally implies the conclusion. === Lemma === Let <math>G \subseteq \mathbb{C}</math> be open, and <math>f: G \to \mathbb{C}</math> be [[w:en:Holomorphic function|holomorphic]]. Let <math>z_0 \in G</math> be a local maximum point of <math>|f|</math>. Then <math>f</math> is constant in a neighborhood of <math>z_0</math>. ==Proof of Lemma 1== Let <math>r > 0</math> be chosen such that <math>|f(z)| \leq |f(z_0)|</math> for all <math>z \in \bar{D}r(z_0) \subseteq G</math>. The [[Cauchy's integral formula|Cauchy's integral formula]]'gives, for all <math>\varepsilon \leq r</math>: :<math>f(z_0) = \frac{1}{2\pi i} \int{\partial D_\varepsilon(z_0)} \frac{f(z)}{z-z_0} , dz</math> This allows us to establish the following estimation: ==Proof of Lemma 2== We derive the following estimation: :<math> \begin{array}{rl} |f(z_0)| &= \frac 1{2\pi} \left| \int_{\partial D_\varepsilon(z_0)} \frac{f(z)}{z-z_0}\, dz \right| \\ &= \frac 1{2\pi} \left| \int_0^{2\pi} \frac {f(z_0 + \varepsilon e^{it})}{\varepsilon \cdot e^{it}} \varepsilon \cdot i \cdot e^{it}\, dt\right|\\ &\le \frac 1{2\pi} \int_0^{2\pi} |f(z_0 + \varepsilon \cdot e^{it})|\, dt\\ &\le \sup_{t \in[0,2\pi]} |f(z_0 + \varepsilon \cdot e^{it})|\\ &\le |f(z_0)| \end{array} </math> ==Proof of Lemma 3== It follows that the inequality <math>\leq</math> must be an equality chain, implying :<math> \begin{array}{rl} |f(z_0)| = & \int_0 ^{2 \pi}|f(z_0)| \cdot \frac 1 {2 \pi} dt = \frac 1{2\pi} \int_0^{2\pi} |f(z_0 + \varepsilon \cdot e^{it})| \, dt\\ \Rightarrow & 0 = \int_0 ^{2 \pi} (| f(z)| - |f(z_0)|)dt \\ \Rightarrow & |f(z)| = |f(z_0)| \end{array} </math>. === Proof of Lemma 4 === Thus, we establish the constancy of <math>|f|</math> using the property: :<math>|f(z)| = |f(z_0)|</math> for all <math>z \in \bar D_r(z_0)</math>, i.e., <math>|f|</math> is constant on <math>D_r(z_0)</math>. === Proof of Lemma 5 === If <math>|f|=\sqrt{\mathfrak{Re}(f)^2+\mathfrak{Im}(f)^2}</math> is constant on <math>D_r(z_0)</math>, then <math>\mathfrak{Re}(f)^2+\mathfrak{Im}(f)^2 = c</math> must also be constant, where <math>c \in \mathbb{R}</math> is a constant. === Proof of Lemma 6 === Since <math>f</math> is holomorphic on <math>D_r(z_0)</math>, the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]'apply: :<math> u := \mathfrak{Re}(f), \ v := \mathfrak{Im}(f), \ \text{and} \ f(z) := u(x,y) + i \cdot v(x,y)</math>, and the following holds: :<math>\frac{\partial u(x,y)^2 + v(x,y)^2}{\partial x} = 0 \ \text{and} \ \frac{\partial u(x,y)^2 + v(x,y)^2}{\partial y} = 0</math>. === Proof of Lemma 7 === Let <math>u_x = \frac{\partial u}{\partial x}</math> and <math>u_y = \frac{\partial u}{\partial y}</math>. Applying the chain rule to the partial derivatives, we obtain: :<math>0 = 2 u u_x + 2 v v_x</math> and <math>0 = 2 u u_y + 2 v v_y</math>. Using the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]', replace the partial derivatives of <math>v</math> with those of <math>u</math>: :<math>u_x = v_y</math> and <math>u_y = -v_x</math>, leading to: :<math>0 = 2 \cdot (u u_x - v u_y)</math> and <math>0 = 2 \cdot (u u_y + v u_x)</math>. === Proof of Lemma 8 === Squaring the above equations yields: :<math>0 = (u u_x - v u_y)^2 = u^2 u_x^2 - 2 u u_x \cdot v u_y + v^2 u_y^2</math>, :<math>0 = (u u_y + v u_x)^2 = u^2 u_y^2 + 2 u u_y \cdot v u_x + v^2 u_x^2</math>. Adding these equations gives: :<math>0 = u^2 u_x^2 + v^2 u_y^2 + u^2 u_y^2 + v^2 u_x^2</math>. === Proof of Lemma 9 === Factoring out <math>u^2</math> and <math>v^2</math>: :<math>0 = u^2 (u_x^2 + u_y^2) + v^2 (u_x^2 + u_y^2) = (u^2 + v^2) \cdot (u_x^2 + u_y^2)</math>. Thus, :<math>0 = u^2 + v^2</math> or <math>0 = u_x^2 + u_y^2</math>. === Proof of Lemma 10 === With <math>u^2 = -v^2</math>, it follows that <math>u = v = 0</math> since <math>u(x,y)</math> and <math>v(x,y)</math> are real-valued, implying <math>f = u + iv = 0</math>. If <math>u_x^2 = -u_y^2</math>, then <math>u_x^2 = u_y^2 = 0</math>, and <math>u_x = u_y = 0</math>. By the [[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]], <math>f' = 0</math>. Thus, <math>f</math> is constant on <math>D_r(z_0)</math>. == Proof == Let <math>z_0 \in G</math> be a local maximum point of <math>|f|</math> in the domain <math>G</math>. Define <math>V := {z \in G : f(z) = f(z_0)}</math> as the set of all <math>z \in G</math> mapped to <math>w := f(z_0) \in \mathbb{C}</math> (level set). === Proof 1: V is closed === Since <math>f</math> is continuous, preimages of open sets are open, and preimages of closed sets are closed (in the relative topology of <math>G</math>). Thus, <math>V = f^{-1}({w})</math> is closed in <math>G</math>. === Proof 2: V is open === Using the lemma, <math>V</math> can also be represented as a union of open disks, and unions of open sets are open. === Proof 3: Connectivity === Thus, <math>V = G</math> due to the connectivity of <math>U</math>, i.e., <math>f</math> is constant. === Proof 4: G is bounded === If <math>G</math> is bounded, then <math>\bar G</math> is compact. Therefore, the continuous function <math>f</math> attains its maximum on <math>\bar G</math>, say at <math>z_0 \in \bar G</math>. If <math>z_0 \in G</math>, then <math>f</math> is constant on <math>G</math> (by the lemma) and hence on <math>\bar G</math>, so <math>f</math> also attains its maximum on <math>\partial G</math>. Otherwise, <math>z_0 \in \partial G</math>, completing the proof. == See Also == *[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]' *[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]''' === Wiki2Reveal === The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Maximum%20Principle https://en.wikiversity.org/wiki/Complex%20Analysis/Maximum%20Principle] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Maximum%20Principle This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/MaximumprinzipWikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Maximumprinzip|Kurs:Funktionentheorie/Maximumprinzip]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Maximumprinzip * Date: 12/26/2024 <span 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104<br>[[File:Venn 0001 0110.svg|30px]] |class="matrix"| [[File:3T super-family 104.svg|200px]] |class="matrix"| [[File:3Z super-family 104.svg|200px]] |class="village"| [6, 14] |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' |class="quaestor-weight"| 0 |class="rep"| 66<br>[[File:Venn 0100 0010.svg|30px]]<br><br>Ж 106<br>[[File:Venn 0101 0110.svg|30px]] |class="matrix"| [[File:3T super-family 106.svg|200px]] |class="matrix"| [[File:3Z super-family 106.svg|200px]] |class="village"| [6, 14] |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 128<br>[[File:Venn 0000 0001.svg|30px]]<br><br>Ж 128<br>[[File:Venn 0000 0001.svg|30px]] |class="matrix"| [[File:3T super-family 128.svg|200px]] |class="matrix"| [[File:3Z super-family 128.svg|200px]] |class="village"| [1, 3, 5, 15] |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 42<br>[[File:Venn 0101 0100.svg|30px]]<br><br>Ж 130<br>[[File:Venn 0100 0001.svg|30px]] |class="matrix"| [[File:3T super-family 130.svg|200px]] |class="matrix"| [[File:3Z super-family 130.svg|200px]] |class="village"| [1, 3, 5, 15] |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 76<br>[[File:Venn 0011 0010.svg|30px]]<br><br>Ж 132<br>[[File:Venn 0010 0001.svg|30px]] |class="matrix"| [[File:3T super-family 132.svg|200px]] |class="matrix"| [[File:3Z super-family 132.svg|200px]] |class="village"| [1, 3, 5, 15] |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 230<br>[[File:Venn 0110 0111.svg|30px]]<br><br>Ж 134<br>[[File:Venn 0110 0001.svg|30px]] |class="matrix"| [[File:3T super-family 134.svg|200px]] |class="matrix"| [[File:3Z super-family 134.svg|200px]] |class="village"| [1, 3, 5, 15] |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 112<br>[[File:Venn 0000 1110.svg|30px]]<br><br>Ж 144<br>[[File:Venn 0000 1001.svg|30px]] |class="matrix"| [[File:3T super-family 144.svg|200px]] |class="matrix"| [[File:3Z super-family 144.svg|200px]] |class="village"| [7, 9, 11, 13] |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 218<br>[[File:Venn 0101 1011.svg|30px]]<br><br>Ж 146<br>[[File:Venn 0100 1001.svg|30px]] |class="matrix"| [[File:3T super-family 146.svg|200px]] |class="matrix"| [[File:3Z super-family 146.svg|200px]] |class="village"| [7, 9, 11, 13] |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 188<br>[[File:Venn 0011 1101.svg|30px]]<br><br>Ж 148<br>[[File:Venn 0010 1001.svg|30px]] |class="matrix"| [[File:3T super-family 148.svg|200px]] |class="matrix"| [[File:3Z super-family 148.svg|200px]] |class="village"| [7, 9, 11, 13] |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 22<br>[[File:Venn 0110 1000.svg|30px]]<br><br>Ж 150<br>[[File:Venn 0110 1001.svg|30px]] |class="matrix"| [[File:3T super-family 150.svg|200px]] |class="matrix"| [[File:3Z super-family 150.svg|200px]] |class="village"| [7, 9, 11, 13] |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> ryb9wdpj203vhwol9ak5ynvbyekrgcx 2693416 2693400 2024-12-26T23:08:30Z Watchduck 137431 2693416 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-families/style.css" /> {| class="wikitable sortable" id="super-families" |- !rowspan="2"| <abbr title="super-family">s.-f.</abbr><br>size !rowspan="2"| family<br>size !rowspan="2"| weight !rowspan="2"| tribe !rowspan="2"| quaestor<br>weight !rowspan="2"| <abbr title="representative">rep.</abbr> !rowspan="2"| truth<br>tables !rowspan="2"| Zhegalkin<br>indices !colspan="3" class="unsortable"| village |- ! size ! set ! junior<br>rep. |- |class="super-family-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 0<br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0<br>[[File:Venn 0000 0000.svg|25px]] |class="matrix"| [[File:3T super-family 0.svg|200px]] |class="matrix"| [[File:3Z super-family 0.svg|200px]] |class="village village-size"| 1 |class="village"| {0} |class="village village-reverse"| T {0}<br>Z {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 170<br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2<br>[[File:Venn 0100 0000.svg|25px]] |class="matrix"| [[File:3T super-family 2.svg|200px]] |class="matrix"| [[File:3Z super-family 2.svg|200px]] |class="village village-size"| 1 |class="village"| {0} |class="village village-reverse"| T {0}<br>Z {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 204<br>[[File:Venn 0011 0011.svg|25px]]<br><br>Ж 4<br>[[File:Venn 0010 0000.svg|25px]] |class="matrix"| [[File:3T super-family 4.svg|200px]] |class="matrix"| [[File:3Z super-family 4.svg|200px]] |class="village village-size"| 1 |class="village"| {0} |class="village village-reverse"| T {0}<br>Z {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 102<br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6<br>[[File:Venn 0110 0000.svg|25px]] |class="matrix"| [[File:3T super-family 6.svg|200px]] |class="matrix"| [[File:3Z super-family 6.svg|200px]] |class="village village-size"| 1 |class="village"| {0} |class="village village-reverse"| T {0}<br>Z {0} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 136<br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8<br>[[File:Venn 0001 0000.svg|25px]] |class="matrix"| [[File:3T super-family 8.svg|200px]] |class="matrix"| [[File:3Z super-family 8.svg|200px]] |class="village village-size"| 1 |class="village"| {0} |class="village village-reverse"| T {0}<br>Z {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 240<br>[[File:Venn 0000 1111.svg|25px]]<br><br>Ж 16<br>[[File:Venn 0000 1000.svg|25px]] |class="matrix"| [[File:3T super-family 16.svg|200px]] |class="matrix"| [[File:3Z super-family 16.svg|200px]] |class="village village-size"| 1 |class="village"| {8} |class="village village-reverse"| T {15}<br>Z {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 90<br>[[File:Venn 0101 1010.svg|25px]]<br><br>Ж 18<br>[[File:Venn 0100 1000.svg|25px]] |class="matrix"| [[File:3T super-family 18.svg|200px]] |class="matrix"| [[File:3Z super-family 18.svg|200px]] |class="village village-size"| 1 |class="village"| {8} |class="village village-reverse"| T {15}<br>Z {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 60<br>[[File:Venn 0011 1100.svg|25px]]<br><br>Ж 20<br>[[File:Venn 0010 1000.svg|25px]] |class="matrix"| [[File:3T super-family 20.svg|200px]] |class="matrix"| [[File:3Z super-family 20.svg|200px]] |class="village village-size"| 1 |class="village"| {8} |class="village village-reverse"| T {15}<br>Z {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 150<br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22<br>[[File:Venn 0110 1000.svg|25px]] |class="matrix"| [[File:3T super-family 22.svg|200px]] |class="matrix"| [[File:3Z super-family 22.svg|200px]] |class="village village-size"| 1 |class="village"| {8} |class="village village-reverse"| T {15}<br>Z {1} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 120<br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24<br>[[File:Venn 0001 1000.svg|25px]] |class="matrix"| [[File:3T super-family 24.svg|200px]] |class="matrix"| [[File:3Z super-family 24.svg|200px]] |class="village village-size"| 1 |class="village"| {8} |class="village village-reverse"| T {15}<br>Z {1} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 160<br>[[File:Venn 0000 0101.svg|25px]]<br><br>Ж 32<br>[[File:Venn 0000 0100.svg|25px]] |class="matrix"| [[File:3T super-family 32.svg|200px]] |class="matrix"| [[File:3Z super-family 32.svg|200px]] |class="village village-size"| 2 |class="village"| {4, 12} |class="village village-reverse"| T {10, 5}<br>Z {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 108<br>[[File:Venn 0011 0110.svg|25px]]<br><br>Ж 36<br>[[File:Venn 0010 0100.svg|25px]] |class="matrix"| [[File:3T super-family 36.svg|200px]] |class="matrix"| [[File:3Z super-family 36.svg|200px]] |class="village village-size"| 2 |class="village"| {4, 12} |class="village village-reverse"| T {10, 5}<br>Z {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 40<br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40<br>[[File:Venn 0001 0100.svg|25px]] |class="matrix"| [[File:3T super-family 40.svg|200px]] |class="matrix"| [[File:3Z super-family 40.svg|200px]] |class="village village-size"| 2 |class="village"| {4, 12} |class="village village-reverse"| T {10, 5}<br>Z {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 228<br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44<br>[[File:Venn 0011 0100.svg|25px]] |class="matrix"| [[File:3T super-family 44.svg|200px]] |class="matrix"| [[File:3Z super-family 44.svg|200px]] |class="village village-size"| 2 |class="village"| {4, 12} |class="village village-reverse"| T {10, 5}<br>Z {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 192<br>[[File:Venn 0000 0011.svg|25px]]<br><br>Ж 64<br>[[File:Venn 0000 0010.svg|25px]] |class="matrix"| [[File:3T super-family 64.svg|200px]] |class="matrix"| [[File:3Z super-family 64.svg|200px]] |class="village village-size"| 2 |class="village"| {2, 10} |class="village village-reverse"| T {12, 3}<br>Z {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 106<br>[[File:Venn 0101 0110.svg|25px]]<br><br>Ж 66<br>[[File:Venn 0100 0010.svg|25px]] |class="matrix"| [[File:3T super-family 66.svg|200px]] |class="matrix"| [[File:3Z super-family 66.svg|200px]] |class="village village-size"| 2 |class="village"| {2, 10} |class="village village-reverse"| T {12, 3}<br>Z {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 72<br>[[File:Venn 0001 0010.svg|25px]]<br><br>Ж 72<br>[[File:Venn 0001 0010.svg|25px]] |class="matrix"| [[File:3T super-family 72.svg|200px]] |class="matrix"| [[File:3Z super-family 72.svg|200px]] |class="village village-size"| 2 |class="village"| {2, 10} |class="village village-reverse"| T {12, 3}<br>Z {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 226<br>[[File:Venn 0100 0111.svg|25px]]<br><br>Ж 74<br>[[File:Venn 0101 0010.svg|25px]] |class="matrix"| [[File:3T super-family 74.svg|200px]] |class="matrix"| [[File:3Z super-family 74.svg|200px]] |class="village village-size"| 2 |class="village"| {2, 10} |class="village village-reverse"| T {12, 3}<br>Z {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 96<br>[[File:Venn 0000 0110.svg|25px]]<br><br>Ж 96<br>[[File:Venn 0000 0110.svg|25px]] |class="matrix"| [[File:3T super-family 96.svg|200px]] |class="matrix"| [[File:3Z super-family 96.svg|200px]] |class="village village-size"| 2 |class="village"| {6, 14} |class="village village-reverse"| T {6, 9}<br>Z {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 202<br>[[File:Venn 0101 0011.svg|25px]]<br><br>Ж 98<br>[[File:Venn 0100 0110.svg|25px]] |class="matrix"| [[File:3T super-family 98.svg|200px]] |class="matrix"| [[File:3Z super-family 98.svg|200px]] |class="village village-size"| 2 |class="village"| {6, 14} |class="village village-reverse"| T {6, 9}<br>Z {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' |class="quaestor-weight"| 4 |class="rep"| 232<br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104<br>[[File:Venn 0001 0110.svg|25px]] |class="matrix"| [[File:3T super-family 104.svg|200px]] |class="matrix"| [[File:3Z super-family 104.svg|200px]] |class="village village-size"| 2 |class="village"| {6, 14} |class="village village-reverse"| T {6, 9}<br>Z {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' |class="quaestor-weight"| 0 |class="rep"| 66<br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106<br>[[File:Venn 0101 0110.svg|25px]] |class="matrix"| [[File:3T super-family 106.svg|200px]] |class="matrix"| [[File:3Z super-family 106.svg|200px]] |class="village village-size"| 2 |class="village"| {6, 14} |class="village village-reverse"| T {6, 9}<br>Z {6, 7} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 128<br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128<br>[[File:Venn 0000 0001.svg|25px]] |class="matrix"| [[File:3T super-family 128.svg|200px]] |class="matrix"| [[File:3Z super-family 128.svg|200px]] |class="village village-size"| 4 |class="village"| {1, 3, 5, 15} |class="village village-reverse"| T {8, 4, 2, 1}<br>Z {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 42<br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130<br>[[File:Venn 0100 0001.svg|25px]] |class="matrix"| [[File:3T super-family 130.svg|200px]] |class="matrix"| [[File:3Z super-family 130.svg|200px]] |class="village village-size"| 4 |class="village"| {1, 3, 5, 15} |class="village village-reverse"| T {8, 4, 2, 1}<br>Z {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 76<br>[[File:Venn 0011 0010.svg|25px]]<br><br>Ж 132<br>[[File:Venn 0010 0001.svg|25px]] |class="matrix"| [[File:3T super-family 132.svg|200px]] |class="matrix"| [[File:3Z super-family 132.svg|200px]] |class="village village-size"| 4 |class="village"| {1, 3, 5, 15} |class="village village-reverse"| T {8, 4, 2, 1}<br>Z {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 230<br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134<br>[[File:Venn 0110 0001.svg|25px]] |class="matrix"| [[File:3T super-family 134.svg|200px]] |class="matrix"| [[File:3Z super-family 134.svg|200px]] |class="village village-size"| 4 |class="village"| {1, 3, 5, 15} |class="village village-reverse"| T {8, 4, 2, 1}<br>Z {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 112<br>[[File:Venn 0000 1110.svg|25px]]<br><br>Ж 144<br>[[File:Venn 0000 1001.svg|25px]] |class="matrix"| [[File:3T super-family 144.svg|200px]] |class="matrix"| [[File:3Z super-family 144.svg|200px]] |class="village village-size"| 4 |class="village"| {7, 9, 11, 13} |class="village village-reverse"| T {14, 7, 11, 13}<br>Z {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 218<br>[[File:Venn 0101 1011.svg|25px]]<br><br>Ж 146<br>[[File:Venn 0100 1001.svg|25px]] |class="matrix"| [[File:3T super-family 146.svg|200px]] |class="matrix"| [[File:3Z super-family 146.svg|200px]] |class="village village-size"| 4 |class="village"| {7, 9, 11, 13} |class="village village-reverse"| T {14, 7, 11, 13}<br>Z {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 188<br>[[File:Venn 0011 1101.svg|25px]]<br><br>Ж 148<br>[[File:Venn 0010 1001.svg|25px]] |class="matrix"| [[File:3T super-family 148.svg|200px]] |class="matrix"| [[File:3Z super-family 148.svg|200px]] |class="village village-size"| 4 |class="village"| {7, 9, 11, 13} |class="village village-reverse"| T {14, 7, 11, 13}<br>Z {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 22<br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150<br>[[File:Venn 0110 1001.svg|25px]] |class="matrix"| [[File:3T super-family 150.svg|200px]] |class="matrix"| [[File:3Z super-family 150.svg|200px]] |class="village village-size"| 4 |class="village"| {7, 9, 11, 13} |class="village village-reverse"| T {14, 7, 11, 13}<br>Z {14, 9, 13, 11} |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> de6gn9devtretax0lq0tdjugbsq4yb1 2693418 2693416 2024-12-26T23:12:03Z Watchduck 137431 2693418 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-families/style.css" /> {| class="wikitable sortable" id="super-families" |- !rowspan="2"| <abbr title="super-family">s.-f.</abbr><br>size !rowspan="2"| family<br>size !rowspan="2"| weight !rowspan="2"| tribe !rowspan="2"| quaestor<br>weight !rowspan="2"| <abbr title="representative">rep.</abbr> !rowspan="2"| truth<br>tables !rowspan="2"| Zhegalkin<br>indices !colspan="3" class="unsortable"| village |- !class="village-small"| size ! !class="village-small"| reverse |- |class="super-family-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 0<br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0<br>[[File:Venn 0000 0000.svg|25px]] |class="matrix"| [[File:3T super-family 0.svg|200px]] |class="matrix"| [[File:3Z super-family 0.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 170<br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2<br>[[File:Venn 0100 0000.svg|25px]] |class="matrix"| [[File:3T super-family 2.svg|200px]] |class="matrix"| [[File:3Z super-family 2.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 204<br>[[File:Venn 0011 0011.svg|25px]]<br><br>Ж 4<br>[[File:Venn 0010 0000.svg|25px]] |class="matrix"| [[File:3T super-family 4.svg|200px]] |class="matrix"| [[File:3Z super-family 4.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 102<br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6<br>[[File:Venn 0110 0000.svg|25px]] |class="matrix"| [[File:3T super-family 6.svg|200px]] |class="matrix"| [[File:3Z super-family 6.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 136<br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8<br>[[File:Venn 0001 0000.svg|25px]] |class="matrix"| [[File:3T super-family 8.svg|200px]] |class="matrix"| [[File:3Z super-family 8.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 240<br>[[File:Venn 0000 1111.svg|25px]]<br><br>Ж 16<br>[[File:Venn 0000 1000.svg|25px]] |class="matrix"| [[File:3T super-family 16.svg|200px]] |class="matrix"| [[File:3Z super-family 16.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 90<br>[[File:Venn 0101 1010.svg|25px]]<br><br>Ж 18<br>[[File:Venn 0100 1000.svg|25px]] |class="matrix"| [[File:3T super-family 18.svg|200px]] |class="matrix"| [[File:3Z super-family 18.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 60<br>[[File:Venn 0011 1100.svg|25px]]<br><br>Ж 20<br>[[File:Venn 0010 1000.svg|25px]] |class="matrix"| [[File:3T super-family 20.svg|200px]] |class="matrix"| [[File:3Z super-family 20.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 150<br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22<br>[[File:Venn 0110 1000.svg|25px]] |class="matrix"| [[File:3T super-family 22.svg|200px]] |class="matrix"| [[File:3Z super-family 22.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 120<br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24<br>[[File:Venn 0001 1000.svg|25px]] |class="matrix"| [[File:3T super-family 24.svg|200px]] |class="matrix"| [[File:3Z super-family 24.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 160<br>[[File:Venn 0000 0101.svg|25px]]<br><br>Ж 32<br>[[File:Venn 0000 0100.svg|25px]] |class="matrix"| [[File:3T super-family 32.svg|200px]] |class="matrix"| [[File:3Z super-family 32.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 108<br>[[File:Venn 0011 0110.svg|25px]]<br><br>Ж 36<br>[[File:Venn 0010 0100.svg|25px]] |class="matrix"| [[File:3T super-family 36.svg|200px]] |class="matrix"| [[File:3Z super-family 36.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 40<br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40<br>[[File:Venn 0001 0100.svg|25px]] |class="matrix"| [[File:3T super-family 40.svg|200px]] |class="matrix"| [[File:3Z super-family 40.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 228<br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44<br>[[File:Venn 0011 0100.svg|25px]] |class="matrix"| [[File:3T super-family 44.svg|200px]] |class="matrix"| [[File:3Z super-family 44.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 192<br>[[File:Venn 0000 0011.svg|25px]]<br><br>Ж 64<br>[[File:Venn 0000 0010.svg|25px]] |class="matrix"| [[File:3T super-family 64.svg|200px]] |class="matrix"| [[File:3Z super-family 64.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 106<br>[[File:Venn 0101 0110.svg|25px]]<br><br>Ж 66<br>[[File:Venn 0100 0010.svg|25px]] |class="matrix"| [[File:3T super-family 66.svg|200px]] |class="matrix"| [[File:3Z super-family 66.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 72<br>[[File:Venn 0001 0010.svg|25px]]<br><br>Ж 72<br>[[File:Venn 0001 0010.svg|25px]] |class="matrix"| [[File:3T super-family 72.svg|200px]] |class="matrix"| [[File:3Z super-family 72.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 226<br>[[File:Venn 0100 0111.svg|25px]]<br><br>Ж 74<br>[[File:Venn 0101 0010.svg|25px]] |class="matrix"| [[File:3T super-family 74.svg|200px]] |class="matrix"| [[File:3Z super-family 74.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 96<br>[[File:Venn 0000 0110.svg|25px]]<br><br>Ж 96<br>[[File:Venn 0000 0110.svg|25px]] |class="matrix"| [[File:3T super-family 96.svg|200px]] |class="matrix"| [[File:3Z super-family 96.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 202<br>[[File:Venn 0101 0011.svg|25px]]<br><br>Ж 98<br>[[File:Venn 0100 0110.svg|25px]] |class="matrix"| [[File:3T super-family 98.svg|200px]] |class="matrix"| [[File:3Z super-family 98.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' |class="quaestor-weight"| 4 |class="rep"| 232<br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104<br>[[File:Venn 0001 0110.svg|25px]] |class="matrix"| [[File:3T super-family 104.svg|200px]] |class="matrix"| [[File:3Z super-family 104.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' |class="quaestor-weight"| 0 |class="rep"| 66<br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106<br>[[File:Venn 0101 0110.svg|25px]] |class="matrix"| [[File:3T super-family 106.svg|200px]] |class="matrix"| [[File:3Z super-family 106.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 128<br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128<br>[[File:Venn 0000 0001.svg|25px]] |class="matrix"| [[File:3T super-family 128.svg|200px]] |class="matrix"| [[File:3Z super-family 128.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 42<br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130<br>[[File:Venn 0100 0001.svg|25px]] |class="matrix"| [[File:3T super-family 130.svg|200px]] |class="matrix"| [[File:3Z super-family 130.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 76<br>[[File:Venn 0011 0010.svg|25px]]<br><br>Ж 132<br>[[File:Venn 0010 0001.svg|25px]] |class="matrix"| [[File:3T super-family 132.svg|200px]] |class="matrix"| [[File:3Z super-family 132.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 230<br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134<br>[[File:Venn 0110 0001.svg|25px]] |class="matrix"| [[File:3T super-family 134.svg|200px]] |class="matrix"| [[File:3Z super-family 134.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 112<br>[[File:Venn 0000 1110.svg|25px]]<br><br>Ж 144<br>[[File:Venn 0000 1001.svg|25px]] |class="matrix"| [[File:3T super-family 144.svg|200px]] |class="matrix"| [[File:3Z super-family 144.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 218<br>[[File:Venn 0101 1011.svg|25px]]<br><br>Ж 146<br>[[File:Venn 0100 1001.svg|25px]] |class="matrix"| [[File:3T super-family 146.svg|200px]] |class="matrix"| [[File:3Z super-family 146.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 188<br>[[File:Venn 0011 1101.svg|25px]]<br><br>Ж 148<br>[[File:Venn 0010 1001.svg|25px]] |class="matrix"| [[File:3T super-family 148.svg|200px]] |class="matrix"| [[File:3Z super-family 148.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 22<br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150<br>[[File:Venn 0110 1001.svg|25px]] |class="matrix"| [[File:3T super-family 150.svg|200px]] |class="matrix"| [[File:3Z super-family 150.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> 78yiqaonjtmd7jszxm3o5dnzlk9rrm6 2693426 2693418 2024-12-26T23:22:01Z Watchduck 137431 2693426 wikitext text/x-wiki <templatestyles src="Template:Families of Boolean functions/table of super-families/style.css" /> {| class="wikitable sortable" id="super-families" |- !rowspan="2"| <abbr title="super-family">s.-f.</abbr><br>size !rowspan="2"| family<br>size !rowspan="2"| weight !rowspan="2"| tribe !rowspan="2"| quaestor<br>weight !rowspan="2"| <abbr title="representative">rep.</abbr> !rowspan="2"| truth<br>tables !rowspan="2"| Zhegalkin<br>indices !colspan="3" class="unsortable"| village |- !class="village-small"| size ! !class="village-small"| reverse |- |class="super-family-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 0<br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0<br>[[File:Venn 0000 0000.svg|25px]] |class="matrix"| [[File:3T super-family 0.svg|200px]] |class="matrix"| [[File:3Z super-family 0.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 170<br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2<br>[[File:Venn 0100 0000.svg|25px]] |class="matrix"| [[File:3T super-family 2.svg|200px]] |class="matrix"| [[File:3Z super-family 2.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 204<br>[[File:Venn 0011 0011.svg|25px]]<br><br>Ж 4<br>[[File:Venn 0010 0000.svg|25px]] |class="matrix"| [[File:3T super-family 4.svg|200px]] |class="matrix"| [[File:3Z super-family 4.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 102<br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6<br>[[File:Venn 0110 0000.svg|25px]] |class="matrix"| [[File:3T super-family 6.svg|200px]] |class="matrix"| [[File:3Z super-family 6.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 136<br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8<br>[[File:Venn 0001 0000.svg|25px]] |class="matrix"| [[File:3T super-family 8.svg|200px]] |class="matrix"| [[File:3Z super-family 8.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 240<br>[[File:Venn 0000 1111.svg|25px]]<br><br>Ж 16<br>[[File:Venn 0000 1000.svg|25px]] |class="matrix"| [[File:3T super-family 16.svg|200px]] |class="matrix"| [[File:3Z super-family 16.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 90<br>[[File:Venn 0101 1010.svg|25px]]<br><br>Ж 18<br>[[File:Venn 0100 1000.svg|25px]] |class="matrix"| [[File:3T super-family 18.svg|200px]] |class="matrix"| [[File:3Z super-family 18.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 0 |class="rep"| 60<br>[[File:Venn 0011 1100.svg|25px]]<br><br>Ж 20<br>[[File:Venn 0010 1000.svg|25px]] |class="matrix"| [[File:3T super-family 20.svg|200px]] |class="matrix"| [[File:3Z super-family 20.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="quaestor-weight"| 4 |class="rep"| 150<br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22<br>[[File:Venn 0110 1000.svg|25px]] |class="matrix"| [[File:3T super-family 22.svg|200px]] |class="matrix"| [[File:3Z super-family 22.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 120<br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24<br>[[File:Venn 0001 1000.svg|25px]] |class="matrix"| [[File:3T super-family 24.svg|200px]] |class="matrix"| [[File:3Z super-family 24.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 160<br>[[File:Venn 0000 0101.svg|25px]]<br><br>Ж 32<br>[[File:Venn 0000 0100.svg|25px]] |class="matrix"| [[File:3T super-family 32.svg|200px]] |class="matrix"| [[File:3Z super-family 32.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 108<br>[[File:Venn 0011 0110.svg|25px]]<br><br>Ж 36<br>[[File:Venn 0010 0100.svg|25px]] |class="matrix"| [[File:3T super-family 36.svg|200px]] |class="matrix"| [[File:3Z super-family 36.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 40<br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40<br>[[File:Venn 0001 0100.svg|25px]] |class="matrix"| [[File:3T super-family 40.svg|200px]] |class="matrix"| [[File:3Z super-family 40.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 228<br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44<br>[[File:Venn 0011 0100.svg|25px]] |class="matrix"| [[File:3T super-family 44.svg|200px]] |class="matrix"| [[File:3Z super-family 44.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 192<br>[[File:Venn 0000 0011.svg|25px]]<br><br>Ж 64<br>[[File:Venn 0000 0010.svg|25px]] |class="matrix"| [[File:3T super-family 64.svg|200px]] |class="matrix"| [[File:3Z super-family 64.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="quaestor-weight"| 2 |class="rep"| 106<br>[[File:Venn 0101 0110.svg|25px]]<br><br>Ж 66<br>[[File:Venn 0100 0010.svg|25px]] |class="matrix"| [[File:3T super-family 66.svg|200px]] |class="matrix"| [[File:3Z super-family 66.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 72<br>[[File:Venn 0001 0010.svg|25px]]<br><br>Ж 72<br>[[File:Venn 0001 0010.svg|25px]] |class="matrix"| [[File:3T super-family 72.svg|200px]] |class="matrix"| [[File:3Z super-family 72.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 226<br>[[File:Venn 0100 0111.svg|25px]]<br><br>Ж 74<br>[[File:Venn 0101 0010.svg|25px]] |class="matrix"| [[File:3T super-family 74.svg|200px]] |class="matrix"| [[File:3Z super-family 74.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 96<br>[[File:Venn 0000 0110.svg|25px]]<br><br>Ж 96<br>[[File:Venn 0000 0110.svg|25px]] |class="matrix"| [[File:3T super-family 96.svg|200px]] |class="matrix"| [[File:3Z super-family 96.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="quaestor-weight"| 2 |class="rep"| 202<br>[[File:Venn 0101 0011.svg|25px]]<br><br>Ж 98<br>[[File:Venn 0100 0110.svg|25px]] |class="matrix"| [[File:3T super-family 98.svg|200px]] |class="matrix"| [[File:3Z super-family 98.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' |class="quaestor-weight"| 4 |class="rep"| 232<br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104<br>[[File:Venn 0001 0110.svg|25px]] |class="matrix"| [[File:3T super-family 104.svg|200px]] |class="matrix"| [[File:3Z super-family 104.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' |class="quaestor-weight"| 0 |class="rep"| 66<br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106<br>[[File:Venn 0101 0110.svg|25px]] |class="matrix"| [[File:3T super-family 106.svg|200px]] |class="matrix"| [[File:3Z super-family 106.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 128<br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128<br>[[File:Venn 0000 0001.svg|25px]] |class="matrix"| [[File:3T super-family 128.svg|200px]] |class="matrix"| [[File:3Z super-family 128.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 42<br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130<br>[[File:Venn 0100 0001.svg|25px]] |class="matrix"| [[File:3T super-family 130.svg|200px]] |class="matrix"| [[File:3Z super-family 130.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 76<br>[[File:Venn 0011 0010.svg|25px]]<br><br>Ж 132<br>[[File:Venn 0010 0001.svg|25px]] |class="matrix"| [[File:3T super-family 132.svg|200px]] |class="matrix"| [[File:3Z super-family 132.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 230<br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134<br>[[File:Venn 0110 0001.svg|25px]] |class="matrix"| [[File:3T super-family 134.svg|200px]] |class="matrix"| [[File:3Z super-family 134.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 112<br>[[File:Venn 0000 1110.svg|25px]]<br><br>Ж 144<br>[[File:Venn 0000 1001.svg|25px]] |class="matrix"| [[File:3T super-family 144.svg|200px]] |class="matrix"| [[File:3Z super-family 144.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 218<br>[[File:Venn 0101 1011.svg|25px]]<br><br>Ж 146<br>[[File:Venn 0100 1001.svg|25px]] |class="matrix"| [[File:3T super-family 146.svg|200px]] |class="matrix"| [[File:3Z super-family 146.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 1 |class="rep"| 188<br>[[File:Venn 0011 1101.svg|25px]]<br><br>Ж 148<br>[[File:Venn 0010 1001.svg|25px]] |class="matrix"| [[File:3T super-family 148.svg|200px]] |class="matrix"| [[File:3Z super-family 148.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="quaestor-weight"| 3 |class="rep"| 22<br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150<br>[[File:Venn 0110 1001.svg|25px]] |class="matrix"| [[File:3T super-family 150.svg|200px]] |class="matrix"| [[File:3Z super-family 150.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> ccppmdhfcle2kxgju4ezovp4l747wfc 2693468 2693426 2024-12-27T00:05:38Z Watchduck 137431 2693468 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-families/style.css" /> {| class="wikitable sortable" id="super-families" |- !rowspan="2"| <abbr title="super-family">s.-f.</abbr><br>size !rowspan="2"| family<br>size !rowspan="2"| weight !rowspan="2"| tribe !rowspan="2"| twin<br>prefect !rowspan="2"| solidity !rowspan="2"| quaestor<br>weight !rowspan="2"| <abbr title="super-clan representative">s.-c.<br>rep.</abbr> !rowspan="2"| <abbr title="super-family representative">s.-f.<br>rep.</abbr> !rowspan="2" class="unsortable"| truth<br>tables !rowspan="2" class="unsortable"| Zhegalkin<br>indices !colspan="3" class="unsortable"| village |- !class="village-small"| size ! !class="village-small unsortable"| reverse |- |class="super-family-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' |class="twin-prefect"| <span class="sortkey">0 0</span>0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 0 |class="sf-rep"| <span class="sortkey">0</span>0<br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0 |class="matrix"| [[File:3T super-family 0.svg|200px]] |class="matrix"| [[File:3Z super-family 0.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="twin-prefect"| <span class="sortkey">0 0</span>0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">2</span>170<br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2 |class="matrix"| [[File:3T super-family 2.svg|200px]] |class="matrix"| [[File:3Z super-family 2.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="twin-prefect"| <span class="sortkey">0 0</span>0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">4</span>204<br>[[File:Venn 0011 0011.svg|25px]]<br><br>Ж 4 |class="matrix"| [[File:3T super-family 4.svg|200px]] |class="matrix"| [[File:3Z super-family 4.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="twin-prefect"| <span class="sortkey">0 0</span>0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">6</span>102<br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6 |class="matrix"| [[File:3T super-family 6.svg|200px]] |class="matrix"| [[File:3Z super-family 6.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="twin-prefect"| <span class="sortkey">4 1</span>¬4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">8</span>136<br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8 |class="matrix"| [[File:3T super-family 8.svg|200px]] |class="matrix"| [[File:3Z super-family 8.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="twin-prefect"| <span class="sortkey">0 0</span>0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">16</span>240<br>[[File:Venn 0000 1111.svg|25px]]<br><br>Ж 16 |class="matrix"| [[File:3T super-family 16.svg|200px]] |class="matrix"| [[File:3Z super-family 16.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="twin-prefect"| <span class="sortkey">0 0</span>0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">18</span>90<br>[[File:Venn 0101 1010.svg|25px]]<br><br>Ж 18 |class="matrix"| [[File:3T super-family 18.svg|200px]] |class="matrix"| [[File:3Z super-family 18.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="twin-prefect"| <span class="sortkey">0 0</span>0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">20</span>60<br>[[File:Venn 0011 1100.svg|25px]]<br><br>Ж 20 |class="matrix"| [[File:3T super-family 20.svg|200px]] |class="matrix"| [[File:3Z super-family 20.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="twin-prefect"| <span class="sortkey">0 0</span>0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 22 |class="sf-rep"| <span class="sortkey">22</span>150<br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22 |class="matrix"| [[File:3T super-family 22.svg|200px]] |class="matrix"| [[File:3Z super-family 22.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="twin-prefect"| <span class="sortkey">4 1</span>¬4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">24</span>120<br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24 |class="matrix"| [[File:3T super-family 24.svg|200px]] |class="matrix"| [[File:3Z super-family 24.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="twin-prefect"| <span class="sortkey">2 1</span>¬2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">32</span>160<br>[[File:Venn 0000 0101.svg|25px]]<br><br>Ж 32 |class="matrix"| [[File:3T super-family 32.svg|200px]] |class="matrix"| [[File:3Z super-family 32.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="twin-prefect"| <span class="sortkey">2 1</span>¬2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">36</span>108<br>[[File:Venn 0011 0110.svg|25px]]<br><br>Ж 36 |class="matrix"| [[File:3T super-family 36.svg|200px]] |class="matrix"| [[File:3Z super-family 36.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="twin-prefect"| <span class="sortkey">6 0</span>6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">40</span>40<br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40 |class="matrix"| [[File:3T super-family 40.svg|200px]] |class="matrix"| [[File:3Z super-family 40.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="twin-prefect"| <span class="sortkey">6 0</span>6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">44</span>228<br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44 |class="matrix"| [[File:3T super-family 44.svg|200px]] |class="matrix"| [[File:3Z super-family 44.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="twin-prefect"| <span class="sortkey">1 1</span>¬1 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">64</span>192<br>[[File:Venn 0000 0011.svg|25px]]<br><br>Ж 64 |class="matrix"| [[File:3T super-family 64.svg|200px]] |class="matrix"| [[File:3Z super-family 64.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="twin-prefect"| <span class="sortkey">1 1</span>¬1 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">66</span>106<br>[[File:Venn 0101 0110.svg|25px]]<br><br>Ж 66 |class="matrix"| [[File:3T super-family 66.svg|200px]] |class="matrix"| [[File:3Z super-family 66.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="twin-prefect"| <span class="sortkey">5 0</span>5 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">72</span>72<br>[[File:Venn 0001 0010.svg|25px]]<br><br>Ж 72 |class="matrix"| [[File:3T super-family 72.svg|200px]] |class="matrix"| [[File:3Z super-family 72.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="twin-prefect"| <span class="sortkey">5 0</span>5 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">74</span>226<br>[[File:Venn 0100 0111.svg|25px]]<br><br>Ж 74 |class="matrix"| [[File:3T super-family 74.svg|200px]] |class="matrix"| [[File:3Z super-family 74.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="twin-prefect"| <span class="sortkey">3 0</span>3 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">96</span>96<br>[[File:Venn 0000 0110.svg|25px]]<br><br>Ж 96 |class="matrix"| [[File:3T super-family 96.svg|200px]] |class="matrix"| [[File:3Z super-family 96.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="twin-prefect"| <span class="sortkey">3 0</span>3 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">98</span>202<br>[[File:Venn 0101 0011.svg|25px]]<br><br>Ж 98 |class="matrix"| [[File:3T super-family 98.svg|200px]] |class="matrix"| [[File:3Z super-family 98.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' |class="twin-prefect"| <span class="sortkey">7 1</span>¬7 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 104 |class="sf-rep"| <span class="sortkey">104</span>232<br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104 |class="matrix"| [[File:3T super-family 104.svg|200px]] |class="matrix"| [[File:3Z super-family 104.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' |class="twin-prefect"| <span class="sortkey">7 1</span>¬7 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 106 |class="sf-rep"| <span class="sortkey">106</span>66<br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106 |class="matrix"| [[File:3T super-family 106.svg|200px]] |class="matrix"| [[File:3Z super-family 106.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp |class="twin-prefect"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 128 |class="sf-rep"| <span class="sortkey">128</span>128<br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128 |class="matrix"| [[File:3T super-family 128.svg|200px]] |class="matrix"| [[File:3Z super-family 128.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="twin-prefect"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">130</span>42<br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130 |class="matrix"| [[File:3T super-family 130.svg|200px]] |class="matrix"| [[File:3Z super-family 130.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="twin-prefect"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">132</span>76<br>[[File:Venn 0011 0010.svg|25px]]<br><br>Ж 132 |class="matrix"| [[File:3T super-family 132.svg|200px]] |class="matrix"| [[File:3Z super-family 132.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="twin-prefect"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">134</span>230<br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134 |class="matrix"| [[File:3T super-family 134.svg|200px]] |class="matrix"| [[File:3Z super-family 134.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="twin-prefect"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">144</span>112<br>[[File:Venn 0000 1110.svg|25px]]<br><br>Ж 144 |class="matrix"| [[File:3T super-family 144.svg|200px]] |class="matrix"| [[File:3Z super-family 144.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="twin-prefect"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">146</span>218<br>[[File:Venn 0101 1011.svg|25px]]<br><br>Ж 146 |class="matrix"| [[File:3T super-family 146.svg|200px]] |class="matrix"| [[File:3Z super-family 146.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="twin-prefect"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">148</span>188<br>[[File:Venn 0011 1101.svg|25px]]<br><br>Ж 148 |class="matrix"| [[File:3T super-family 148.svg|200px]] |class="matrix"| [[File:3Z super-family 148.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="twin-prefect"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 150 |class="sf-rep"| <span class="sortkey">150</span>22<br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150 |class="matrix"| [[File:3T super-family 150.svg|200px]] |class="matrix"| [[File:3Z super-family 150.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> 4gkad0os312dl7trkfemd74g1p039h8 2693498 2693468 2024-12-27T00:33:49Z Watchduck 137431 2693498 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-families/style.css" /> {| class="wikitable sortable" id="super-families" |- !rowspan="2"| <abbr title="super-family">s.-f.</abbr><br>size !rowspan="2"| family<br>size !rowspan="2"| weight !rowspan="2"| tribe !rowspan="2"| consul !rowspan="2"| solidity !rowspan="2"| quaestor<br>weight !rowspan="2"| <abbr title="super-clan representative">s.-c.<br>rep.</abbr> !rowspan="2"| <abbr title="super-family representative">s.-f.<br>rep.</abbr> !rowspan="2" class="unsortable"| truth<br>tables !rowspan="2" class="unsortable"| Zhegalkin<br>indices !colspan="3" class="unsortable"| village |- !class="village-small"| size ! !class="village-small unsortable"| reverse |- |class="super-family-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 0 |class="sf-rep"| <span class="sortkey">0</span>0<br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0 |class="matrix"| [[File:3T super-family 0.svg|200px]] |class="matrix"| [[File:3Z super-family 0.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">2</span>170<br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2 |class="matrix"| [[File:3T super-family 2.svg|200px]] |class="matrix"| [[File:3Z super-family 2.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">4</span>204<br>[[File:Venn 0011 0011.svg|25px]]<br><br>Ж 4 |class="matrix"| [[File:3T super-family 4.svg|200px]] |class="matrix"| [[File:3Z super-family 4.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">6</span>102<br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6 |class="matrix"| [[File:3T super-family 6.svg|200px]] |class="matrix"| [[File:3Z super-family 6.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">8</span>136<br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8 |class="matrix"| [[File:3T super-family 8.svg|200px]] |class="matrix"| [[File:3Z super-family 8.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">16</span>240<br>[[File:Venn 0000 1111.svg|25px]]<br><br>Ж 16 |class="matrix"| [[File:3T super-family 16.svg|200px]] |class="matrix"| [[File:3Z super-family 16.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">18</span>90<br>[[File:Venn 0101 1010.svg|25px]]<br><br>Ж 18 |class="matrix"| [[File:3T super-family 18.svg|200px]] |class="matrix"| [[File:3Z super-family 18.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">20</span>60<br>[[File:Venn 0011 1100.svg|25px]]<br><br>Ж 20 |class="matrix"| [[File:3T super-family 20.svg|200px]] |class="matrix"| [[File:3Z super-family 20.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 22 |class="sf-rep"| <span class="sortkey">22</span>150<br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22 |class="matrix"| [[File:3T super-family 22.svg|200px]] |class="matrix"| [[File:3Z super-family 22.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">24</span>120<br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24 |class="matrix"| [[File:3T super-family 24.svg|200px]] |class="matrix"| [[File:3Z super-family 24.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">32</span>160<br>[[File:Venn 0000 0101.svg|25px]]<br><br>Ж 32 |class="matrix"| [[File:3T super-family 32.svg|200px]] |class="matrix"| [[File:3Z super-family 32.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">36</span>108<br>[[File:Venn 0011 0110.svg|25px]]<br><br>Ж 36 |class="matrix"| [[File:3T super-family 36.svg|200px]] |class="matrix"| [[File:3Z super-family 36.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">40</span>40<br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40 |class="matrix"| [[File:3T super-family 40.svg|200px]] |class="matrix"| [[File:3Z super-family 40.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">44</span>228<br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44 |class="matrix"| [[File:3T super-family 44.svg|200px]] |class="matrix"| [[File:3Z super-family 44.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 1 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">64</span>192<br>[[File:Venn 0000 0011.svg|25px]]<br><br>Ж 64 |class="matrix"| [[File:3T super-family 64.svg|200px]] |class="matrix"| [[File:3Z super-family 64.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 1 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">66</span>106<br>[[File:Venn 0101 0110.svg|25px]]<br><br>Ж 66 |class="matrix"| [[File:3T super-family 66.svg|200px]] |class="matrix"| [[File:3Z super-family 66.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 5 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">72</span>72<br>[[File:Venn 0001 0010.svg|25px]]<br><br>Ж 72 |class="matrix"| [[File:3T super-family 72.svg|200px]] |class="matrix"| [[File:3Z super-family 72.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 5 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">74</span>226<br>[[File:Venn 0100 0111.svg|25px]]<br><br>Ж 74 |class="matrix"| [[File:3T super-family 74.svg|200px]] |class="matrix"| [[File:3Z super-family 74.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 3 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">96</span>96<br>[[File:Venn 0000 0110.svg|25px]]<br><br>Ж 96 |class="matrix"| [[File:3T super-family 96.svg|200px]] |class="matrix"| [[File:3Z super-family 96.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 3 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">98</span>202<br>[[File:Venn 0101 0011.svg|25px]]<br><br>Ж 98 |class="matrix"| [[File:3T super-family 98.svg|200px]] |class="matrix"| [[File:3Z super-family 98.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' |class="consul"| 7 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 104 |class="sf-rep"| <span class="sortkey">104</span>232<br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104 |class="matrix"| [[File:3T super-family 104.svg|200px]] |class="matrix"| [[File:3Z super-family 104.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' |class="consul"| 7 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 106 |class="sf-rep"| <span class="sortkey">106</span>66<br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106 |class="matrix"| [[File:3T super-family 106.svg|200px]] |class="matrix"| [[File:3Z super-family 106.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 128 |class="sf-rep"| <span class="sortkey">128</span>128<br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128 |class="matrix"| [[File:3T super-family 128.svg|200px]] |class="matrix"| [[File:3Z super-family 128.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">130</span>42<br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130 |class="matrix"| [[File:3T super-family 130.svg|200px]] |class="matrix"| [[File:3Z super-family 130.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">132</span>76<br>[[File:Venn 0011 0010.svg|25px]]<br><br>Ж 132 |class="matrix"| [[File:3T super-family 132.svg|200px]] |class="matrix"| [[File:3Z super-family 132.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">134</span>230<br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134 |class="matrix"| [[File:3T super-family 134.svg|200px]] |class="matrix"| [[File:3Z super-family 134.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">144</span>112<br>[[File:Venn 0000 1110.svg|25px]]<br><br>Ж 144 |class="matrix"| [[File:3T super-family 144.svg|200px]] |class="matrix"| [[File:3Z super-family 144.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">146</span>218<br>[[File:Venn 0101 1011.svg|25px]]<br><br>Ж 146 |class="matrix"| [[File:3T super-family 146.svg|200px]] |class="matrix"| [[File:3Z super-family 146.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">148</span>188<br>[[File:Venn 0011 1101.svg|25px]]<br><br>Ж 148 |class="matrix"| [[File:3T super-family 148.svg|200px]] |class="matrix"| [[File:3Z super-family 148.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 150 |class="sf-rep"| <span class="sortkey">150</span>22<br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150 |class="matrix"| [[File:3T super-family 150.svg|200px]] |class="matrix"| [[File:3Z super-family 150.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> ay60jnfmdop3uuvdbwhp46see0nlpxm 2693515 2693498 2024-12-27T00:46:23Z Watchduck 137431 2693515 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-families/style.css" /> {| class="wikitable sortable" id="super-families" |- !rowspan="2"| <abbr title="super-family">s.-f.</abbr><br>size !rowspan="2"| family<br>size !rowspan="2"| weight !rowspan="2"| tribe !rowspan="2"| consul !rowspan="2"| solidity !rowspan="2"| quaestor<br>weight !rowspan="2"| <abbr title="super-clan representative">s.-c.<br>rep.</abbr> !rowspan="2"| <abbr title="super-family representative">s.-f.<br>rep.</abbr> !rowspan="2" class="unsortable"| truth<br>tables !rowspan="2" class="unsortable"| Zhegalkin<br>indices !colspan="3" class="unsortable"| village |- !class="village-small"| size ! !class="village-small unsortable"| reverse |- |class="super-family-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 0 |class="sf-rep"| <span class="sortkey">0</span><small>0</small><br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0 |class="matrix"| [[File:3T super-family 0.svg|200px]] |class="matrix"| [[File:3Z super-family 0.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">2</span><small>170</small><br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2 |class="matrix"| [[File:3T super-family 2.svg|200px]] |class="matrix"| [[File:3Z super-family 2.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">4</span><small>204</small><br>[[File:Venn 0011 0011.svg|25px]]<br><br>Ж 4 |class="matrix"| [[File:3T super-family 4.svg|200px]] |class="matrix"| [[File:3Z super-family 4.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">6</span><small>102</small><br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6 |class="matrix"| [[File:3T super-family 6.svg|200px]] |class="matrix"| [[File:3Z super-family 6.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">8</span><small>136</small><br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8 |class="matrix"| [[File:3T super-family 8.svg|200px]] |class="matrix"| [[File:3Z super-family 8.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">16</span><small>240</small><br>[[File:Venn 0000 1111.svg|25px]]<br><br>Ж 16 |class="matrix"| [[File:3T super-family 16.svg|200px]] |class="matrix"| [[File:3Z super-family 16.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">18</span><small>90</small><br>[[File:Venn 0101 1010.svg|25px]]<br><br>Ж 18 |class="matrix"| [[File:3T super-family 18.svg|200px]] |class="matrix"| [[File:3Z super-family 18.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">20</span><small>60</small><br>[[File:Venn 0011 1100.svg|25px]]<br><br>Ж 20 |class="matrix"| [[File:3T super-family 20.svg|200px]] |class="matrix"| [[File:3Z super-family 20.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 22 |class="sf-rep"| <span class="sortkey">22</span><small>150</small><br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22 |class="matrix"| [[File:3T super-family 22.svg|200px]] |class="matrix"| [[File:3Z super-family 22.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">24</span><small>120</small><br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24 |class="matrix"| [[File:3T super-family 24.svg|200px]] |class="matrix"| [[File:3Z super-family 24.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">32</span><small>160</small><br>[[File:Venn 0000 0101.svg|25px]]<br><br>Ж 32 |class="matrix"| [[File:3T super-family 32.svg|200px]] |class="matrix"| [[File:3Z super-family 32.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">36</span><small>108</small><br>[[File:Venn 0011 0110.svg|25px]]<br><br>Ж 36 |class="matrix"| [[File:3T super-family 36.svg|200px]] |class="matrix"| [[File:3Z super-family 36.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">40</span><small>40</small><br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40 |class="matrix"| [[File:3T super-family 40.svg|200px]] |class="matrix"| [[File:3Z super-family 40.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">44</span><small>228</small><br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44 |class="matrix"| [[File:3T super-family 44.svg|200px]] |class="matrix"| [[File:3Z super-family 44.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 1 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">64</span><small>192</small><br>[[File:Venn 0000 0011.svg|25px]]<br><br>Ж 64 |class="matrix"| [[File:3T super-family 64.svg|200px]] |class="matrix"| [[File:3Z super-family 64.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 1 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">66</span><small>106</small><br>[[File:Venn 0101 0110.svg|25px]]<br><br>Ж 66 |class="matrix"| [[File:3T super-family 66.svg|200px]] |class="matrix"| [[File:3Z super-family 66.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 5 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">72</span><small>72</small><br>[[File:Venn 0001 0010.svg|25px]]<br><br>Ж 72 |class="matrix"| [[File:3T super-family 72.svg|200px]] |class="matrix"| [[File:3Z super-family 72.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 5 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">74</span><small>226</small><br>[[File:Venn 0100 0111.svg|25px]]<br><br>Ж 74 |class="matrix"| [[File:3T super-family 74.svg|200px]] |class="matrix"| [[File:3Z super-family 74.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 3 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">96</span><small>96</small><br>[[File:Venn 0000 0110.svg|25px]]<br><br>Ж 96 |class="matrix"| [[File:3T super-family 96.svg|200px]] |class="matrix"| [[File:3Z super-family 96.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 3 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">98</span><small>202</small><br>[[File:Venn 0101 0011.svg|25px]]<br><br>Ж 98 |class="matrix"| [[File:3T super-family 98.svg|200px]] |class="matrix"| [[File:3Z super-family 98.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' |class="consul"| 7 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 104 |class="sf-rep"| <span class="sortkey">104</span><small>232</small><br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104 |class="matrix"| [[File:3T super-family 104.svg|200px]] |class="matrix"| [[File:3Z super-family 104.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' |class="consul"| 7 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 106 |class="sf-rep"| <span class="sortkey">106</span><small>66</small><br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106 |class="matrix"| [[File:3T super-family 106.svg|200px]] |class="matrix"| [[File:3Z super-family 106.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 128 |class="sf-rep"| <span class="sortkey">128</span><small>128</small><br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128 |class="matrix"| [[File:3T super-family 128.svg|200px]] |class="matrix"| [[File:3Z super-family 128.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">130</span><small>42</small><br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130 |class="matrix"| [[File:3T super-family 130.svg|200px]] |class="matrix"| [[File:3Z super-family 130.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">132</span><small>76</small><br>[[File:Venn 0011 0010.svg|25px]]<br><br>Ж 132 |class="matrix"| [[File:3T super-family 132.svg|200px]] |class="matrix"| [[File:3Z super-family 132.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">134</span><small>230</small><br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134 |class="matrix"| [[File:3T super-family 134.svg|200px]] |class="matrix"| [[File:3Z super-family 134.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">144</span><small>112</small><br>[[File:Venn 0000 1110.svg|25px]]<br><br>Ж 144 |class="matrix"| [[File:3T super-family 144.svg|200px]] |class="matrix"| [[File:3Z super-family 144.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">146</span><small>218</small><br>[[File:Venn 0101 1011.svg|25px]]<br><br>Ж 146 |class="matrix"| [[File:3T super-family 146.svg|200px]] |class="matrix"| [[File:3Z super-family 146.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">148</span><small>188</small><br>[[File:Venn 0011 1101.svg|25px]]<br><br>Ж 148 |class="matrix"| [[File:3T super-family 148.svg|200px]] |class="matrix"| [[File:3Z super-family 148.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 150 |class="sf-rep"| <span class="sortkey">150</span><small>22</small><br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150 |class="matrix"| [[File:3T super-family 150.svg|200px]] |class="matrix"| [[File:3Z super-family 150.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> 6f0ybnmkzh9w22xuxw7ylmoeyhr50c0 2693536 2693515 2024-12-27T00:51:57Z Watchduck 137431 2693536 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-families/style.css" /> {| class="wikitable sortable" id="super-families" |- !rowspan="2"| <abbr title="super-family">s.-f.</abbr><br>size !rowspan="2"| family<br>size !rowspan="2"| weight !rowspan="2"| tribe !rowspan="2"| consul !rowspan="2"| solidity !rowspan="2"| quaestor<br>weight !rowspan="2"| <abbr title="super-clan representative">s.-c.<br>rep.</abbr> !rowspan="2"| <abbr title="super-family representative">s.-f.<br>rep.</abbr> !rowspan="2" class="unsortable"| truth<br>tables !rowspan="2" class="unsortable"| Zhegalkin<br>indices !colspan="3" class="unsortable"| village |- !class="village-small"| size ! !class="village-small unsortable"| reverse |- |class="super-family-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 0 |class="sf-rep"| <span class="sortkey">0<br></span><small>0</small><br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0 |class="matrix"| [[File:3T super-family 0.svg|200px]] |class="matrix"| [[File:3Z super-family 0.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">2<br></span><small>170</small><br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2 |class="matrix"| [[File:3T super-family 2.svg|200px]] |class="matrix"| [[File:3Z super-family 2.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">4<br></span><small>204</small><br>[[File:Venn 0011 0011.svg|25px]]<br><br>Ж 4 |class="matrix"| [[File:3T super-family 4.svg|200px]] |class="matrix"| [[File:3Z super-family 4.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">6<br></span><small>102</small><br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6 |class="matrix"| [[File:3T super-family 6.svg|200px]] |class="matrix"| [[File:3Z super-family 6.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">8<br></span><small>136</small><br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8 |class="matrix"| [[File:3T super-family 8.svg|200px]] |class="matrix"| [[File:3Z super-family 8.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">16<br></span><small>240</small><br>[[File:Venn 0000 1111.svg|25px]]<br><br>Ж 16 |class="matrix"| [[File:3T super-family 16.svg|200px]] |class="matrix"| [[File:3Z super-family 16.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">18<br></span><small>90</small><br>[[File:Venn 0101 1010.svg|25px]]<br><br>Ж 18 |class="matrix"| [[File:3T super-family 18.svg|200px]] |class="matrix"| [[File:3Z super-family 18.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">20<br></span><small>60</small><br>[[File:Venn 0011 1100.svg|25px]]<br><br>Ж 20 |class="matrix"| [[File:3T super-family 20.svg|200px]] |class="matrix"| [[File:3Z super-family 20.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 22 |class="sf-rep"| <span class="sortkey">22<br></span><small>150</small><br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22 |class="matrix"| [[File:3T super-family 22.svg|200px]] |class="matrix"| [[File:3Z super-family 22.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">24<br></span><small>120</small><br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24 |class="matrix"| [[File:3T super-family 24.svg|200px]] |class="matrix"| [[File:3Z super-family 24.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">32<br></span><small>160</small><br>[[File:Venn 0000 0101.svg|25px]]<br><br>Ж 32 |class="matrix"| [[File:3T super-family 32.svg|200px]] |class="matrix"| [[File:3Z super-family 32.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">36<br></span><small>108</small><br>[[File:Venn 0011 0110.svg|25px]]<br><br>Ж 36 |class="matrix"| [[File:3T super-family 36.svg|200px]] |class="matrix"| [[File:3Z super-family 36.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">40<br></span><small>40</small><br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40 |class="matrix"| [[File:3T super-family 40.svg|200px]] |class="matrix"| [[File:3Z super-family 40.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">44<br></span><small>228</small><br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44 |class="matrix"| [[File:3T super-family 44.svg|200px]] |class="matrix"| [[File:3Z super-family 44.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 1 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">64<br></span><small>192</small><br>[[File:Venn 0000 0011.svg|25px]]<br><br>Ж 64 |class="matrix"| [[File:3T super-family 64.svg|200px]] |class="matrix"| [[File:3Z super-family 64.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 1 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">66<br></span><small>106</small><br>[[File:Venn 0101 0110.svg|25px]]<br><br>Ж 66 |class="matrix"| [[File:3T super-family 66.svg|200px]] |class="matrix"| [[File:3Z super-family 66.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 5 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">72<br></span><small>72</small><br>[[File:Venn 0001 0010.svg|25px]]<br><br>Ж 72 |class="matrix"| [[File:3T super-family 72.svg|200px]] |class="matrix"| [[File:3Z super-family 72.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 5 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">74<br></span><small>226</small><br>[[File:Venn 0100 0111.svg|25px]]<br><br>Ж 74 |class="matrix"| [[File:3T super-family 74.svg|200px]] |class="matrix"| [[File:3Z super-family 74.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 3 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">96<br></span><small>96</small><br>[[File:Venn 0000 0110.svg|25px]]<br><br>Ж 96 |class="matrix"| [[File:3T super-family 96.svg|200px]] |class="matrix"| [[File:3Z super-family 96.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 3 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">98<br></span><small>202</small><br>[[File:Venn 0101 0011.svg|25px]]<br><br>Ж 98 |class="matrix"| [[File:3T super-family 98.svg|200px]] |class="matrix"| [[File:3Z super-family 98.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' |class="consul"| 7 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 104 |class="sf-rep"| <span class="sortkey">104<br></span><small>232</small><br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104 |class="matrix"| [[File:3T super-family 104.svg|200px]] |class="matrix"| [[File:3Z super-family 104.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' |class="consul"| 7 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 106 |class="sf-rep"| <span class="sortkey">106<br></span><small>66</small><br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106 |class="matrix"| [[File:3T super-family 106.svg|200px]] |class="matrix"| [[File:3Z super-family 106.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 128 |class="sf-rep"| <span class="sortkey">128<br></span><small>128</small><br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128 |class="matrix"| [[File:3T super-family 128.svg|200px]] |class="matrix"| [[File:3Z super-family 128.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">130<br></span><small>42</small><br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130 |class="matrix"| [[File:3T super-family 130.svg|200px]] |class="matrix"| [[File:3Z super-family 130.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">132<br></span><small>76</small><br>[[File:Venn 0011 0010.svg|25px]]<br><br>Ж 132 |class="matrix"| [[File:3T super-family 132.svg|200px]] |class="matrix"| [[File:3Z super-family 132.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">134<br></span><small>230</small><br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134 |class="matrix"| [[File:3T super-family 134.svg|200px]] |class="matrix"| [[File:3Z super-family 134.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">144<br></span><small>112</small><br>[[File:Venn 0000 1110.svg|25px]]<br><br>Ж 144 |class="matrix"| [[File:3T super-family 144.svg|200px]] |class="matrix"| [[File:3Z super-family 144.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">146<br></span><small>218</small><br>[[File:Venn 0101 1011.svg|25px]]<br><br>Ж 146 |class="matrix"| [[File:3T super-family 146.svg|200px]] |class="matrix"| [[File:3Z super-family 146.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">148<br></span><small>188</small><br>[[File:Venn 0011 1101.svg|25px]]<br><br>Ж 148 |class="matrix"| [[File:3T super-family 148.svg|200px]] |class="matrix"| [[File:3Z super-family 148.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 150 |class="sf-rep"| <span class="sortkey">150<br></span><small>22</small><br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150 |class="matrix"| [[File:3T super-family 150.svg|200px]] |class="matrix"| [[File:3Z super-family 150.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> 9n0v5x0v5zxklqg2k0m0lgflayk0xl6 2693587 2693536 2024-12-27T10:36:38Z Watchduck 137431 2693587 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-families/style.css" /> {| class="wikitable sortable" id="super-families" |- !rowspan="2"| <abbr title="super-family">s.-f.</abbr><br>size !rowspan="2"| family<br>size !rowspan="2"| weight !rowspan="2"| tribe !rowspan="2"| consul !rowspan="2"| solidity !rowspan="2"| quaestor<br>weight !colspan="3" class="unsortable"| representatives !rowspan="2" class="unsortable"| truth<br>tables !rowspan="2" class="unsortable"| Zhegalkin<br>indices !colspan="3" class="unsortable"| village |- ! <abbr title="super-clan">s.-c.</abbr> ! <abbr title="super-family">s.-f.</abbr> ! <abbr title="ultra-family">u.-f.</abbr> !class="village-small"| size ! !class="village-small unsortable"| reverse |- |class="super-family-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 0 |class="sf-rep"| <span class="sortkey">0<br></span><small>0</small><br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0 |class="uf-rep"| Ж 0 |class="matrix"| [[File:3T super-family 0.svg|200px]] |class="matrix"| [[File:3Z super-family 0.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">2<br></span><small>170</small><br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2 |class="uf-rep"| Ж 2 |class="matrix"| [[File:3T super-family 2.svg|200px]] |class="matrix"| [[File:3Z super-family 2.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">4<br></span><small>204</small><br>[[File:Venn 0011 0011.svg|25px]]<br><br>Ж 4 |class="uf-rep"| Ж 4 |class="matrix"| [[File:3T super-family 4.svg|200px]] |class="matrix"| [[File:3Z super-family 4.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">6<br></span><small>102</small><br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6 |class="uf-rep"| Ж 6 |class="matrix"| [[File:3T super-family 6.svg|200px]] |class="matrix"| [[File:3Z super-family 6.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">8<br></span><small>136</small><br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8 |class="uf-rep"| Ж 8 |class="matrix"| [[File:3T super-family 8.svg|200px]] |class="matrix"| [[File:3Z super-family 8.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">16<br></span><small>240</small><br>[[File:Venn 0000 1111.svg|25px]]<br><br>Ж 16 |class="uf-rep"| Ж 0 |class="matrix"| [[File:3T super-family 16.svg|200px]] |class="matrix"| [[File:3Z super-family 16.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">18<br></span><small>90</small><br>[[File:Venn 0101 1010.svg|25px]]<br><br>Ж 18 |class="uf-rep"| Ж 2 |class="matrix"| [[File:3T super-family 18.svg|200px]] |class="matrix"| [[File:3Z super-family 18.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">20<br></span><small>60</small><br>[[File:Venn 0011 1100.svg|25px]]<br><br>Ж 20 |class="uf-rep"| Ж 4 |class="matrix"| [[File:3T super-family 20.svg|200px]] |class="matrix"| [[File:3Z super-family 20.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 22 |class="sf-rep"| <span class="sortkey">22<br></span><small>150</small><br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22 |class="uf-rep"| Ж 6 |class="matrix"| [[File:3T super-family 22.svg|200px]] |class="matrix"| [[File:3Z super-family 22.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">24<br></span><small>120</small><br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24 |class="uf-rep"| Ж 8 |class="matrix"| [[File:3T super-family 24.svg|200px]] |class="matrix"| [[File:3Z super-family 24.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">32<br></span><small>160</small><br>[[File:Venn 0000 0101.svg|25px]]<br><br>Ж 32 |class="uf-rep"| Ж 32 |class="matrix"| [[File:3T super-family 32.svg|200px]] |class="matrix"| [[File:3Z super-family 32.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">36<br></span><small>108</small><br>[[File:Venn 0011 0110.svg|25px]]<br><br>Ж 36 |class="uf-rep"| Ж 36 |class="matrix"| [[File:3T super-family 36.svg|200px]] |class="matrix"| [[File:3Z super-family 36.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">40<br></span><small>40</small><br>[[File:Venn 0001 0100.svg|25px]]<br><br>Ж 40 |class="uf-rep"| Ж 40 |class="matrix"| [[File:3T super-family 40.svg|200px]] |class="matrix"| [[File:3Z super-family 40.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">44<br></span><small>228</small><br>[[File:Venn 0010 0111.svg|25px]]<br><br>Ж 44 |class="uf-rep"| Ж 40 |class="matrix"| [[File:3T super-family 44.svg|200px]] |class="matrix"| [[File:3Z super-family 44.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 1 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">64<br></span><small>192</small><br>[[File:Venn 0000 0011.svg|25px]]<br><br>Ж 64 |class="uf-rep"| Ж 64 |class="matrix"| [[File:3T super-family 64.svg|200px]] |class="matrix"| [[File:3Z super-family 64.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 1 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">66<br></span><small>106</small><br>[[File:Venn 0101 0110.svg|25px]]<br><br>Ж 66 |class="uf-rep"| Ж 66 |class="matrix"| [[File:3T super-family 66.svg|200px]] |class="matrix"| [[File:3Z super-family 66.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 5 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">72<br></span><small>72</small><br>[[File:Venn 0001 0010.svg|25px]]<br><br>Ж 72 |class="uf-rep"| Ж 72 |class="matrix"| [[File:3T super-family 72.svg|200px]] |class="matrix"| [[File:3Z super-family 72.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 5 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">74<br></span><small>226</small><br>[[File:Venn 0100 0111.svg|25px]]<br><br>Ж 74 |class="uf-rep"| Ж 72 |class="matrix"| [[File:3T super-family 74.svg|200px]] |class="matrix"| [[File:3Z super-family 74.svg|200px]] |class="village village-small"| 2 |class="village"| {2, 10} |class="village village-small"| <span class="letter">T</span> {12, 3}<br><span class="letter">Z</span> {4, 5} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 3 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span class="sortkey">96<br></span><small>96</small><br>[[File:Venn 0000 0110.svg|25px]]<br><br>Ж 96 |class="uf-rep"| Ж 96 |class="matrix"| [[File:3T super-family 96.svg|200px]] |class="matrix"| [[File:3Z super-family 96.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''2''' |class="consul"| 3 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 44 |class="sf-rep"| <span class="sortkey">98<br></span><small>202</small><br>[[File:Venn 0101 0011.svg|25px]]<br><br>Ж 98 |class="uf-rep"| Ж 98 |class="matrix"| [[File:3T super-family 98.svg|200px]] |class="matrix"| [[File:3Z super-family 98.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''3''' |class="consul"| 7 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 104 |class="sf-rep"| <span class="sortkey">104<br></span><small>232</small><br>[[File:Venn 0001 0111.svg|25px]]<br><br>Ж 104 |class="uf-rep"| Ж 104 |class="matrix"| [[File:3T super-family 104.svg|200px]] |class="matrix"| [[File:3Z super-family 104.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''3''' |class="consul"| 7 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 106 |class="sf-rep"| <span class="sortkey">106<br></span><small>66</small><br>[[File:Venn 0100 0010.svg|25px]]<br><br>Ж 106 |class="uf-rep"| Ж 104 |class="matrix"| [[File:3T super-family 106.svg|200px]] |class="matrix"| [[File:3Z super-family 106.svg|200px]] |class="village village-small"| 2 |class="village"| {6, 14} |class="village village-small"| <span class="letter">T</span> {6, 9}<br><span class="letter">Z</span> {6, 7} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 1, 7 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 128 |class="sf-rep"| <span class="sortkey">128<br></span><small>128</small><br>[[File:Venn 0000 0001.svg|25px]]<br><br>Ж 128 |class="uf-rep"| Ж 128 |class="matrix"| [[File:3T super-family 128.svg|200px]] |class="matrix"| [[File:3Z super-family 128.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">130<br></span><small>42</small><br>[[File:Venn 0101 0100.svg|25px]]<br><br>Ж 130 |class="uf-rep"| Ж 130 |class="matrix"| [[File:3T super-family 130.svg|200px]] |class="matrix"| [[File:3Z super-family 130.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">132<br></span><small>76</small><br>[[File:Venn 0011 0010.svg|25px]]<br><br>Ж 132 |class="uf-rep"| Ж 132 |class="matrix"| [[File:3T super-family 132.svg|200px]] |class="matrix"| [[File:3Z super-family 132.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">134<br></span><small>230</small><br>[[File:Venn 0110 0111.svg|25px]]<br><br>Ж 134 |class="uf-rep"| Ж 134 |class="matrix"| [[File:3T super-family 134.svg|200px]] |class="matrix"| [[File:3Z super-family 134.svg|200px]] |class="village village-small"| 4 |class="village"| {1, 3, 5, 15} |class="village village-small"| <span class="letter">T</span> {8, 4, 2, 1}<br><span class="letter">Z</span> {8, 12, 10, 15} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 130 |class="sf-rep"| <span class="sortkey">144<br></span><small>112</small><br>[[File:Venn 0000 1110.svg|25px]]<br><br>Ж 144 |class="uf-rep"| Ж 128 |class="matrix"| [[File:3T super-family 144.svg|200px]] |class="matrix"| [[File:3Z super-family 144.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">146<br></span><small>218</small><br>[[File:Venn 0101 1011.svg|25px]]<br><br>Ж 146 |class="uf-rep"| Ж 130 |class="matrix"| [[File:3T super-family 146.svg|200px]] |class="matrix"| [[File:3Z super-family 146.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">148<br></span><small>188</small><br>[[File:Venn 0011 1101.svg|25px]]<br><br>Ж 148 |class="uf-rep"| Ж 132 |class="matrix"| [[File:3T super-family 148.svg|200px]] |class="matrix"| [[File:3Z super-family 148.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 150 |class="sf-rep"| <span class="sortkey">150<br></span><small>22</small><br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150 |class="uf-rep"| Ж 134 |class="matrix"| [[File:3T super-family 150.svg|200px]] |class="matrix"| [[File:3Z super-family 150.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> gemv1mg50d3b4itwhlygpxwyz0pv5jh 2693589 2693587 2024-12-27T10:41:03Z Watchduck 137431 2693589 wikitext text/x-wiki <templatestyles src="Families of Boolean functions/table of super-families/style.css" /> {| class="wikitable sortable" id="super-families" |- !colspan="2"| sizes !rowspan="2"| weight !rowspan="2"| tribe !rowspan="2"| consul !rowspan="2"| solidity !rowspan="2"| quaestor<br>weight !colspan="3" class="unsortable"| representatives !rowspan="2" class="unsortable"| truth<br>tables !rowspan="2" class="unsortable"| Zhegalkin<br>indices !colspan="3" class="unsortable"| village |- ! <abbr title="super-family">sf</abbr> ! <abbr title="family">f</abbr> ! <abbr title="super-clan">sc</abbr> ! <abbr title="super-family">sf</abbr> ! <abbr title="ultra-family">uf</abbr> !class="village-small"| size ! !class="village-small unsortable"| reverse |- |class="super-family-size"| 2 |class="family-size"| 1 |class="weight"| 0, 8 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 0 |class="sf-rep"| <span class="sortkey">0<br></span><small>0</small><br>[[File:Venn 0000 0000.svg|25px]]<br><br>Ж 0 |class="uf-rep"| Ж 0 |class="matrix"| [[File:3T super-family 0.svg|200px]] |class="matrix"| [[File:3Z super-family 0.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">2<br></span><small>170</small><br>[[File:Venn 0101 0101.svg|25px]]<br><br>Ж 2 |class="uf-rep"| Ж 2 |class="matrix"| [[File:3T super-family 2.svg|200px]] |class="matrix"| [[File:3Z super-family 2.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">4<br></span><small>204</small><br>[[File:Venn 0011 0011.svg|25px]]<br><br>Ж 4 |class="uf-rep"| Ж 4 |class="matrix"| [[File:3T super-family 4.svg|200px]] |class="matrix"| [[File:3Z super-family 4.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">6<br></span><small>102</small><br>[[File:Venn 0110 0110.svg|25px]]<br><br>Ж 6 |class="uf-rep"| Ж 6 |class="matrix"| [[File:3T super-family 6.svg|200px]] |class="matrix"| [[File:3Z super-family 6.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">8<br></span><small>136</small><br>[[File:Venn 0001 0001.svg|25px]]<br><br>Ж 8 |class="uf-rep"| Ж 8 |class="matrix"| [[File:3T super-family 8.svg|200px]] |class="matrix"| [[File:3Z super-family 8.svg|200px]] |class="village village-small"| 1 |class="village"| {0} |class="village village-small"| <span class="letter">T</span> {0}<br><span class="letter">Z</span> {0} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 2 |class="sf-rep"| <span class="sortkey">16<br></span><small>240</small><br>[[File:Venn 0000 1111.svg|25px]]<br><br>Ж 16 |class="uf-rep"| Ж 0 |class="matrix"| [[File:3T super-family 16.svg|200px]] |class="matrix"| [[File:3Z super-family 16.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">18<br></span><small>90</small><br>[[File:Venn 0101 1010.svg|25px]]<br><br>Ж 18 |class="uf-rep"| Ж 2 |class="matrix"| [[File:3T super-family 18.svg|200px]] |class="matrix"| [[File:3Z super-family 18.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 0 |class="sc-rep"| Ж 6 |class="sf-rep"| <span class="sortkey">20<br></span><small>60</small><br>[[File:Venn 0011 1100.svg|25px]]<br><br>Ж 20 |class="uf-rep"| Ж 4 |class="matrix"| [[File:3T super-family 20.svg|200px]] |class="matrix"| [[File:3Z super-family 20.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 2 |class="family-size"| 2 |class="weight"| 4 |class="tribe"| blunt '''0''' |class="consul"| 0 |class="solidity"| fluid |class="quaestor-weight"| 4 |class="sc-rep"| Ж 22 |class="sf-rep"| <span class="sortkey">22<br></span><small>150</small><br>[[File:Venn 0110 1001.svg|25px]]<br><br>Ж 22 |class="uf-rep"| Ж 6 |class="matrix"| [[File:3T super-family 22.svg|200px]] |class="matrix"| [[File:3Z super-family 22.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 4 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">24<br></span><small>120</small><br>[[File:Venn 0001 1110.svg|25px]]<br><br>Ж 24 |class="uf-rep"| Ж 8 |class="matrix"| [[File:3T super-family 24.svg|200px]] |class="matrix"| [[File:3Z super-family 24.svg|200px]] |class="village village-small"| 1 |class="village"| {8} |class="village village-small"| <span class="letter">T</span> {15}<br><span class="letter">Z</span> {1} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''1''' |class="consul"| 2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 8 |class="sf-rep"| <span class="sortkey">32<br></span><small>160</small><br>[[File:Venn 0000 0101.svg|25px]]<br><br>Ж 32 |class="uf-rep"| Ж 32 |class="matrix"| [[File:3T super-family 32.svg|200px]] |class="matrix"| [[File:3Z super-family 32.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 8 |class="weight"| 4 |class="tribe"| blunt '''1''' |class="consul"| 2 |class="solidity"| solid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 24 |class="sf-rep"| <span class="sortkey">36<br></span><small>108</small><br>[[File:Venn 0011 0110.svg|25px]]<br><br>Ж 36 |class="uf-rep"| Ж 36 |class="matrix"| [[File:3T super-family 36.svg|200px]] |class="matrix"| [[File:3Z super-family 36.svg|200px]] |class="village village-small"| 2 |class="village"| {4, 12} |class="village village-small"| <span class="letter">T</span> {10, 5}<br><span class="letter">Z</span> {2, 3} |- |class="super-family-size"| 8 |class="family-size"| 4 |class="weight"| 2, 6 |class="tribe"| blunt '''2''' |class="consul"| 6 |class="solidity"| fluid |class="quaestor-weight"| 2 |class="sc-rep"| Ж 40 |class="sf-rep"| <span 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|class="quaestor-weight"| 1 |class="sc-rep"| Ж 134 |class="sf-rep"| <span class="sortkey">148<br></span><small>188</small><br>[[File:Venn 0011 1101.svg|25px]]<br><br>Ж 148 |class="uf-rep"| Ж 132 |class="matrix"| [[File:3T super-family 148.svg|200px]] |class="matrix"| [[File:3Z super-family 148.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |- |class="super-family-size"| 16 |class="family-size"| 8 |class="weight"| 3, 5 |class="tribe"| sharp |class="consul"| |class="solidity"| |class="quaestor-weight"| 3 |class="sc-rep"| Ж 150 |class="sf-rep"| <span class="sortkey">150<br></span><small>22</small><br>[[File:Venn 0110 1000.svg|25px]]<br><br>Ж 150 |class="uf-rep"| Ж 134 |class="matrix"| [[File:3T super-family 150.svg|200px]] |class="matrix"| [[File:3Z super-family 150.svg|200px]] |class="village village-small"| 4 |class="village"| {7, 9, 11, 13} |class="village village-small"| <span class="letter">T</span> {14, 7, 11, 13}<br><span class="letter">Z</span> {14, 9, 13, 11} |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> q9kfg0cmb1cksiy0citqeqh1o8hfjdm 0 317535 2693402 2024-12-26T22:38:07Z Eshaa2024 2993595 New resource with "==Introduction== The Cauchy integral theorem is one of the central results of [[Complex Analysis]]. It exists in various versions, and in this article, we aim to present a basic one for convex regions and a relatively general one for [[w:en:Homology (mathematics)|nullhomologous cycles]]. ==For Convex Regions== === Statement === Let <math>G \subseteq \mathbb{C}</math> be a convex region, and let <math>\gamma</math> be a closed Complex Analysis/rectifiable curve|rectif..." 2693402 wikitext text/x-wiki ==Introduction== The Cauchy integral theorem is one of the central results of [[Complex Analysis]]. It exists in various versions, and in this article, we aim to present a basic one for convex regions and a relatively general one for [[w:en:Homology (mathematics)|nullhomologous cycles]]. ==For Convex Regions== === Statement === Let <math>G \subseteq \mathbb{C}</math> be a convex region, and let <math>\gamma</math> be a closed [[Complex Analysis/rectifiable curve|rectifiable curve]] [[Complex Analysis/Trace|Trace of Curve]] in <math>G</math>. Then, for every holomorphic function <math>f \colon G \to \mathbb{C}</math>, the following holds: <center><math>\int_\gamma f(z)\, dz = 0</math></center> === Proof 1: Antiderivatives of f === First, we observe that <math>f</math> has a antiderivative in <math>G</math>. Fix a point <math>z_0 \in G</math>. For any point <math>z \in G</math>, let <math>[z_0, z]</math> denote the straight-line segment connecting <math>z_0</math> and <math>z</math> as path. == Proof 2: Definition of the Antiderivative == Define <math>F \colon G \to \mathbb{C}</math> by: :<math>F(z) := \int_{[z_0, z]} f(\zeta), d\zeta</math>. Due to the convexity of <math>G</math>, the triangle <math>D</math> with vertices <math>z_0, z, w</math> lies entirely within <math>G</math> for <math>z, w \in G</math>. === Proof 3: Application of Goursat’s Lemma === By [[Complex Analysis/Goursat's Lemma|Goursat's Lemma]] for the boundary <math>\partial \Delta</math> of a triangle <math>\Delta</math> with vertices <math>z_0, z, w \in \mathbb{C}</math>, we have: :<math> \begin{array}{rl} 0 &= \int_{\partial \Delta} f(z)\, dz \\ &= \int_{[z_{0},z]} f(\zeta)\, d\zeta - \int_{[z_{0},w]} f(\zeta)\, d\zeta + \int_{[z,w]} f(\zeta)\, d\zeta \\ & = F(z) - F(w) + \int_{[z,w]} f(\zeta)\, d\zeta \end{array} </math> === Proof 4: Conclusion Using Goursat's Lemma === This leads to: :<math>\begin{array}{rl} F(z) - F(w) &= \int_{[w,z]} f(\zeta)\,d\zeta\\ &= \int_0^1 f\bigl(w + t(z-w)\bigr)\cdot (z-w)\, dt\\ &= \underbrace{\int_0^1 f\bigl(w+t(z-w)\bigr)\, dt}_{A(z):=} \cdot (z-w) \end{array}</math> Thus, we have: :<math>A(z)=\frac{F(z) - F(w)}{(z-w)}</math> === Proof 5: Limit Process === Since <math>A</math> is continuous in <math>w</math>, taking the limit as <math>z \to w</math> gives: :<math>A(w) = \lim_{z \to w} A(z) = \lim_{z \to w} \frac{F(z) - F(w)}{(z-w)} = F'(w).</math> === Proof 5: Differentiability of <math>F</math> === Therefore, <math>A \colon G \to \mathbb{C}</math> is continuous, and <math>F</math> is differentiable in <math>w \in G</math>, with: <center><math>F'(w) = A(w) = f(w).</math></center> Since <math>w \in G</math> was arbitrary, we conclude <math>F' = f</math>, proving that <math>f</math> has a antiderivative. === Proof 6: Path Integration === Now, let <math>\gamma \colon [a, b] \to G</math> be a piecewise continuously differentiable, closed curve. Then: <center><math> \begin{array}{rl} \int_\gamma f(z)\, dz &= \int_a^b f(\gamma(t)) \gamma'(t)\, dt \\ &= \int_a^b F'(\gamma(t)) \gamma'(t)\, dt \\ &= \int_a^b (F \circ \gamma)'(t)\, dt \\ &= F(\gamma(b)) - F(\gamma(a)) = 0. \end{array} </math></center> === Proof 7: === Let <math>\gamma \colon [a, b] \to G</math> be an arbitrary integration path in <math>G</math>, and let <math>\epsilon > 0</math>. As shown [[Complex Analysis/Curve Integral #Approximation by polygonal paths|here]], we choose a polygonal path <math>\hat{\gamma} \colon [a, b] \to \mathbb{C}</math> such that <math>\hat{\gamma}(a) = \gamma(a)</math>, <math>\hat{\gamma}(b) = \gamma(b)</math>, and :<math>\left|\int_{\hat{\gamma}} f(z), dz - \int_{\gamma} f(z), dz\right| < \epsilon.</math> Since polygonal paths are piecewise continuously differentiable, the above result implies <math>\int_{\hat{\gamma}} f(z), dz = 0</math>. Consequently, :<math>\left|\int_{\gamma} f(z), dz\right| < \epsilon.</math> As <math>\epsilon > 0</math> was arbitrary, the claim follows. == For Cycles in Arbitrary Open Sets == In arbitrary open sets, one must ensure that cycles do not enclose singularities or poles in the complement of the domain. Enclosing such singularities may contribute a non-zero value to the integral (e.g., the function <math>f(z) = \frac{1}{z}</math> and <math>\gamma(t) := e^{it}</math> in a domain <math>G = \mathbb{C}\setminus \{0\}</math> . Even though <math>f</math> is holomorphic in <math>G</math>, the integral is not zero but <math>2\pi i</math> (see [[w:en:Homology (mathematics)#Nullhomologous cycle|nullhomologous cycle]]). === Statement === Let <math>G \subseteq \mathbb{C}</math> be open, and let <math>\Gamma</math> be a [[w:en:Homology (mathematics)#Nullhomologous cycle|nullhomologous]] cycle in <math>G</math>. Then, for every holomorphic function <math>f \colon G \to \mathbb{C}</math>, the following holds: <center><math>\int_\Gamma f(z)\, dz = 0</math></center> === Proof === Let <math>w \in G \setminus \text{trace}(\Gamma)</math>, and define <math>g \colon G \to \mathbb{C}</math> by <center><math>g(z) := (z-w) \cdot f(z).</math></center> Then, <math>g</math> is holomorphic, and by the [[w:en:Cauchy integral formula#For cycles in arbitrary open sets|global integral formula]], we have: :<math> \int_\Gamma f(z)\, dz = \int_{\Gamma} \frac{g(z)}{z-w}\, dz = 2\pi i \cdot n(\Gamma, w) \cdot g(w) = 0. </math> == See Also == *[[w:en:Complex Analysis|Complex Analysis]] *[[w:en:Cauchy's integral theorem|Cauchy integral theorem for disks]] == Page Information == You can display this page as 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This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Cauchy%20Integral%20Theorem * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Integralsatz von Cauchy Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Integralsatz von Cauchy|Integralsatz von Cauchy]] - URL: https://de.wikiversity.org/wiki/Integralsatz_von_Cauchy * Date: 12/18/2024 <span type="translate" src="Integralsatz von Cauchy" srclang="de" date="12/18/2024" time="09:15" status="inprogress"></span> <noinclude> [[de:Integralsatz von Cauchy]] </noinclude> [[Category:Wiki2Reveal]] 4l34dc1dy3bzv55d8nq0lbdx3baw1fo 0 317536 2693409 2024-12-26T22:54:35Z Tule-hog 2984180 adapt item from [[Network+/Operations/Performance]] 2693409 wikitext text/x-wiki * Test your internet connection using any of the public available services: [https://fast.com fast.com] or [http://speedtest.net speedtest.net]. ** Install and test [https://www.speedtest.net/apps/cli speedtest-cli], [[:w:Speedtest.net|Speedtest.net]]'s [[:w:command-line interface|command-line interface]]. mm7sw00u0nty34oxw15w649h5lp4cyt 2693411 2693409 2024-12-26T22:55:19Z Tule-hog 2984180 add {[[template:bookcat|bookcat]] 2693411 wikitext text/x-wiki * Test your internet connection using any of the public available services: [https://fast.com fast.com] or [http://speedtest.net speedtest.net]. ** Install and test [https://www.speedtest.net/apps/cli speedtest-cli], [[:w:Speedtest.net|Speedtest.net]]'s [[:w:command-line interface|command-line interface]]. {{bookcat}} qf9dm9c62crt704cq1x4b2lzp1hi4xi 2693499 2693411 2024-12-27T00:34:55Z Tule-hog 2984180 mv item from [[Network+/Architecture/Services/Reverse Proxy]] 2693499 wikitext text/x-wiki * Test your internet connection using any of the public available services: [https://fast.com fast.com] or [http://speedtest.net speedtest.net]. ** Install and test [https://www.speedtest.net/apps/cli speedtest-cli], [[:w:Speedtest.net|Speedtest.net]]'s [[:w:command-line interface|command-line interface]]. * Browse [[/Review list of available proxy servers/]] {{bookcat}} qb1xidp6t0vw4pw1tfkkqlvhn8nr4kj 2693512 2693499 2024-12-27T00:45:08Z Tule-hog 2984180 mv from old guide 2693512 wikitext text/x-wiki * Test your internet connection using any of the public available services: [https://fast.com fast.com] or [http://speedtest.net speedtest.net]. ** Install and test [https://www.speedtest.net/apps/cli speedtest-cli], [[:w:Speedtest.net|Speedtest.net]]'s [[:w:command-line interface|command-line interface]]. * Browse [[/Review list of available proxy servers/]] *Work through the following activities taken from the [[Computer Networks]] resource. ** [[Computer Networks/Arp|Arp]] ** [[Computer Networks/Ipconfig|Ipconfig]] ** [[Computer Networks/Nbtstat|Nbtstat]] ** [[Computer Networks/Netstat|Netstat]] ** [[Computer Networks/Nslookup|Nslookup]] ** [[Computer Networks/Ping|Ping]] ** [[Computer Networks/Route|Route]] ** [[Computer Networks/Tracert|Tracert]] {{bookcat}} iljoemnhm6sh634gvxc1ris0hooc46f 2693513 2693512 2024-12-27T00:45:26Z Tule-hog 2984180 clarify item 2693513 wikitext text/x-wiki * Test your internet connection using any of the public available services: [https://fast.com fast.com] or [http://speedtest.net speedtest.net]. ** Install and test [https://www.speedtest.net/apps/cli speedtest-cli], [[:w:Speedtest.net|Speedtest.net]]'s [[:w:command-line interface|command-line interface]]. * Browse [[/Review list of available proxy servers/]] *Work through the following activities on networking tools taken from the [[Computer Networks]] resource. ** [[Computer Networks/Arp|Arp]] ** [[Computer Networks/Ipconfig|Ipconfig]] ** [[Computer Networks/Nbtstat|Nbtstat]] ** [[Computer Networks/Netstat|Netstat]] ** [[Computer Networks/Nslookup|Nslookup]] ** [[Computer Networks/Ping|Ping]] ** [[Computer Networks/Route|Route]] ** [[Computer Networks/Tracert|Tracert]] {{bookcat}} h656wlu1jbcbr43f8x1q27e9u55nxn2 10 317537 2693410 2024-12-26T22:55:03Z Tule-hog 2984180 mk tre 2693410 wikitext text/x-wiki #REDIRECT [[Template:BookCat]] kf7npwfgnbkxnupxjbfgt01m4brlkj4 0 317538 2693419 2024-12-26T23:14:38Z Tule-hog 2984180 adapt from [[Network+/Troubleshooting/Tools]] 2693419 wikitext text/x-wiki Work through the following activities taken from the [[Computer Networks]] resource. * [[Computer Networks/Arp|Arp]] * [[Computer Networks/Ipconfig|Ipconfig]] * [[Computer Networks/Nbtstat|Nbtstat]] * [[Computer Networks/Netstat|Netstat]] * [[Computer Networks/Nslookup|Nslookup]] * [[Computer Networks/Ping|Ping]] * [[Computer Networks/Route|Route]] * [[Computer Networks/Tracert|Tracert]] htv0fj4kmhz3krcsr9f6q60jfxim2h7 2693420 2693419 2024-12-26T23:14:54Z Tule-hog 2984180 add {[[template:bookcat|bookcat]] 2693420 wikitext text/x-wiki Work through the following activities taken from the [[Computer Networks]] resource. * [[Computer Networks/Arp|Arp]] * [[Computer Networks/Ipconfig|Ipconfig]] * [[Computer Networks/Nbtstat|Nbtstat]] * [[Computer Networks/Netstat|Netstat]] * [[Computer Networks/Nslookup|Nslookup]] * [[Computer Networks/Ping|Ping]] * [[Computer Networks/Route|Route]] * [[Computer Networks/Tracert|Tracert]] {{bookcat}} ny5t0000nv1r1uf9wxoev0mxjr8akqe 2693511 2693420 2024-12-27T00:44:55Z Tule-hog 2984180 mv to activities, nominate speedy 2693511 wikitext text/x-wiki {{speedy|C7}} *Work through the following activities taken from the [[Computer Networks]] resource. ** [[Computer Networks/Arp|Arp]] ** [[Computer Networks/Ipconfig|Ipconfig]] ** [[Computer Networks/Nbtstat|Nbtstat]] ** [[Computer Networks/Netstat|Netstat]] ** [[Computer Networks/Nslookup|Nslookup]] ** [[Computer Networks/Ping|Ping]] ** [[Computer Networks/Route|Route]] ** [[Computer Networks/Tracert|Tracert]] {{bookcat}} 2a1htzxfp5i9hz9mep8a2gofzjmaeo6 0 317539 2693421 2024-12-26T23:16:03Z Eshaa2024 2993595 New resource with "== To the Learning Unit == This learning unit addresses singularities of complex functions. For singularities in real analysis, these are referred to as [[w:en:Removable singularity|removable singularities]]. In complex analysis, singularities hold particular significance for the value of contour integrals. With the residue theorem, we find that only the coefficients of the Laurent series preceding <math>(z-z_0)^{-1}</math> contribute meaningfully to the contour integral..." 2693421 wikitext text/x-wiki == To the Learning Unit == This learning unit addresses singularities of complex functions. For singularities in real analysis, these are referred to as [[w:en:Removable singularity|removable singularities]]. In complex analysis, singularities hold particular significance for the value of contour integrals. With the residue theorem, we find that only the coefficients of the Laurent series preceding <math>(z-z_0)^{-1}</math> contribute meaningfully to the contour integral. The integration of other summands from the Laurent series results in a contribution of 0 to the contour integral. To prove this residue theorem, singularities must first be classified. == Introduction == '''Isolated singularities''' are studied in the [[w:en:Branch of mathematics|branch of mathematics]] known as [[w:en:Complex analysis|complex analysis]]. Isolated singularities are special [[w:en:Isolated point|isolated points]] in the [[w:en:Domain (mathematics)|domain]] of a [[w:en:Holomorphic function|holomorphic function]]. Isolated singularities are classified into [[w:en:Removable singularity|removable singularities]], [[w:en:Pole (complex analysis)|poles]], and '''essential singularities'''. == Definition == Let <math>\Omega \subseteq \mathbb C</math> be an [[w:en:Open set|open subset]] and <math>z_0 \in \Omega</math>. Further, let <math>f\colon \Omega \setminus {z_0} \to \mathbb C</math> be a holomorphic [[w:en:Complex-valued function|complex-valued function]]. Then, <math>z_0</math> is called an ''isolated singularity'' of <math>f</math>. == Classification == *Each isolated singularity belongs to one of the following three classes: *The point <math>z_0</math> is called a ''removable singularity'' if <math>f</math> can be extended to be holomorphic on <math>\Omega</math>. According to the [[w:en:Riemann's removable singularity theorem|Riemann removable singularity theorem]], this is, for example, the case if <math>f</math> is bounded in a neighborhood of <math>z_0</math>. *The point <math>z_0</math> is called a ''[[w:en:Pole (complex analysis)|pole]]'' if <math>z_0</math> is not a removable singularity and there exists a natural number <math>k</math> such that <math>(z-z_0)^k \cdot f(z)</math> has a removable singularity at <math>z_0</math>. If <math>k</math> is chosen minimally, <math>f</math> is said to have a pole of order <math>k</math> at <math>z_0</math>. *Otherwise, <math>z_0</math> is called an ''essential singularity'' of <math>f</math>. == Isolated Singularities and the Laurent Series == The type of singularity can also be determined from the [[Laurent Series|Laurent Series]]: : <math>\sum_{n=-\infty}^\infty a_n(z-z_0)^n</math> of <math>f</math> at <math>z_0</math>. === Removable Singularities and the Laurent Series === A singularity is removable if and only if the principal part vanishes, i.e., <math>a_n=0</math> for all negative integers <math>n</math>. : <math>\sum_{n=-\infty}^\infty a_n(z-z_0)^n</math> of <math>f</math> at <math>z_0</math>. === Pole and the Laurent Series === A pole of order <math>k</math> occurs if and only if the principal part terminates after <math>k</math> terms, i.e., <math>a_{-k} \neq 0</math> and <math>a_n=0</math> for all <math>n < -k</math>. === Essential Singularity and the Laurent Series === An essential singularity occurs if infinitely many terms with negative exponents are nonzero. Statements about the properties of holomorphic functions at essential singularities are made by the [[w:en:Picard theorem|Great Picard theorem]] and, as a simpler special case, the [[w:en:Casorati-Weierstrass theorem|Casorati-Weierstrass theorem]]. == Examples == [[File:Essential singularity.png|300px|Plot of the function <math>\exp(1/z)</math>. It has an essential singularity at the origin (center). The hue corresponds to the complex argument of the function value, while the brightness represents its magnitude. Here, the behavior of the essential singularity varies depending on the approach (in contrast to a pole, which would appear uniformly white).]] Plot of the function <math>\exp(1/z)</math>. It has an essential singularity at the origin (center). The hue corresponds to the complex argument of the function value, while the brightness represents its magnitude. Here, the behavior of the essential singularity varies depending on the approach (in contrast to a pole, which would appear uniformly white). === Properties === Let <math>\Omega=\mathbb C</math> and <math>z_0=0.</math> *<math>f\colon \Omega \setminus {0}\to\mathbb C,,z\mapsto \tfrac{\sin(z)}{z}</math> can be continuously extended to <math>\Omega</math> by defining <math>f(0)=1</math>. Thus, <math>f</math> has a ''removable'' singularity at <math>0</math>. *<math>f\colon \Omega\setminus {0}\to\mathbb C,,z\mapsto \tfrac{1}{z}</math> has a pole of order 1 at <math>0</math> because <math>g(z)=z^1\cdot f(z)</math> can be continuously extended to <math>\Omega</math> by defining <math>g(0)=1</math>. *<math>f\colon \Omega\setminus {0}\to\mathbb C,,z\mapsto\exp\left(\tfrac{1}{z}\right)</math> has an ''essential singularity'' at <math>0</math> because <math>z^k \exp\left(\tfrac{1}{z}\right)</math> is unbounded as <math>z\to 0</math> for fixed <math>k\in\mathbb N</math>, or because the Laurent series at <math>z_0</math> contains infinitely many nonzero terms in the principal part: ::<math>f(z)=\sum_{n=0}^{\infty}\frac{1}{n!,z^n}</math>. == References == [[w:en:Eberhard Freitag|Eberhard Freitag]], Rolf Busam: ''Complex Analysis 1.'' Springer-Verlag, Berlin, ISBN 3-540-67641-4. [[Category:Complex Analysis]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis//Singularities https://en.wikiversity.org/wiki/Complex%20Analysis//Singularities] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis//Singularities This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Singularitäten Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Singularitäten|Kurs:Funktionentheorie/Singularitäten]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Singularitäten * Date: 12/26/2024 <span type="translate" src="Kurs:Funktionentheorie/Singularitäten" srclang="de" date="12/26/2024" time="12:18" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Singularitäten]] </noinclude> [[Category:Wiki2Reveal]] 3b6kl48huq92ncho9aeuekgz6fk0wv4 2693438 2693421 2024-12-26T23:27:29Z Eshaa2024 2993595 2693438 wikitext text/x-wiki == To the Learning Unit == This learning unit addresses singularities of complex functions. For singularities in real analysis, these are referred to as [[w:en:Singularity (mathematics)|Singularity]]. In complex analysis, singularities hold particular significance for the value of contour integrals. With the residue theorem, we find that only the coefficients of the Laurent series preceding <math>(z-z_0)^{-1}</math> contribute meaningfully to the contour integral. The integration of other summands from the Laurent series results in a contribution of 0 to the contour integral. To prove this residue theorem, singularities must first be classified. == Introduction == '''Isolated singularities''' are studied in the [[w:en:Branch of mathematics|branch of mathematics]] known as [[Complex Analysis|Complex Analysis]]. Isolated singularities are special [[w:en:Isolated point|isolated points]] in the [[w:en:Domain (mathematics)|domain]] of a [[w:en:Holomorphic function|holomorphic function]]. Isolated singularities are classified into [[w:en:Singularity (mathematics)|Singularity]], [[w:en:Pole (complex analysis)|poles]], and '''essential singularities'''. == Definition == Let <math>\Omega \subseteq \mathbb C</math> be an [[w:en:Open set|open subset]] and <math>z_0 \in \Omega</math>. Further, let <math>f\colon \Omega \setminus {z_0} \to \mathbb C</math> be a holomorphic [[w:en:Complex-valued function|complex-valued function]]. Then, <math>z_0</math> is called an ''isolated singularity'' of <math>f</math>. == Classification == *Each isolated singularity belongs to one of the following three classes: *The point <math>z_0</math> is called a ''removable singularity'' if <math>f</math> can be extended to be holomorphic on <math>\Omega</math>. According to the [[w:en:Removable singularity|Removable singularity]], this is, for example, the case if <math>f</math> is bounded in a neighborhood of <math>z_0</math>. *The point <math>z_0</math> is called a ''[[w:en:Pole (complex analysis)|pole]]'' if <math>z_0</math> is not a removable singularity and there exists a natural number <math>k</math> such that <math>(z-z_0)^k \cdot f(z)</math> has a removable singularity at <math>z_0</math>. If <math>k</math> is chosen minimally, <math>f</math> is said to have a pole of order <math>k</math> at <math>z_0</math>. *Otherwise, <math>z_0</math> is called an ''essential singularity'' of <math>f</math>. == Isolated Singularities and the Laurent Series == The type of singularity can also be determined from the [[Laurent Series|Laurent Series]]: : <math>\sum_{n=-\infty}^\infty a_n(z-z_0)^n</math> of <math>f</math> at <math>z_0</math>. === Removable Singularities and the Laurent Series === A singularity is removable if and only if the principal part vanishes, i.e., <math>a_n=0</math> for all negative integers <math>n</math>. : <math>\sum_{n=-\infty}^\infty a_n(z-z_0)^n</math> of <math>f</math> at <math>z_0</math>. === Pole and the Laurent Series === A pole of order <math>k</math> occurs if and only if the principal part terminates after <math>k</math> terms, i.e., <math>a_{-k} \neq 0</math> and <math>a_n=0</math> for all <math>n < -k</math>. === Essential Singularity and the Laurent Series === An essential singularity occurs if infinitely many terms with negative exponents are nonzero. Statements about the properties of holomorphic functions at essential singularities are made by the [[w:en:Picard theorem|Great Picard theorem]] and, as a simpler special case, the [[w:en:Casorati-Weierstrass theorem|Casorati-Weierstrass theorem]]. == Examples == [[File:Essential singularity.png|300px|Plot of the function <math>\exp(1/z)</math>. It has an essential singularity at the origin (center). The hue corresponds to the complex argument of the function value, while the brightness represents its magnitude. Here, the behavior of the essential singularity varies depending on the approach (in contrast to a pole, which would appear uniformly white).]] Plot of the function <math>\exp(1/z)</math>. It has an essential singularity at the origin (center). The hue corresponds to the complex argument of the function value, while the brightness represents its magnitude. Here, the behavior of the essential singularity varies depending on the approach (in contrast to a pole, which would appear uniformly white). === Properties === Let <math>\Omega=\mathbb C</math> and <math>z_0=0.</math> *<math>f\colon \Omega \setminus {0}\to\mathbb C,,z\mapsto \tfrac{\sin(z)}{z}</math> can be continuously extended to <math>\Omega</math> by defining <math>f(0)=1</math>. Thus, <math>f</math> has a ''removable'' singularity at <math>0</math>. *<math>f\colon \Omega\setminus {0}\to\mathbb C,,z\mapsto \tfrac{1}{z}</math> has a pole of order 1 at <math>0</math> because <math>g(z)=z^1\cdot f(z)</math> can be continuously extended to <math>\Omega</math> by defining <math>g(0)=1</math>. *<math>f\colon \Omega\setminus {0}\to\mathbb C,,z\mapsto\exp\left(\tfrac{1}{z}\right)</math> has an ''essential singularity'' at <math>0</math> because <math>z^k \exp\left(\tfrac{1}{z}\right)</math> is unbounded as <math>z\to 0</math> for fixed <math>k\in\mathbb N</math>, or because the Laurent series at <math>z_0</math> contains infinitely many nonzero terms in the principal part: ::<math>f(z)=\sum_{n=0}^{\infty}\frac{1}{n!,z^n}</math>. == References == [[w:en:Eberhard Freitag|Eberhard Freitag]], Rolf Busam: ''Complex Analysis 1.'' Springer-Verlag, Berlin, ISBN 3-540-67641-4. [[Category:Complex Analysis]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis//Singularities https://en.wikiversity.org/wiki/Complex%20Analysis//Singularities] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis//Singularities This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Singularitäten Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Singularitäten|Kurs:Funktionentheorie/Singularitäten]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Singularitäten * Date: 12/26/2024 <span type="translate" src="Kurs:Funktionentheorie/Singularitäten" srclang="de" date="12/26/2024" time="12:18" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Singularitäten]] </noinclude> [[Category:Wiki2Reveal]] f5n92d7pf4iqzo3koncuyvrjb2tn3jq 1 317540 2693422 2024-12-26T23:16:06Z Juandev 2651 /* English Wikiversity */ new section 2693422 wikitext text/x-wiki == English Wikiversity == English Wikiversity "...has become a haven for users banned from other Wikimedia projects". Its kind of a wierd fact. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 23:16, 26 December 2024 (UTC) 2z54mmd0dnslk68ae5zd4erft0dzwxm 0 317541 2693428 2024-12-26T23:22:31Z Tule-hog 2984180 Tule-hog moved page [[Network+/Standards/Theory and Concepts/Introduction]] to [[Network+/Old guides/Ethernet]]: alter parent 2693428 wikitext text/x-wiki #REDIRECT [[Network+/Old guides/Ethernet]] 1z06d00vcbn0x4gpbl3tx6flqlz67go 2693431 2693428 2024-12-26T23:22:56Z Tule-hog 2984180 nominate speedy 2693431 wikitext text/x-wiki {{speedy|C10}} #REDIRECT [[Network+/Old guides/Ethernet]] 16hbb2cnnnenk3p016vprgd7fvbz2dm 1 317542 2693430 2024-12-26T23:22:31Z Tule-hog 2984180 Tule-hog moved page [[Talk:Network+/Standards/Theory and Concepts/Introduction]] to [[Talk:Network+/Old guides/Ethernet]]: alter parent 2693430 wikitext text/x-wiki #REDIRECT [[Talk:Network+/Old guides/Ethernet]] toex16qlaa1ta1fa0utkv95b8f7vo6x 15 317543 2693437 2024-12-26T23:27:21Z Tule-hog 2984180 /* Merge proposal */ new section 2693437 wikitext text/x-wiki == Merge proposal == I am [[wv:bold|boldly]] removing the {{tlx|merge}} tag to merge [[:Category:Computer Networks]], which is specific to the learning resource [[Computer Networks]]. This category should be used as a top-level view of resources involving computer networks. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:27, 26 December 2024 (UTC) 8k5yce1o0esraf173hx870quyqmzu7k 2693441 2693437 2024-12-26T23:29:00Z Tule-hog 2984180 /* Merge proposal */ Reply 2693441 wikitext text/x-wiki == Merge proposal == I am [[wv:bold|boldly]] removing the {{tlx|merge}} tag to merge [[:Category:Computer Networks]], which is specific to the learning resource [[Computer Networks]]. This category should be used as a top-level view of resources involving computer networks. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:27, 26 December 2024 (UTC) :With that said, it seems this category is redundant with [[:Category:Networking]] - so I may propose a merge to that. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:29, 26 December 2024 (UTC) do48jhh703z51425makgf8xdm9catb7 2 317544 2693449 2024-12-26T23:37:38Z Atcovi 276019 New resource with "* Disability affects 15% of the world's population. == Raw Textbook Page == Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disa..." 2693449 wikitext text/x-wiki * Disability affects 15% of the world's population. == Raw Textbook Page == Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? read more What are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. a1pwyviyu5uzr1givwf6vlfcxy1hp0z 2693450 2693449 2024-12-26T23:37:53Z Atcovi 276019 /* Raw Textbook Page */ 2693450 wikitext text/x-wiki * Disability affects 15% of the world's population. == Raw Textbook Page == Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? rWhat are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. njxtnv632feip4vuzpsa0uuin6odw1k 2693451 2693450 2024-12-26T23:38:02Z Atcovi 276019 /* Raw Textbook Page */ 2693451 wikitext text/x-wiki * Disability affects 15% of the world's population. == Raw Textbook Page == Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? What are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. 2tojoj6vjgdcz257xvuoda5soxhficc 2693452 2693451 2024-12-26T23:39:47Z Atcovi 276019 2693452 wikitext text/x-wiki * Disability affects 15% of the world's population. '''Disability Models''' This passage explores disability through three main models: moral, medical, and social. The '''moral model''' associates disability with divine intervention or punishment, emphasizing cultural and religious beliefs, and remains prevalent globally, though less so in Western cultures. The '''medical model''', dominant in the West, views disability as a biological dysfunction to be treated or cured, focusing responsibility on medical professionals and individuals. In contrast, the '''social model''' attributes disability to societal structures and attitudes, urging systemic changes to dismantle ableism. The '''biopsychosocial model''', adopted by the World Health Organization (WHO) in the International Classification of Functioning (ICF), integrates aspects of all three models, highlighting how physical, personal, and societal factors interact to shape disability experiences. Disability dimensions, including visibility, functionality, and progression, influence individual adaptation and societal responses. Chronic illnesses like cardiovascular disease and diabetes are prevalent today due to increased life expectancy and medical advancements. The discussion also underscores the impact of cultural norms, health equity, and chronic conditions on life expectancy and societal roles in health and disability. == Raw Textbook Page == Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? What are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. f747v8bfyz8xs547hcgqeiuabkqqdh4 2693454 2693452 2024-12-26T23:42:52Z Atcovi 276019 2693454 wikitext text/x-wiki == 11.1 - What is Disability and Chronic Illness? == * Disability affects 15% of the world's population. '''Disability Models''' This passage explores disability through three main models: moral, medical, and social. The '''moral model''' associates disability with divine intervention or punishment, emphasizing cultural and religious beliefs, and remains prevalent globally, though less so in Western cultures. The '''medical model''', dominant in the West, views disability as a biological dysfunction to be treated or cured, focusing responsibility on medical professionals and individuals. In contrast, the '''social model''' attributes disability to societal structures and attitudes, urging systemic changes to dismantle ableism and emphasizing a possible "cure" and "return to normalcy" for disabled people. The '''biopsychosocial model''', adopted by the World Health Organization (WHO) in the International Classification of Functioning (ICF), integrates aspects of all three models, highlighting how physical, personal, and societal factors interact to shape disability experiences. Disability dimensions, including visibility, functionality, and progression, influence individual adaptation and societal responses. Chronic illnesses like cardiovascular disease and diabetes are prevalent today due to increased life expectancy and medical advancements. The discussion also underscores the impact of cultural norms, health equity, and chronic conditions on life expectancy and societal roles in health and disability. === Raw Textbook Page === Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? What are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. 0c4jkyks7z0z0lvlszjaj2east5sthk 2693457 2693454 2024-12-26T23:44:45Z Atcovi 276019 2693457 wikitext text/x-wiki == 11.1 - What is Disability and Chronic Illness? == * Disability affects 15% of the world's population. '''Disability Models''' This passage explores disability through three main models: moral, medical, and social. The '''moral model''' associates disability with divine intervention or punishment, emphasizing cultural and religious beliefs, and remains prevalent globally, though less so in Western cultures. The '''medical model''', dominant in the West, views disability as a biological dysfunction to be treated or cured, focusing responsibility on medical professionals and individuals. In contrast, the '''social model''' attributes disability to societal structures and attitudes, urging systemic changes to dismantle ableism and emphasizing a possible "cure" and "return to normalcy" for disabled people. The '''biopsychosocial model''', adopted by the World Health Organization (WHO) in the International Classification of Functioning (ICF), integrates aspects of all three models, highlighting how physical, personal, and societal factors interact to shape disability experiences. Disability dimensions, including visibility, functionality, and progression, influence individual adaptation and societal responses. Chronic illnesses like cardiovascular disease and diabetes are prevalent today due to increased life expectancy and medical advancements. The discussion also underscores the impact of cultural norms, health equity, and chronic conditions on life expectancy and societal roles in health and disability. === Raw Textbook Page === Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? What are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. == 11.2 - Coping With Disability and Chronic Illness == === Raw Textbook Page === Improving one’s diet, refraining from smoking, and consuming minimal alcoholic beverages (see Chapter 7) may help prevent chronic illnesses, but do not guarantee avoiding these illnesses. Goals of Treatment Before we discuss how one can cope with having a chronic illness, it is important to consider some goals for treatment. Science has made many advances in the treatment of cancer and HIV infection, and some research suggests that illnesses such as CHD and diabetes can be reversed (e.g., Campbell & Campbell, 2016; Ornish et al., 1998); however, we still cannot cure these illnesses. Therefore, helping people cope with having these illnesses becomes very important. Adaptation to disability can be defined as a dynamic process of affective, cognitive, and behavioral changes that gradually approach an optimal state of well-being (Livneh & Antonak, 1997). Five major forms of adaptation are the successful performance of daily tasks, the minimizing psychological disorders, low levels of negative affect and high levels of positive affect, good functional status, and the experience of satisfaction in different areas of life (Stanton et al., 2001). Of all of these, the most common psychological outcome studied is the quality of life (Morgan & McGee, 2016). Quality of Life The most commonly used measure of how someone is coping with a chronic illness is a measure of their quality of life (QOL). Sometimes called health-related quality of life (HRQOL) or discussed as well-being, the past 40 years has seen an upsurge in research on QOL (Morgan & McGee, 2016). QOL features prominently in the study of how patients cope with diseases and is important for planning further treatment (Brodsky et al., 2017). QOL was originally a measure made by the physician, purely by whether the disease was present or absent. If the disease presence was strong, it was assumed that QOL would be low. It is now clear that patients are the best judges of their own QOL. Asking patients how much pain they are experiencing and how they feel (e.g., assessing depression and anxiety) is a valuable way to determine how well they are coping (Morrow et al., 2012). Assess your own QOL with one of the most common measures of QOL (Table 11.2). *See Procedures Manual, pages 13–15. Quality of life includes several components. Similar to measures of adaptation, QOL includes a measure of physical status and functioning, psychological status, social functioning, and the presence of the disease- or treatment-related symptoms. A wide array of measures assesses QOL. Figure 11.4 lists some of the major assessment tools, ranging from the generic to the specific, and separated by disease- or patient-specific types. In the age of the internet, can you Google for some? Yes indeed. The good news is you can find a number of credible collections of measures with accompanying psychometric information (Morgan & McGee, 2016). These include the Patient-Reported Outcomes Measurement System (www.promishealth.org), Patient Reported Outcomes and Quality of Life Instruments database (PROQOLID; Emery et al., 2005), the On-Line Guide to Quality-of-Life Assessment database (OLGA; http://www.olga-qol.com), and Optum (http://www.optum.com). A table depicting major measures of quality of life with column heads type, focus, and examples of assessment tools. Source: Reprinted with permission from Assessment in Health Psychology by Y. Benyamini, M. Johnston, & E. C. Karademas, ISBN 978-0-88937-452-2 ©2016 Hogrefe. www.hogrefe.com. Let’s take a look at the different biological, psychological, and sociocultural factors that can influence QOL and adaptation. Biopsychosocial Components of Adaptation Most of us will experience a disability or chronic illness at some point in our lives. However, it is clear that changing health behaviors can greatly reduce the chance of contracting some chronic illnesses (LaCaille & Hooker, 2019; Mermelstein & Brikmanis, 2019). Furthermore, psychological strategies can help one cope with chronic illness. For example, in a longitudinal study of patients with inflammatory bowel disease and arthritis, patients who displayed gratitude were less depressed later in the study (Sirois & Wood, 2017). In fact, feeling grateful was a significant predictor of lowered depression even after controlling for other psychological variables such as illness cognitions discussed in Chapter 10. Similarly, two other psychological variables, optimism and hope, are strong aids to helping patients cope with chronic diseases (Schiavon et al., 2017). Adaptation to chronic illnesses has many different components. Patients need to cope with not only their own affect, behaviors, and cognitions concerning the illness but also with revising their lifestyles to accommodate the treatment and coping with how others in their social networks respond to them because of their illness (Day, 2019; Hoyt & Stanton, 2019). They may experience many different feelings including anxiety, depression, and frustration, and may not be able to perform activities they used to, such as going to work or shopping for groceries. Daily tasks, changing symptoms, and fluctuating emotions can be overwhelming (Emery, 2019). There are numerous challenges to the process of integration; successful self-management with psychosocial, vocational, and existential support is critical. Next, I will discuss some of the different components of adaptation using the major approaches in health psychology. Biological Issues Biologically, different chronic illnesses will have different courses. For example, coronary heart disease (CHD) and cancer, the two leading causes of death for Americans, inflict significant changes in the body. Cancer causes cells to grow uncontrollably, harming surrounding tissue and limiting normal function. In CHD, the blood vessels around the heart are clogged with plaque and fat, changing blood flow and possibly leading to a heart attack. Other chronic illnesses such as diabetes and asthma similarly have physiological correlates, such as changes in insulin sensitivity and the blocking of breathing channels (Kalyva et al., 2016). The slow physiological changes limit functioning in many areas and are often accompanied by pain (Hoyt & Stanton, 2012). Consequently, physical rehabilitation is a big component of any treatment of chronic illnesses. The loss of function and increase in pain also have major consequences for how the patient views the world, and psychological issues need to be considered as well. Psychological Issues There has been growing interest in the role of psychological factors in adaptation to chronic illnesses and disability (Samson & Siam, 2008). In a review of both theoretical and empirical literature on adjusting to chronic illnesses, Stanton et al. (2001) identified two key multidimensional psychological aspects. First, the individual has to go through adaptations, which include cognitive aspects such as intrusive thoughts and changing views of the self, emotional aspects such as depression and anxiety, and behavioral and physical aspects such as dealing with pain or not being able to perform daily activities. Second, the person must make interpersonal adaptations, negotiating personal relationships with friends and family as well as professional relationships with health-care providers. Positive adaptation includes the mastery of illness-related tasks, the minimizing psychological disorder and negative feelings, perceptions of high quality of life, and the maintenance of adequate functional status and social roles (Hoyt & Stanton, 2019). Perhaps one of the most effective psychological resources that a person with a chronic illness or disability has is their mental approach to the situation and appraisals (see Chapter 5). Patients’ primary and secondary appraisals of the illness can correspondingly influence how they fare. If the illness is seen as a challenge (primary appraisal) and they believe they have a lot of social support to cope with it (secondary appraisal), they will probably have a higher QOL (Gatchel & Oordt, 2003b). For example, in a study of colorectal cancer-specific concerns in a population-based sample of colorectal cancer survivors, patients’ threat appraisals significantly predicted their quality of life up to 2 years after treatment (Steginga et al., 2009). A number of health psychologists have modified cognitive appraisal theory from its original context (i.e., stress) and have adapted it to help explain coping with chronic illnesses such as arthritis, breast cancer, prostate cancer, and AIDS (Merz et al., 2011; Schwartz & Rapkin, 2012). In fact, new work teases apart appraisals from personality factors. The Quality of Life Appraisal Profile–Version 2 (QOLAPv2) helps assess individual differences and is useful in explaining why people experiencing very different health states may report the same QOL (Rapkin et al., 2017). First developed with 4,173 respondents, the QOLAPv2 is useful across populations and provides better predictions of QOL than measures of personality alone. A caring healthcare nurse conducting physical therapy exercises with a senior adult patient at home. The nurse is assisting the patient who is holding dumbbells with her hands raised. Physical Therapy. Physical therapy is an important part of coping with the biological aspects of chronic illnesses, but mobility can influence the psychosocial aspects as well. Another common psychological reaction to a positive test result or even experiencing symptoms of a chronic illness is anxiety. Anxiety interferes with healthy functioning, causing a person to cope poorly and to delay the recognition and reporting of symptoms. Anxiety is often high when the patient is waiting for test results, receiving a diagnosis, and awaiting invasive medical procedures. Not knowing about the course of the illness or not having enough information about what the illness entails is especially anxiety provoking. Such lack of information-induced anxiety is more pronounced in populations of lower SES and in some ethnic groups. The most common negative reaction to a chronic illness is depression (Giardini et al., 2017). Depression can be either biological or psychological in nature and often goes undiagnosed because its symptoms are shadowed by the symptoms of the chronic illness. Unlike anxiety, depression tends to be a long-term reaction and increases as pain increases. When patients get depressed, they are less motivated to cope actively with the illness, and tend to interpret any bodily change negatively. Interestingly, the most important predictor of one’s mental health after acquiring a disability or chronic illness is their premorbid mental health. The form of psychological reaction varies also depending on the illness and varies considerably across individuals with the same illness. Personality factors, the amount of social support one receives or perceives to have, and cultural beliefs surrounding the illness can all influence coping with the illness and can alleviate depression and anxiety. Chapter 6 included details about the ways different personalities influence coping. The same relationships that link stress and coping link chronic illnesses and coping. The Big Five personality variables (conscientiousness, agreeableness, neuroticism, openness to experience, and extroversion; see Chapter 6) have been linked to coping in general (Smith, 2019) and coping with chronic illnesses in particular (Sirois, 2015). Similarly, being high in positive affect is also a good thing for those with chronic illnesses. Positive affect was significantly associated with having a lower risk of dying from any cause (i.e., all-cause mortality) in people with diabetes (Moskowitz et al., 2008). In a study of the role of religious involvement, spirituality, and physical or emotional functioning in a sample of African American men and women with cancer, positive affect was a key factor in predicting better adaptation (Holt et al., 2011). Optimism is another powerful personality characteristic in coping with chronic illnesses (Giardini et al., 2017). Carver et al. (1993) first demonstrated convincingly the role of optimism in women coping with breast cancer. When measured before surgery, the optimistic women were those using more active coping and facing the disease, and those with less distress. This pattern held for three further assessments at 3, 6, and 12 months after surgery. Optimism is also helpful in coping with diabetes mellitus, rheumatoid arthritis, and multiple sclerosis (Fournier et al., 2002a), breast cancer (Sohl et al., 2012), coronary bypass surgery (Tindle et al., 2012), and HIV infection (Peterson et al., 2012). Building optimism can go a long way. For example, falling, common in older chronically ill adults, predicts poorer physical health and greater negative emotions among the group (Ruthig et al., 2007). Falling also causes drops in optimism, which mediates the effects of falling on health and well-being. Recovery from falling can be enhanced by bolstering optimism (Ruthig et al., 2007). In general, different personality characteristics can greatly help coping (Smith, 2019). Another important component of psychological coping is related to how patients compare themselves with others with the disease and how much meaning they derive from the illness. For example, studies on upward and downward social comparison show that people can sometimes compare themselves with those better off than they are (“Boy, my coworker has the same problem, and he is doing so much better than I am.”) or worse off than themselves (“Oh, at least I am doing better than my neighbor who has the same illness.”). Women who cope better with breast cancer make comparisons with people who are faring less well than they are to enhance their own self-esteem (Wood et al., 1985). Chinese women facing breast cancer were also found to make the best of it. The essences of Chinese women’s experiences were that they faced the reality of the cancer diagnosis, took an active part in the cancer treatment, sustained an optimistic spirit, maintained physical activity, reflected, and then moved on (Fu et al., 2008). Finding meaning in your illness can often be beneficial, leading to lower mortality and morbidity (Hooker et al., 2018), but in some cases it can be detrimental to well-being as well. Originally, research documented that finding meaning in your experience can lead to positive well-being and better adaptation to the disease (Taylor, 1983). There are some important qualifications to this early finding. Tomich and Helgeson (2004) examined the consequences of finding meaning (they called it benefit finding) on QOL in 364 women diagnosed with stage I, II, or III breast cancer. Benefit finding and QOL were measured 4 months post diagnosis (Tl), 3 months after Tl (T2), and 6 months after T2 (T3). Women with lower socioeconomic status, minority women, and those with more severe levels of the disease perceived more benefits at baseline. Benefit finding was associated with more negative affect at baseline and also interacted with the stage of disease, such that negative relationships to QOL across time were limited to those with more severe disease. Findings suggest that there are qualifiers as to whether finding something good in the bad is, in itself, good or bad (Yanez et al., 2011). We discuss this further in Chapter 13. chklml8cp3eei3p1hym9khp7epi1fp0 2693460 2693457 2024-12-26T23:49:25Z Atcovi 276019 /* 11.2 - Coping With Disability and Chronic Illness */ 2693460 wikitext text/x-wiki == 11.1 - What is Disability and Chronic Illness? == * Disability affects 15% of the world's population. '''Disability Models''' This passage explores disability through three main models: moral, medical, and social. The '''moral model''' associates disability with divine intervention or punishment, emphasizing cultural and religious beliefs, and remains prevalent globally, though less so in Western cultures. The '''medical model''', dominant in the West, views disability as a biological dysfunction to be treated or cured, focusing responsibility on medical professionals and individuals. In contrast, the '''social model''' attributes disability to societal structures and attitudes, urging systemic changes to dismantle ableism and emphasizing a possible "cure" and "return to normalcy" for disabled people. The '''biopsychosocial model''', adopted by the World Health Organization (WHO) in the International Classification of Functioning (ICF), integrates aspects of all three models, highlighting how physical, personal, and societal factors interact to shape disability experiences. Disability dimensions, including visibility, functionality, and progression, influence individual adaptation and societal responses. Chronic illnesses like cardiovascular disease and diabetes are prevalent today due to increased life expectancy and medical advancements. The discussion also underscores the impact of cultural norms, health equity, and chronic conditions on life expectancy and societal roles in health and disability. === Raw Textbook Page === Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? What are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. == 11.2 - Coping With Disability and Chronic Illness == '''Goals of Treatment''' * Adaptation/coping is key. Can be defined as the affective, mental, and behavior chances that allow a disabled person to embrace their life despite their disability. The five major areas are success in performing daily tasks, reducing psych. disorders, reduce negative affect and improve positive affect, maintain a satisfactory functional status, and experience happiness in other areas of life. '''Quality of Life''' * '''QOL''' measures how well someone copes with chronic illness, considering physical, psychological, and social factors. * Initially assessed by physicians, QOL is now better evaluated by patients themselves based on their experiences with pain, emotional well-being, and functional status. * Tools like PROMIS, PROQOLID, OLGA, and Optum provide assessments for QOL. '''Biopsychosocial components of adaptation:''' # '''Biological Issues''': #* Chronic illnesses like cancer, coronary heart disease (CHD), and diabetes impact physical functioning and often involve pain, necessitating physical rehabilitation. #* Such conditions also influence psychological outlooks. # '''Psychological Issues''': #* Adaptation involves cognitive, emotional, and interpersonal adjustments, with key factors including: #** '''Appraisals''': Seeing illness as a challenge and perceiving social support improves QOL. #** '''Personality''': Traits like optimism and positive affect aid in coping, while depression and anxiety hinder it. #** '''Comparison''': Social comparisons (upward or downward) can impact self-esteem. #** '''Meaning''': Finding meaning in illness can enhance or detract from well-being, depending on context. '''Research Highlights''' * '''Optimism''': Strong predictor of better adaptation, linked to active coping and less distress across illnesses. * '''Gratitude''': Reduces depression and promotes better outcomes. * '''Social Support''': Critical for managing emotional and practical challenges. * '''Cultural Variations''': Different coping mechanisms observed across cultures (e.g., Chinese women with breast cancer). '''Challenges and Strategies:''' * Psychological reactions vary by illness and individual differences (e.g., premorbid mental health, cultural beliefs). * Interventions that foster optimism, gratitude, and supportive relationships are effective in improving QOL. === Raw Textbook Page === Improving one’s diet, refraining from smoking, and consuming minimal alcoholic beverages (see Chapter 7) may help prevent chronic illnesses, but do not guarantee avoiding these illnesses. Goals of Treatment Before we discuss how one can cope with having a chronic illness, it is important to consider some goals for treatment. Science has made many advances in the treatment of cancer and HIV infection, and some research suggests that illnesses such as CHD and diabetes can be reversed (e.g., Campbell & Campbell, 2016; Ornish et al., 1998); however, we still cannot cure these illnesses. Therefore, helping people cope with having these illnesses becomes very important. Adaptation to disability can be defined as a dynamic process of affective, cognitive, and behavioral changes that gradually approach an optimal state of well-being (Livneh & Antonak, 1997). Five major forms of adaptation are the successful performance of daily tasks, the minimizing psychological disorders, low levels of negative affect and high levels of positive affect, good functional status, and the experience of satisfaction in different areas of life (Stanton et al., 2001). Of all of these, the most common psychological outcome studied is the quality of life (Morgan & McGee, 2016). Quality of Life The most commonly used measure of how someone is coping with a chronic illness is a measure of their quality of life (QOL). Sometimes called health-related quality of life (HRQOL) or discussed as well-being, the past 40 years has seen an upsurge in research on QOL (Morgan & McGee, 2016). QOL features prominently in the study of how patients cope with diseases and is important for planning further treatment (Brodsky et al., 2017). QOL was originally a measure made by the physician, purely by whether the disease was present or absent. If the disease presence was strong, it was assumed that QOL would be low. It is now clear that patients are the best judges of their own QOL. Asking patients how much pain they are experiencing and how they feel (e.g., assessing depression and anxiety) is a valuable way to determine how well they are coping (Morrow et al., 2012). Assess your own QOL with one of the most common measures of QOL (Table 11.2). *See Procedures Manual, pages 13–15. Quality of life includes several components. Similar to measures of adaptation, QOL includes a measure of physical status and functioning, psychological status, social functioning, and the presence of the disease- or treatment-related symptoms. A wide array of measures assesses QOL. Figure 11.4 lists some of the major assessment tools, ranging from the generic to the specific, and separated by disease- or patient-specific types. In the age of the internet, can you Google for some? Yes indeed. The good news is you can find a number of credible collections of measures with accompanying psychometric information (Morgan & McGee, 2016). These include the Patient-Reported Outcomes Measurement System (www.promishealth.org), Patient Reported Outcomes and Quality of Life Instruments database (PROQOLID; Emery et al., 2005), the On-Line Guide to Quality-of-Life Assessment database (OLGA; http://www.olga-qol.com), and Optum (http://www.optum.com). A table depicting major measures of quality of life with column heads type, focus, and examples of assessment tools. Source: Reprinted with permission from Assessment in Health Psychology by Y. Benyamini, M. Johnston, & E. C. Karademas, ISBN 978-0-88937-452-2 ©2016 Hogrefe. www.hogrefe.com. Let’s take a look at the different biological, psychological, and sociocultural factors that can influence QOL and adaptation. Biopsychosocial Components of Adaptation Most of us will experience a disability or chronic illness at some point in our lives. However, it is clear that changing health behaviors can greatly reduce the chance of contracting some chronic illnesses (LaCaille & Hooker, 2019; Mermelstein & Brikmanis, 2019). Furthermore, psychological strategies can help one cope with chronic illness. For example, in a longitudinal study of patients with inflammatory bowel disease and arthritis, patients who displayed gratitude were less depressed later in the study (Sirois & Wood, 2017). In fact, feeling grateful was a significant predictor of lowered depression even after controlling for other psychological variables such as illness cognitions discussed in Chapter 10. Similarly, two other psychological variables, optimism and hope, are strong aids to helping patients cope with chronic diseases (Schiavon et al., 2017). Adaptation to chronic illnesses has many different components. Patients need to cope with not only their own affect, behaviors, and cognitions concerning the illness but also with revising their lifestyles to accommodate the treatment and coping with how others in their social networks respond to them because of their illness (Day, 2019; Hoyt & Stanton, 2019). They may experience many different feelings including anxiety, depression, and frustration, and may not be able to perform activities they used to, such as going to work or shopping for groceries. Daily tasks, changing symptoms, and fluctuating emotions can be overwhelming (Emery, 2019). There are numerous challenges to the process of integration; successful self-management with psychosocial, vocational, and existential support is critical. Next, I will discuss some of the different components of adaptation using the major approaches in health psychology. Biological Issues Biologically, different chronic illnesses will have different courses. For example, coronary heart disease (CHD) and cancer, the two leading causes of death for Americans, inflict significant changes in the body. Cancer causes cells to grow uncontrollably, harming surrounding tissue and limiting normal function. In CHD, the blood vessels around the heart are clogged with plaque and fat, changing blood flow and possibly leading to a heart attack. Other chronic illnesses such as diabetes and asthma similarly have physiological correlates, such as changes in insulin sensitivity and the blocking of breathing channels (Kalyva et al., 2016). The slow physiological changes limit functioning in many areas and are often accompanied by pain (Hoyt & Stanton, 2012). Consequently, physical rehabilitation is a big component of any treatment of chronic illnesses. The loss of function and increase in pain also have major consequences for how the patient views the world, and psychological issues need to be considered as well. Psychological Issues There has been growing interest in the role of psychological factors in adaptation to chronic illnesses and disability (Samson & Siam, 2008). In a review of both theoretical and empirical literature on adjusting to chronic illnesses, Stanton et al. (2001) identified two key multidimensional psychological aspects. First, the individual has to go through adaptations, which include cognitive aspects such as intrusive thoughts and changing views of the self, emotional aspects such as depression and anxiety, and behavioral and physical aspects such as dealing with pain or not being able to perform daily activities. Second, the person must make interpersonal adaptations, negotiating personal relationships with friends and family as well as professional relationships with health-care providers. Positive adaptation includes the mastery of illness-related tasks, the minimizing psychological disorder and negative feelings, perceptions of high quality of life, and the maintenance of adequate functional status and social roles (Hoyt & Stanton, 2019). Perhaps one of the most effective psychological resources that a person with a chronic illness or disability has is their mental approach to the situation and appraisals (see Chapter 5). Patients’ primary and secondary appraisals of the illness can correspondingly influence how they fare. If the illness is seen as a challenge (primary appraisal) and they believe they have a lot of social support to cope with it (secondary appraisal), they will probably have a higher QOL (Gatchel & Oordt, 2003b). For example, in a study of colorectal cancer-specific concerns in a population-based sample of colorectal cancer survivors, patients’ threat appraisals significantly predicted their quality of life up to 2 years after treatment (Steginga et al., 2009). A number of health psychologists have modified cognitive appraisal theory from its original context (i.e., stress) and have adapted it to help explain coping with chronic illnesses such as arthritis, breast cancer, prostate cancer, and AIDS (Merz et al., 2011; Schwartz & Rapkin, 2012). In fact, new work teases apart appraisals from personality factors. The Quality of Life Appraisal Profile–Version 2 (QOLAPv2) helps assess individual differences and is useful in explaining why people experiencing very different health states may report the same QOL (Rapkin et al., 2017). First developed with 4,173 respondents, the QOLAPv2 is useful across populations and provides better predictions of QOL than measures of personality alone. A caring healthcare nurse conducting physical therapy exercises with a senior adult patient at home. The nurse is assisting the patient who is holding dumbbells with her hands raised. Physical Therapy. Physical therapy is an important part of coping with the biological aspects of chronic illnesses, but mobility can influence the psychosocial aspects as well. Another common psychological reaction to a positive test result or even experiencing symptoms of a chronic illness is anxiety. Anxiety interferes with healthy functioning, causing a person to cope poorly and to delay the recognition and reporting of symptoms. Anxiety is often high when the patient is waiting for test results, receiving a diagnosis, and awaiting invasive medical procedures. Not knowing about the course of the illness or not having enough information about what the illness entails is especially anxiety provoking. Such lack of information-induced anxiety is more pronounced in populations of lower SES and in some ethnic groups. The most common negative reaction to a chronic illness is depression (Giardini et al., 2017). Depression can be either biological or psychological in nature and often goes undiagnosed because its symptoms are shadowed by the symptoms of the chronic illness. Unlike anxiety, depression tends to be a long-term reaction and increases as pain increases. When patients get depressed, they are less motivated to cope actively with the illness, and tend to interpret any bodily change negatively. Interestingly, the most important predictor of one’s mental health after acquiring a disability or chronic illness is their premorbid mental health. The form of psychological reaction varies also depending on the illness and varies considerably across individuals with the same illness. Personality factors, the amount of social support one receives or perceives to have, and cultural beliefs surrounding the illness can all influence coping with the illness and can alleviate depression and anxiety. Chapter 6 included details about the ways different personalities influence coping. The same relationships that link stress and coping link chronic illnesses and coping. The Big Five personality variables (conscientiousness, agreeableness, neuroticism, openness to experience, and extroversion; see Chapter 6) have been linked to coping in general (Smith, 2019) and coping with chronic illnesses in particular (Sirois, 2015). Similarly, being high in positive affect is also a good thing for those with chronic illnesses. Positive affect was significantly associated with having a lower risk of dying from any cause (i.e., all-cause mortality) in people with diabetes (Moskowitz et al., 2008). In a study of the role of religious involvement, spirituality, and physical or emotional functioning in a sample of African American men and women with cancer, positive affect was a key factor in predicting better adaptation (Holt et al., 2011). Optimism is another powerful personality characteristic in coping with chronic illnesses (Giardini et al., 2017). Carver et al. (1993) first demonstrated convincingly the role of optimism in women coping with breast cancer. When measured before surgery, the optimistic women were those using more active coping and facing the disease, and those with less distress. This pattern held for three further assessments at 3, 6, and 12 months after surgery. Optimism is also helpful in coping with diabetes mellitus, rheumatoid arthritis, and multiple sclerosis (Fournier et al., 2002a), breast cancer (Sohl et al., 2012), coronary bypass surgery (Tindle et al., 2012), and HIV infection (Peterson et al., 2012). Building optimism can go a long way. For example, falling, common in older chronically ill adults, predicts poorer physical health and greater negative emotions among the group (Ruthig et al., 2007). Falling also causes drops in optimism, which mediates the effects of falling on health and well-being. Recovery from falling can be enhanced by bolstering optimism (Ruthig et al., 2007). In general, different personality characteristics can greatly help coping (Smith, 2019). Another important component of psychological coping is related to how patients compare themselves with others with the disease and how much meaning they derive from the illness. For example, studies on upward and downward social comparison show that people can sometimes compare themselves with those better off than they are (“Boy, my coworker has the same problem, and he is doing so much better than I am.”) or worse off than themselves (“Oh, at least I am doing better than my neighbor who has the same illness.”). Women who cope better with breast cancer make comparisons with people who are faring less well than they are to enhance their own self-esteem (Wood et al., 1985). Chinese women facing breast cancer were also found to make the best of it. The essences of Chinese women’s experiences were that they faced the reality of the cancer diagnosis, took an active part in the cancer treatment, sustained an optimistic spirit, maintained physical activity, reflected, and then moved on (Fu et al., 2008). Finding meaning in your illness can often be beneficial, leading to lower mortality and morbidity (Hooker et al., 2018), but in some cases it can be detrimental to well-being as well. Originally, research documented that finding meaning in your experience can lead to positive well-being and better adaptation to the disease (Taylor, 1983). There are some important qualifications to this early finding. Tomich and Helgeson (2004) examined the consequences of finding meaning (they called it benefit finding) on QOL in 364 women diagnosed with stage I, II, or III breast cancer. Benefit finding and QOL were measured 4 months post diagnosis (Tl), 3 months after Tl (T2), and 6 months after T2 (T3). Women with lower socioeconomic status, minority women, and those with more severe levels of the disease perceived more benefits at baseline. Benefit finding was associated with more negative affect at baseline and also interacted with the stage of disease, such that negative relationships to QOL across time were limited to those with more severe disease. Findings suggest that there are qualifiers as to whether finding something good in the bad is, in itself, good or bad (Yanez et al., 2011). We discuss this further in Chapter 13. s2smpd3cb8nm67vayxvhpl0soxzuay6 2693462 2693460 2024-12-26T23:53:33Z Atcovi 276019 2693462 wikitext text/x-wiki == 11.1 - What is Disability and Chronic Illness? == * Disability affects 15% of the world's population. '''Disability Models''' This passage explores disability through three main models: moral, medical, and social. The '''moral model''' associates disability with divine intervention or punishment, emphasizing cultural and religious beliefs, and remains prevalent globally, though less so in Western cultures. The '''medical model''', dominant in the West, views disability as a biological dysfunction to be treated or cured, focusing responsibility on medical professionals and individuals. In contrast, the '''social model''' attributes disability to societal structures and attitudes, urging systemic changes to dismantle ableism and emphasizing a possible "cure" and "return to normalcy" for disabled people. The '''biopsychosocial model''', adopted by the World Health Organization (WHO) in the International Classification of Functioning (ICF), integrates aspects of all three models, highlighting how physical, personal, and societal factors interact to shape disability experiences. Disability dimensions, including visibility, functionality, and progression, influence individual adaptation and societal responses. Chronic illnesses like cardiovascular disease and diabetes are prevalent today due to increased life expectancy and medical advancements. The discussion also underscores the impact of cultural norms, health equity, and chronic conditions on life expectancy and societal roles in health and disability. === Raw Textbook Page === Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? What are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. == 11.2 - Coping With Disability and Chronic Illness == '''Goals of Treatment''' * Adaptation/coping is key. Can be defined as the affective, mental, and behavior chances that allow a disabled person to embrace their life despite their disability. The five major areas are success in performing daily tasks, reducing psych. disorders, reduce negative affect and improve positive affect, maintain a satisfactory functional status, and experience happiness in other areas of life. '''Quality of Life''' * '''QOL''' measures how well someone copes with chronic illness, considering physical, psychological, and social factors. * Initially assessed by physicians, QOL is now better evaluated by patients themselves based on their experiences with pain, emotional well-being, and functional status. * Tools like PROMIS, PROQOLID, OLGA, and Optum provide assessments for QOL. '''Biopsychosocial components of adaptation:''' # '''Biological Issues''': #* Chronic illnesses like cancer, coronary heart disease (CHD), and diabetes impact physical functioning and often involve pain, necessitating physical rehabilitation. #* Such conditions also influence psychological outlooks. # '''Psychological Issues''': #* Adaptation involves cognitive, emotional, and interpersonal adjustments, with key factors including: #** '''Appraisals''': Seeing illness as a challenge and perceiving social support improves QOL. #** '''Personality''': Traits like optimism and positive affect aid in coping, while depression and anxiety hinder it. #** '''Comparison''': Social comparisons (upward or downward) can impact self-esteem. #** '''Meaning''': Finding meaning in illness can enhance or detract from well-being, depending on context. '''Research Highlights''' * '''Optimism''': Strong predictor of better adaptation, linked to active coping and less distress across illnesses. * '''Gratitude''': Reduces depression and promotes better outcomes. * '''Social Support''': Critical for managing emotional and practical challenges. * '''Cultural Variations''': Different coping mechanisms observed across cultures (e.g., Chinese women with breast cancer). '''Challenges and Strategies:''' * Psychological reactions vary by illness and individual differences (e.g., premorbid mental health, cultural beliefs). * Interventions that foster optimism, gratitude, and supportive relationships are effective in improving QOL. === Raw Textbook Page === Improving one’s diet, refraining from smoking, and consuming minimal alcoholic beverages (see Chapter 7) may help prevent chronic illnesses, but do not guarantee avoiding these illnesses. Goals of Treatment Before we discuss how one can cope with having a chronic illness, it is important to consider some goals for treatment. Science has made many advances in the treatment of cancer and HIV infection, and some research suggests that illnesses such as CHD and diabetes can be reversed (e.g., Campbell & Campbell, 2016; Ornish et al., 1998); however, we still cannot cure these illnesses. Therefore, helping people cope with having these illnesses becomes very important. Adaptation to disability can be defined as a dynamic process of affective, cognitive, and behavioral changes that gradually approach an optimal state of well-being (Livneh & Antonak, 1997). Five major forms of adaptation are the successful performance of daily tasks, the minimizing psychological disorders, low levels of negative affect and high levels of positive affect, good functional status, and the experience of satisfaction in different areas of life (Stanton et al., 2001). Of all of these, the most common psychological outcome studied is the quality of life (Morgan & McGee, 2016). Quality of Life The most commonly used measure of how someone is coping with a chronic illness is a measure of their quality of life (QOL). Sometimes called health-related quality of life (HRQOL) or discussed as well-being, the past 40 years has seen an upsurge in research on QOL (Morgan & McGee, 2016). QOL features prominently in the study of how patients cope with diseases and is important for planning further treatment (Brodsky et al., 2017). QOL was originally a measure made by the physician, purely by whether the disease was present or absent. If the disease presence was strong, it was assumed that QOL would be low. It is now clear that patients are the best judges of their own QOL. Asking patients how much pain they are experiencing and how they feel (e.g., assessing depression and anxiety) is a valuable way to determine how well they are coping (Morrow et al., 2012). Assess your own QOL with one of the most common measures of QOL (Table 11.2). *See Procedures Manual, pages 13–15. Quality of life includes several components. Similar to measures of adaptation, QOL includes a measure of physical status and functioning, psychological status, social functioning, and the presence of the disease- or treatment-related symptoms. A wide array of measures assesses QOL. Figure 11.4 lists some of the major assessment tools, ranging from the generic to the specific, and separated by disease- or patient-specific types. In the age of the internet, can you Google for some? Yes indeed. The good news is you can find a number of credible collections of measures with accompanying psychometric information (Morgan & McGee, 2016). These include the Patient-Reported Outcomes Measurement System (www.promishealth.org), Patient Reported Outcomes and Quality of Life Instruments database (PROQOLID; Emery et al., 2005), the On-Line Guide to Quality-of-Life Assessment database (OLGA; http://www.olga-qol.com), and Optum (http://www.optum.com). A table depicting major measures of quality of life with column heads type, focus, and examples of assessment tools. Source: Reprinted with permission from Assessment in Health Psychology by Y. Benyamini, M. Johnston, & E. C. Karademas, ISBN 978-0-88937-452-2 ©2016 Hogrefe. www.hogrefe.com. Let’s take a look at the different biological, psychological, and sociocultural factors that can influence QOL and adaptation. Biopsychosocial Components of Adaptation Most of us will experience a disability or chronic illness at some point in our lives. However, it is clear that changing health behaviors can greatly reduce the chance of contracting some chronic illnesses (LaCaille & Hooker, 2019; Mermelstein & Brikmanis, 2019). Furthermore, psychological strategies can help one cope with chronic illness. For example, in a longitudinal study of patients with inflammatory bowel disease and arthritis, patients who displayed gratitude were less depressed later in the study (Sirois & Wood, 2017). In fact, feeling grateful was a significant predictor of lowered depression even after controlling for other psychological variables such as illness cognitions discussed in Chapter 10. Similarly, two other psychological variables, optimism and hope, are strong aids to helping patients cope with chronic diseases (Schiavon et al., 2017). Adaptation to chronic illnesses has many different components. Patients need to cope with not only their own affect, behaviors, and cognitions concerning the illness but also with revising their lifestyles to accommodate the treatment and coping with how others in their social networks respond to them because of their illness (Day, 2019; Hoyt & Stanton, 2019). They may experience many different feelings including anxiety, depression, and frustration, and may not be able to perform activities they used to, such as going to work or shopping for groceries. Daily tasks, changing symptoms, and fluctuating emotions can be overwhelming (Emery, 2019). There are numerous challenges to the process of integration; successful self-management with psychosocial, vocational, and existential support is critical. Next, I will discuss some of the different components of adaptation using the major approaches in health psychology. Biological Issues Biologically, different chronic illnesses will have different courses. For example, coronary heart disease (CHD) and cancer, the two leading causes of death for Americans, inflict significant changes in the body. Cancer causes cells to grow uncontrollably, harming surrounding tissue and limiting normal function. In CHD, the blood vessels around the heart are clogged with plaque and fat, changing blood flow and possibly leading to a heart attack. Other chronic illnesses such as diabetes and asthma similarly have physiological correlates, such as changes in insulin sensitivity and the blocking of breathing channels (Kalyva et al., 2016). The slow physiological changes limit functioning in many areas and are often accompanied by pain (Hoyt & Stanton, 2012). Consequently, physical rehabilitation is a big component of any treatment of chronic illnesses. The loss of function and increase in pain also have major consequences for how the patient views the world, and psychological issues need to be considered as well. Psychological Issues There has been growing interest in the role of psychological factors in adaptation to chronic illnesses and disability (Samson & Siam, 2008). In a review of both theoretical and empirical literature on adjusting to chronic illnesses, Stanton et al. (2001) identified two key multidimensional psychological aspects. First, the individual has to go through adaptations, which include cognitive aspects such as intrusive thoughts and changing views of the self, emotional aspects such as depression and anxiety, and behavioral and physical aspects such as dealing with pain or not being able to perform daily activities. Second, the person must make interpersonal adaptations, negotiating personal relationships with friends and family as well as professional relationships with health-care providers. Positive adaptation includes the mastery of illness-related tasks, the minimizing psychological disorder and negative feelings, perceptions of high quality of life, and the maintenance of adequate functional status and social roles (Hoyt & Stanton, 2019). Perhaps one of the most effective psychological resources that a person with a chronic illness or disability has is their mental approach to the situation and appraisals (see Chapter 5). Patients’ primary and secondary appraisals of the illness can correspondingly influence how they fare. If the illness is seen as a challenge (primary appraisal) and they believe they have a lot of social support to cope with it (secondary appraisal), they will probably have a higher QOL (Gatchel & Oordt, 2003b). For example, in a study of colorectal cancer-specific concerns in a population-based sample of colorectal cancer survivors, patients’ threat appraisals significantly predicted their quality of life up to 2 years after treatment (Steginga et al., 2009). A number of health psychologists have modified cognitive appraisal theory from its original context (i.e., stress) and have adapted it to help explain coping with chronic illnesses such as arthritis, breast cancer, prostate cancer, and AIDS (Merz et al., 2011; Schwartz & Rapkin, 2012). In fact, new work teases apart appraisals from personality factors. The Quality of Life Appraisal Profile–Version 2 (QOLAPv2) helps assess individual differences and is useful in explaining why people experiencing very different health states may report the same QOL (Rapkin et al., 2017). First developed with 4,173 respondents, the QOLAPv2 is useful across populations and provides better predictions of QOL than measures of personality alone. A caring healthcare nurse conducting physical therapy exercises with a senior adult patient at home. The nurse is assisting the patient who is holding dumbbells with her hands raised. Physical Therapy. Physical therapy is an important part of coping with the biological aspects of chronic illnesses, but mobility can influence the psychosocial aspects as well. Another common psychological reaction to a positive test result or even experiencing symptoms of a chronic illness is anxiety. Anxiety interferes with healthy functioning, causing a person to cope poorly and to delay the recognition and reporting of symptoms. Anxiety is often high when the patient is waiting for test results, receiving a diagnosis, and awaiting invasive medical procedures. Not knowing about the course of the illness or not having enough information about what the illness entails is especially anxiety provoking. Such lack of information-induced anxiety is more pronounced in populations of lower SES and in some ethnic groups. The most common negative reaction to a chronic illness is depression (Giardini et al., 2017). Depression can be either biological or psychological in nature and often goes undiagnosed because its symptoms are shadowed by the symptoms of the chronic illness. Unlike anxiety, depression tends to be a long-term reaction and increases as pain increases. When patients get depressed, they are less motivated to cope actively with the illness, and tend to interpret any bodily change negatively. Interestingly, the most important predictor of one’s mental health after acquiring a disability or chronic illness is their premorbid mental health. The form of psychological reaction varies also depending on the illness and varies considerably across individuals with the same illness. Personality factors, the amount of social support one receives or perceives to have, and cultural beliefs surrounding the illness can all influence coping with the illness and can alleviate depression and anxiety. Chapter 6 included details about the ways different personalities influence coping. The same relationships that link stress and coping link chronic illnesses and coping. The Big Five personality variables (conscientiousness, agreeableness, neuroticism, openness to experience, and extroversion; see Chapter 6) have been linked to coping in general (Smith, 2019) and coping with chronic illnesses in particular (Sirois, 2015). Similarly, being high in positive affect is also a good thing for those with chronic illnesses. Positive affect was significantly associated with having a lower risk of dying from any cause (i.e., all-cause mortality) in people with diabetes (Moskowitz et al., 2008). In a study of the role of religious involvement, spirituality, and physical or emotional functioning in a sample of African American men and women with cancer, positive affect was a key factor in predicting better adaptation (Holt et al., 2011). Optimism is another powerful personality characteristic in coping with chronic illnesses (Giardini et al., 2017). Carver et al. (1993) first demonstrated convincingly the role of optimism in women coping with breast cancer. When measured before surgery, the optimistic women were those using more active coping and facing the disease, and those with less distress. This pattern held for three further assessments at 3, 6, and 12 months after surgery. Optimism is also helpful in coping with diabetes mellitus, rheumatoid arthritis, and multiple sclerosis (Fournier et al., 2002a), breast cancer (Sohl et al., 2012), coronary bypass surgery (Tindle et al., 2012), and HIV infection (Peterson et al., 2012). Building optimism can go a long way. For example, falling, common in older chronically ill adults, predicts poorer physical health and greater negative emotions among the group (Ruthig et al., 2007). Falling also causes drops in optimism, which mediates the effects of falling on health and well-being. Recovery from falling can be enhanced by bolstering optimism (Ruthig et al., 2007). In general, different personality characteristics can greatly help coping (Smith, 2019). Another important component of psychological coping is related to how patients compare themselves with others with the disease and how much meaning they derive from the illness. For example, studies on upward and downward social comparison show that people can sometimes compare themselves with those better off than they are (“Boy, my coworker has the same problem, and he is doing so much better than I am.”) or worse off than themselves (“Oh, at least I am doing better than my neighbor who has the same illness.”). Women who cope better with breast cancer make comparisons with people who are faring less well than they are to enhance their own self-esteem (Wood et al., 1985). Chinese women facing breast cancer were also found to make the best of it. The essences of Chinese women’s experiences were that they faced the reality of the cancer diagnosis, took an active part in the cancer treatment, sustained an optimistic spirit, maintained physical activity, reflected, and then moved on (Fu et al., 2008). Finding meaning in your illness can often be beneficial, leading to lower mortality and morbidity (Hooker et al., 2018), but in some cases it can be detrimental to well-being as well. Originally, research documented that finding meaning in your experience can lead to positive well-being and better adaptation to the disease (Taylor, 1983). There are some important qualifications to this early finding. Tomich and Helgeson (2004) examined the consequences of finding meaning (they called it benefit finding) on QOL in 364 women diagnosed with stage I, II, or III breast cancer. Benefit finding and QOL were measured 4 months post diagnosis (Tl), 3 months after Tl (T2), and 6 months after T2 (T3). Women with lower socioeconomic status, minority women, and those with more severe levels of the disease perceived more benefits at baseline. Benefit finding was associated with more negative affect at baseline and also interacted with the stage of disease, such that negative relationships to QOL across time were limited to those with more severe disease. Findings suggest that there are qualifiers as to whether finding something good in the bad is, in itself, good or bad (Yanez et al., 2011). We discuss this further in Chapter 13. == 11.3 - Culture, Community, Chronic Illness, and Disability == The text explores how chronic illness and disability intersect with various social and cultural factors, influencing how individuals cope and adapt. Marginalized groups, such as those who are older, low-income, or from ethnic minorities, face unique challenges that are compounded by social discrimination. The environment, including family dynamics, neighborhood safety, and social support, significantly affects the management of chronic illnesses. Cultural beliefs and practices also play a role, with different groups relying on religion, community support, or traditional medicine for coping. Social support, both formal and informal, is crucial but must be tailored to individual needs, as excessive or mismatched support can sometimes be counterproductive. The importance of close relationships, such as family and community, is highlighted as a sustainable approach to managing chronic illnesses. Additionally, interventions like support groups and positive psychology strategies offer varying degrees of effectiveness, underscoring the need for personalized approaches to care. === Raw Textbook Page === Disability intersects with all other social groups, and in fact, is more likely to occur within other marginalized groups. Disabled people are more likely to also be older, female, Black, Hispanic, or indigenous, LGBTQ+, and low income (Okoro et al., 2018; Varadaraj et al., 2021). A person’s intersectionality has many implications for how they cope with chronic illnesses or disability. Jose, who lives with a large extended Mexican American family, is going to cope with a diagnosis of cancer differently from how Joshua copes, who lives alone and far away from his European American family. Jessica, a devout Catholic, may face breast cancer very differently from Carmel, who is agnostic. Friends, family, and society can make a big difference in how one copes. If you get a chronic illness that is stigmatized in your society, you are likely to be discriminated against for having the disease, and this discrimination can negatively influence your ability to cope with it. Family and Neighborhoods The environment in which you live can accentuate a disease or help control it (Gurung et al., 2004). Stressful events influence anxiety levels, thereby influencing adaptation to the disease (Lepore & Evans, 1996). In a major review of the ways that sociocultural factors can affect a patient, Taylor et al., (1997) traced the different ways that unhealthy environments—stressful work or family situations, living in a neighborhood with a high crime rate, being unemployed, or having multiple chronic burdens—can reduce social support and hurt adaptation to illness. As shown in Figure 11.5 each of these different elements plays a role in influencing perceptions and the availability of coping resources. The importance of social factors such as the family and community structures increases when the person with the chronic illness is a child (Lyon et al., 2011). For example, the family dynamics can change significantly when a child is diagnosed with diabetes. Some families become more protective and controlling when an adolescent has diabetes (if you thought an early curfew was bad when you were young, imagine your parents wanting complete control over what you eat and drink). In such situations, families may get overtaxed and summon help from the extended family, neighbors, or the community. The neighborhood may be key, as a supportive community is proven to be advantageous (Waverijn et al., 2017). Statistics show that adolescents living in dangerous neighborhoods are more susceptible to engaging in risky behaviors, hence accelerating the course of their chronic illnesses (Obeidallah et al., 2001). Chronic Illness and Ethnicity Your cultural environment is important as well. The experience and outcomes of an illness are shaped by cultural factors that influence how it is perceived, labeled, and explained, and how the experience is valued (Broadbent, 2019). For example, African Americans with chronic illness have poorer outcomes than European Americans in the United States (Lederer et al., 2008). Therefore, we actually learn ways of being ill that depend on our cultural backgrounds (see Chapter 3). Someone coming from a self-reliant farm family may be taught to downplay illness and put on a brave face and keep on working. Someone else who grew up in a city may be more likely to follow the complete bed rest prescriptions of a doctor. Both the patients’ and the providers’ cultural approaches to the source of disease and illness affect patients’ care-seeking behavior and treatment opinions, choices, and compliance (Turner, 1996). In the past few years, practitioners have been sensitized to the role of cultural factors, especially acculturation, in rates of life-threatening chronic illnesses. For example, when treating migrants, practitioners are now more aware of casual factors of illnesses such as having to deal with changing diets and stress from their new environment, stressors that often lead to certain diseases like obesity and prostate cancer (Jasso et al., 2004). There is also extensive research on the role of linguistic competency (e.g., Ngo-Metzger et al., 2003) and ethnic match of patient and practitioner in coping with chronic illnesses (e.g., Tarn et al., 2005; see Chapter 9 for more on this topic). Some cultural groups react to chronic illnesses differently from others (Galanti, 2014). Many collectivist groups see chronic illnesses as something that an entire family or community, not just the individual, is responsible for and has to cope with. In the last section of Chapter 9, we discussed how the Hmong family rallied around the sick child with epilepsy and endured personal hardships to take care of her. There are similar cultural patterns across different religious and ethnic groups. For example, many church groups have organized programs to take care of chronically ill worshippers. Are there cultural differences in how ethnic groups cope with specific illnesses? Research in this area is growing. For example, Culver et al., (2004) tested for differences in coping responses in middle-class African American, Latinas, and European American women with early stage breast cancer. They found only two differences in coping (controlling for medical variables, education, and distress). Compared with European American women, the other two groups both reported using humor-based coping less and religion-based coping more. There was one difference in how coping related to distress: Venting related more strongly to elevated distress among Latinas than among non-Latinas (Culver et al., 2004). Religion (as seen in coping with breast cancer) plays a key role in understanding cultural differences in coping with chronic illnesses (Park & Carney, 2019). Spirituality in particular plays an especially strong role in the self-management of chronic illness among older women (Harvey, 2008). In a study directly testing the role of religion in coping with pain and psychological adaptation, Abraído--Lanza et al. (2004) found that Latinos with arthritis reported using high levels of religious coping. Further analysis indicated that religious coping was correlated with active but not passive coping and directly related to psychological well-being. Passive coping was associated with greater pain and worse adjustment. Findings such as this, together with similar work in other ethnic groups, such as African American (Holt et al., 2011), suggest that interventions and community-based outreach approaches should embrace an appreciation for expressions and experiences of spirituality for both patients and caregivers. The cultural group’s beliefs about health and illness are important as well (Arellano-Morales & Sosa, 2018). For many chronic medical problems, a patient’s coping behaviors and adherence to treatment will depend on the quality of the patient–practitioner interaction (see Chapter 8). Some earlier studies suggest that patients whose beliefs favor folk medicine are healthier when they seek treatment from folk healers rather than biomedical doctors (Kleinman et al., 1978; Mehl-Medrona, 1998; also see Chapter 3). This could be because of the corresponding belief systems as well as the relative closeness in social class between patient and practitioner. In other cases, it may be because the doctor’s own cultural identity may influence how they treat a patient from a similar culture (Gurung & Mehta, 2001). In many folk and traditional medical systems, a greater emphasis is also placed on communication, which can increase patient satisfaction and adherence to treatment. Prejudice and discrimination account for many negative outcomes for certain cultural groups. Lederer et al. (2008) conducted a retrospective cohort study of 280 non-Hispanic African American and 5,272 non-Hispanic European American adults age 40 years and older with chronic obstructive pulmonary disease (COPD). The patients were listed for lung transplantation in the United States between 1995 and 2004. After listing for lung transplantation, African American patients were less likely to undergo transplantation and more likely to die or to be removed from the list compared with the non-Hispanic European American patients. Unequal access to care may have contributed to these differences. African Americans in the study were more likely to have pulmonary hypertension, to be obese and diabetic, to lack private health insurance, and to live in poorer neighborhoods. Social Support Compared to individuals without disabilities, those with disabilities are more likely to experience social isolation and less likely to report obtaining enough social support (Bryson et al., 2019; Havercamp et al., 2004; Krahn et al., 2015). One strategy for enhancing support is through support groups and group psychotherapy, which have been shown to have positive effects on a range of psychosocial outcomes (Bardon et al., 2022). For those who have only recently been diagnosed with a disability or illness, support groups may be very helpful in fostering knowledge, coping mechanisms, and self-efficacy. Empirical studies and reviews show that people with more social support have more positive adjustment to chronic illnesses. Illnesses studied ranged from cancer (Rogers et al., 2017) to rheumatic diseases (Shim et al., 2017). Having a socially supportive environment often makes the patient more actively cope with the illness and less likely to disengage and get worse (Rosland et al., 2012). In the case of a chronic illness such as coronary heart disease, social network size and having a stressed partner can influence morbidity and mortality by influencing whether patients attend rehabilitation (Molloy et al., 2008). Social networks also help maintain quality of life and are particularly important for low SES individuals (Barden et al., 2016; Ruiz et al., 2019). Groups do not always work for everyone. Helgeson et al.(2000) determined the extent to which individual difference variables moderated the effects of an information-based educational group and how an emotion-focused peer discussion group helped women with breast cancer. Women who needed outside support (e.g., did not have strong personal connections) benefited the most from the educational group, and peer discussion groups were helpful for women who lacked support from their partners or doctors. Surprisingly, however, too much support can be detrimental. Helgeson et al. (2000) found that the discussion groups were harmful for women who already had high levels of personal support. These groups frequently use a medical model approach, taking place in clinics under the direction of professionals in the medical or mental health fields. However, there is also a need for less formal ways to connect and socialize with the disability community. Additional studies have focused on the experience of connecting with others with the same disability. Because it is unlikely that people with rare disorders will encounter others in their local community with the same disability, support conferences are sometimes offered by patient organizations as a way for people to connect. In a study of individuals with the rare condition Moebius syndrome, attending conferences more frequently was linked to higher social support and lower stigma (Bogart & Hemmesch, 2016). In surveys of people with physical disabilities, Silverman et al., (2017) found that nearly half of participants did not have any friends who also had physical disabilities. However, friendships between people with the same disabilities or friendships across different types of disabilities were linked to greater life satisfaction. There was additional evidence that friendships among people with disabilities reduced the negative effects of functional impairment on wellbeing, perhaps by learning adaptive strategies from each other. The cross-disability findings are particularly encouraging since people with disabilities, even rare ones, can benefit from relationships with persons who have any disability. '''Hospital Chapel'''. Many hospitals have chapels such as the one shown here. Caregivers can come and pray for their loved ones and even patients who are too sick to go to church can go say a prayer. There are some important cultural differences in how social support is used (Taylor et al., 2007; Wong & Lu, 2017). For example, a review of studies on culture and social support shows that Asians and Asian Americans are more reluctant to explicitly ask for support from others than are European Americans (Kim et al., 2008). This is likely due to their concern about the potentially negative relational consequences of such behaviors. Asians and Asian Americans are more likely to use and benefit from forms of support that do not involve explicit disclosure of personal stressful events and feelings of distress. A number of reviews provide important insights into interventions to help people cope with chronic illnesses. In one recent review, Martire and Helgeson (2017) suggest family members are the most important aid in children’s and adults’ illness management. Evidence suggests a dyadic approach to chronic illness management that targets the influence of close relationships may be the most helpful and sustainable method to affect patient behavior. Specifically, dyadic approaches aimed at helping patients and family members to find ways to set goals together may best benefit family members who are ill or are at risk because of poor health behaviors. Ghosh and Deb (2017) systematically reviewed positive psychology interventions in chronic physical illness and found writing is the most commonly used method for administration (see emotional expression in Chapter 6). Positive psychology interventions are considered feasible and acceptable by patients, but findings about their usefulness are inconclusive. n38cr0ncz6i4duyqnqwdrng60ej9l9a 2693463 2693462 2024-12-26T23:56:56Z Atcovi 276019 2693463 wikitext text/x-wiki == 11.1 - What is Disability and Chronic Illness? == * Disability affects 15% of the world's population. '''Disability Models''' This passage explores disability through three main models: moral, medical, and social. The '''moral model''' associates disability with divine intervention or punishment, emphasizing cultural and religious beliefs, and remains prevalent globally, though less so in Western cultures. The '''medical model''', dominant in the West, views disability as a biological dysfunction to be treated or cured, focusing responsibility on medical professionals and individuals. In contrast, the '''social model''' attributes disability to societal structures and attitudes, urging systemic changes to dismantle ableism and emphasizing a possible "cure" and "return to normalcy" for disabled people. The '''biopsychosocial model''', adopted by the World Health Organization (WHO) in the International Classification of Functioning (ICF), integrates aspects of all three models, highlighting how physical, personal, and societal factors interact to shape disability experiences. Disability dimensions, including visibility, functionality, and progression, influence individual adaptation and societal responses. Chronic illnesses like cardiovascular disease and diabetes are prevalent today due to increased life expectancy and medical advancements. The discussion also underscores the impact of cultural norms, health equity, and chronic conditions on life expectancy and societal roles in health and disability. === Raw Textbook Page === Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? What are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. == 11.2 - Coping With Disability and Chronic Illness == '''Goals of Treatment''' * Adaptation/coping is key. Can be defined as the affective, mental, and behavior chances that allow a disabled person to embrace their life despite their disability. The five major areas are success in performing daily tasks, reducing psych. disorders, reduce negative affect and improve positive affect, maintain a satisfactory functional status, and experience happiness in other areas of life. '''Quality of Life''' * '''QOL''' measures how well someone copes with chronic illness, considering physical, psychological, and social factors. * Initially assessed by physicians, QOL is now better evaluated by patients themselves based on their experiences with pain, emotional well-being, and functional status. * Tools like PROMIS, PROQOLID, OLGA, and Optum provide assessments for QOL. '''Biopsychosocial components of adaptation:''' # '''Biological Issues''': #* Chronic illnesses like cancer, coronary heart disease (CHD), and diabetes impact physical functioning and often involve pain, necessitating physical rehabilitation. #* Such conditions also influence psychological outlooks. # '''Psychological Issues''': #* Adaptation involves cognitive, emotional, and interpersonal adjustments, with key factors including: #** '''Appraisals''': Seeing illness as a challenge and perceiving social support improves QOL. #** '''Personality''': Traits like optimism and positive affect aid in coping, while depression and anxiety hinder it. #** '''Comparison''': Social comparisons (upward or downward) can impact self-esteem. #** '''Meaning''': Finding meaning in illness can enhance or detract from well-being, depending on context. '''Research Highlights''' * '''Optimism''': Strong predictor of better adaptation, linked to active coping and less distress across illnesses. * '''Gratitude''': Reduces depression and promotes better outcomes. * '''Social Support''': Critical for managing emotional and practical challenges. * '''Cultural Variations''': Different coping mechanisms observed across cultures (e.g., Chinese women with breast cancer). '''Challenges and Strategies:''' * Psychological reactions vary by illness and individual differences (e.g., premorbid mental health, cultural beliefs). * Interventions that foster optimism, gratitude, and supportive relationships are effective in improving QOL. === Raw Textbook Page === Improving one’s diet, refraining from smoking, and consuming minimal alcoholic beverages (see Chapter 7) may help prevent chronic illnesses, but do not guarantee avoiding these illnesses. Goals of Treatment Before we discuss how one can cope with having a chronic illness, it is important to consider some goals for treatment. Science has made many advances in the treatment of cancer and HIV infection, and some research suggests that illnesses such as CHD and diabetes can be reversed (e.g., Campbell & Campbell, 2016; Ornish et al., 1998); however, we still cannot cure these illnesses. Therefore, helping people cope with having these illnesses becomes very important. Adaptation to disability can be defined as a dynamic process of affective, cognitive, and behavioral changes that gradually approach an optimal state of well-being (Livneh & Antonak, 1997). Five major forms of adaptation are the successful performance of daily tasks, the minimizing psychological disorders, low levels of negative affect and high levels of positive affect, good functional status, and the experience of satisfaction in different areas of life (Stanton et al., 2001). Of all of these, the most common psychological outcome studied is the quality of life (Morgan & McGee, 2016). Quality of Life The most commonly used measure of how someone is coping with a chronic illness is a measure of their quality of life (QOL). Sometimes called health-related quality of life (HRQOL) or discussed as well-being, the past 40 years has seen an upsurge in research on QOL (Morgan & McGee, 2016). QOL features prominently in the study of how patients cope with diseases and is important for planning further treatment (Brodsky et al., 2017). QOL was originally a measure made by the physician, purely by whether the disease was present or absent. If the disease presence was strong, it was assumed that QOL would be low. It is now clear that patients are the best judges of their own QOL. Asking patients how much pain they are experiencing and how they feel (e.g., assessing depression and anxiety) is a valuable way to determine how well they are coping (Morrow et al., 2012). Assess your own QOL with one of the most common measures of QOL (Table 11.2). *See Procedures Manual, pages 13–15. Quality of life includes several components. Similar to measures of adaptation, QOL includes a measure of physical status and functioning, psychological status, social functioning, and the presence of the disease- or treatment-related symptoms. A wide array of measures assesses QOL. Figure 11.4 lists some of the major assessment tools, ranging from the generic to the specific, and separated by disease- or patient-specific types. In the age of the internet, can you Google for some? Yes indeed. The good news is you can find a number of credible collections of measures with accompanying psychometric information (Morgan & McGee, 2016). These include the Patient-Reported Outcomes Measurement System (www.promishealth.org), Patient Reported Outcomes and Quality of Life Instruments database (PROQOLID; Emery et al., 2005), the On-Line Guide to Quality-of-Life Assessment database (OLGA; http://www.olga-qol.com), and Optum (http://www.optum.com). A table depicting major measures of quality of life with column heads type, focus, and examples of assessment tools. Source: Reprinted with permission from Assessment in Health Psychology by Y. Benyamini, M. Johnston, & E. C. Karademas, ISBN 978-0-88937-452-2 ©2016 Hogrefe. www.hogrefe.com. Let’s take a look at the different biological, psychological, and sociocultural factors that can influence QOL and adaptation. Biopsychosocial Components of Adaptation Most of us will experience a disability or chronic illness at some point in our lives. However, it is clear that changing health behaviors can greatly reduce the chance of contracting some chronic illnesses (LaCaille & Hooker, 2019; Mermelstein & Brikmanis, 2019). Furthermore, psychological strategies can help one cope with chronic illness. For example, in a longitudinal study of patients with inflammatory bowel disease and arthritis, patients who displayed gratitude were less depressed later in the study (Sirois & Wood, 2017). In fact, feeling grateful was a significant predictor of lowered depression even after controlling for other psychological variables such as illness cognitions discussed in Chapter 10. Similarly, two other psychological variables, optimism and hope, are strong aids to helping patients cope with chronic diseases (Schiavon et al., 2017). Adaptation to chronic illnesses has many different components. Patients need to cope with not only their own affect, behaviors, and cognitions concerning the illness but also with revising their lifestyles to accommodate the treatment and coping with how others in their social networks respond to them because of their illness (Day, 2019; Hoyt & Stanton, 2019). They may experience many different feelings including anxiety, depression, and frustration, and may not be able to perform activities they used to, such as going to work or shopping for groceries. Daily tasks, changing symptoms, and fluctuating emotions can be overwhelming (Emery, 2019). There are numerous challenges to the process of integration; successful self-management with psychosocial, vocational, and existential support is critical. Next, I will discuss some of the different components of adaptation using the major approaches in health psychology. Biological Issues Biologically, different chronic illnesses will have different courses. For example, coronary heart disease (CHD) and cancer, the two leading causes of death for Americans, inflict significant changes in the body. Cancer causes cells to grow uncontrollably, harming surrounding tissue and limiting normal function. In CHD, the blood vessels around the heart are clogged with plaque and fat, changing blood flow and possibly leading to a heart attack. Other chronic illnesses such as diabetes and asthma similarly have physiological correlates, such as changes in insulin sensitivity and the blocking of breathing channels (Kalyva et al., 2016). The slow physiological changes limit functioning in many areas and are often accompanied by pain (Hoyt & Stanton, 2012). Consequently, physical rehabilitation is a big component of any treatment of chronic illnesses. The loss of function and increase in pain also have major consequences for how the patient views the world, and psychological issues need to be considered as well. Psychological Issues There has been growing interest in the role of psychological factors in adaptation to chronic illnesses and disability (Samson & Siam, 2008). In a review of both theoretical and empirical literature on adjusting to chronic illnesses, Stanton et al. (2001) identified two key multidimensional psychological aspects. First, the individual has to go through adaptations, which include cognitive aspects such as intrusive thoughts and changing views of the self, emotional aspects such as depression and anxiety, and behavioral and physical aspects such as dealing with pain or not being able to perform daily activities. Second, the person must make interpersonal adaptations, negotiating personal relationships with friends and family as well as professional relationships with health-care providers. Positive adaptation includes the mastery of illness-related tasks, the minimizing psychological disorder and negative feelings, perceptions of high quality of life, and the maintenance of adequate functional status and social roles (Hoyt & Stanton, 2019). Perhaps one of the most effective psychological resources that a person with a chronic illness or disability has is their mental approach to the situation and appraisals (see Chapter 5). Patients’ primary and secondary appraisals of the illness can correspondingly influence how they fare. If the illness is seen as a challenge (primary appraisal) and they believe they have a lot of social support to cope with it (secondary appraisal), they will probably have a higher QOL (Gatchel & Oordt, 2003b). For example, in a study of colorectal cancer-specific concerns in a population-based sample of colorectal cancer survivors, patients’ threat appraisals significantly predicted their quality of life up to 2 years after treatment (Steginga et al., 2009). A number of health psychologists have modified cognitive appraisal theory from its original context (i.e., stress) and have adapted it to help explain coping with chronic illnesses such as arthritis, breast cancer, prostate cancer, and AIDS (Merz et al., 2011; Schwartz & Rapkin, 2012). In fact, new work teases apart appraisals from personality factors. The Quality of Life Appraisal Profile–Version 2 (QOLAPv2) helps assess individual differences and is useful in explaining why people experiencing very different health states may report the same QOL (Rapkin et al., 2017). First developed with 4,173 respondents, the QOLAPv2 is useful across populations and provides better predictions of QOL than measures of personality alone. A caring healthcare nurse conducting physical therapy exercises with a senior adult patient at home. The nurse is assisting the patient who is holding dumbbells with her hands raised. Physical Therapy. Physical therapy is an important part of coping with the biological aspects of chronic illnesses, but mobility can influence the psychosocial aspects as well. Another common psychological reaction to a positive test result or even experiencing symptoms of a chronic illness is anxiety. Anxiety interferes with healthy functioning, causing a person to cope poorly and to delay the recognition and reporting of symptoms. Anxiety is often high when the patient is waiting for test results, receiving a diagnosis, and awaiting invasive medical procedures. Not knowing about the course of the illness or not having enough information about what the illness entails is especially anxiety provoking. Such lack of information-induced anxiety is more pronounced in populations of lower SES and in some ethnic groups. The most common negative reaction to a chronic illness is depression (Giardini et al., 2017). Depression can be either biological or psychological in nature and often goes undiagnosed because its symptoms are shadowed by the symptoms of the chronic illness. Unlike anxiety, depression tends to be a long-term reaction and increases as pain increases. When patients get depressed, they are less motivated to cope actively with the illness, and tend to interpret any bodily change negatively. Interestingly, the most important predictor of one’s mental health after acquiring a disability or chronic illness is their premorbid mental health. The form of psychological reaction varies also depending on the illness and varies considerably across individuals with the same illness. Personality factors, the amount of social support one receives or perceives to have, and cultural beliefs surrounding the illness can all influence coping with the illness and can alleviate depression and anxiety. Chapter 6 included details about the ways different personalities influence coping. The same relationships that link stress and coping link chronic illnesses and coping. The Big Five personality variables (conscientiousness, agreeableness, neuroticism, openness to experience, and extroversion; see Chapter 6) have been linked to coping in general (Smith, 2019) and coping with chronic illnesses in particular (Sirois, 2015). Similarly, being high in positive affect is also a good thing for those with chronic illnesses. Positive affect was significantly associated with having a lower risk of dying from any cause (i.e., all-cause mortality) in people with diabetes (Moskowitz et al., 2008). In a study of the role of religious involvement, spirituality, and physical or emotional functioning in a sample of African American men and women with cancer, positive affect was a key factor in predicting better adaptation (Holt et al., 2011). Optimism is another powerful personality characteristic in coping with chronic illnesses (Giardini et al., 2017). Carver et al. (1993) first demonstrated convincingly the role of optimism in women coping with breast cancer. When measured before surgery, the optimistic women were those using more active coping and facing the disease, and those with less distress. This pattern held for three further assessments at 3, 6, and 12 months after surgery. Optimism is also helpful in coping with diabetes mellitus, rheumatoid arthritis, and multiple sclerosis (Fournier et al., 2002a), breast cancer (Sohl et al., 2012), coronary bypass surgery (Tindle et al., 2012), and HIV infection (Peterson et al., 2012). Building optimism can go a long way. For example, falling, common in older chronically ill adults, predicts poorer physical health and greater negative emotions among the group (Ruthig et al., 2007). Falling also causes drops in optimism, which mediates the effects of falling on health and well-being. Recovery from falling can be enhanced by bolstering optimism (Ruthig et al., 2007). In general, different personality characteristics can greatly help coping (Smith, 2019). Another important component of psychological coping is related to how patients compare themselves with others with the disease and how much meaning they derive from the illness. For example, studies on upward and downward social comparison show that people can sometimes compare themselves with those better off than they are (“Boy, my coworker has the same problem, and he is doing so much better than I am.”) or worse off than themselves (“Oh, at least I am doing better than my neighbor who has the same illness.”). Women who cope better with breast cancer make comparisons with people who are faring less well than they are to enhance their own self-esteem (Wood et al., 1985). Chinese women facing breast cancer were also found to make the best of it. The essences of Chinese women’s experiences were that they faced the reality of the cancer diagnosis, took an active part in the cancer treatment, sustained an optimistic spirit, maintained physical activity, reflected, and then moved on (Fu et al., 2008). Finding meaning in your illness can often be beneficial, leading to lower mortality and morbidity (Hooker et al., 2018), but in some cases it can be detrimental to well-being as well. Originally, research documented that finding meaning in your experience can lead to positive well-being and better adaptation to the disease (Taylor, 1983). There are some important qualifications to this early finding. Tomich and Helgeson (2004) examined the consequences of finding meaning (they called it benefit finding) on QOL in 364 women diagnosed with stage I, II, or III breast cancer. Benefit finding and QOL were measured 4 months post diagnosis (Tl), 3 months after Tl (T2), and 6 months after T2 (T3). Women with lower socioeconomic status, minority women, and those with more severe levels of the disease perceived more benefits at baseline. Benefit finding was associated with more negative affect at baseline and also interacted with the stage of disease, such that negative relationships to QOL across time were limited to those with more severe disease. Findings suggest that there are qualifiers as to whether finding something good in the bad is, in itself, good or bad (Yanez et al., 2011). We discuss this further in Chapter 13. == 11.3 - Culture, Community, Chronic Illness, and Disability == The text explores how chronic illness and disability intersect with various social and cultural factors, influencing how individuals cope and adapt. Marginalized groups, such as those who are older, low-income, or from ethnic minorities, face unique challenges that are compounded by social discrimination. The environment, including family dynamics, neighborhood safety, and social support, significantly affects the management of chronic illnesses. Cultural beliefs and practices also play a role, with different groups relying on religion, community support, or traditional medicine for coping. Social support, both formal and informal, is crucial but must be tailored to individual needs, as excessive or mismatched support can sometimes be counterproductive. The importance of close relationships, such as family and community, is highlighted as a sustainable approach to managing chronic illnesses. Additionally, interventions like support groups and positive psychology strategies offer varying degrees of effectiveness, underscoring the need for personalized approaches to care. === Raw Textbook Page === Disability intersects with all other social groups, and in fact, is more likely to occur within other marginalized groups. Disabled people are more likely to also be older, female, Black, Hispanic, or indigenous, LGBTQ+, and low income (Okoro et al., 2018; Varadaraj et al., 2021). A person’s intersectionality has many implications for how they cope with chronic illnesses or disability. Jose, who lives with a large extended Mexican American family, is going to cope with a diagnosis of cancer differently from how Joshua copes, who lives alone and far away from his European American family. Jessica, a devout Catholic, may face breast cancer very differently from Carmel, who is agnostic. Friends, family, and society can make a big difference in how one copes. If you get a chronic illness that is stigmatized in your society, you are likely to be discriminated against for having the disease, and this discrimination can negatively influence your ability to cope with it. Family and Neighborhoods The environment in which you live can accentuate a disease or help control it (Gurung et al., 2004). Stressful events influence anxiety levels, thereby influencing adaptation to the disease (Lepore & Evans, 1996). In a major review of the ways that sociocultural factors can affect a patient, Taylor et al., (1997) traced the different ways that unhealthy environments—stressful work or family situations, living in a neighborhood with a high crime rate, being unemployed, or having multiple chronic burdens—can reduce social support and hurt adaptation to illness. As shown in Figure 11.5 each of these different elements plays a role in influencing perceptions and the availability of coping resources. The importance of social factors such as the family and community structures increases when the person with the chronic illness is a child (Lyon et al., 2011). For example, the family dynamics can change significantly when a child is diagnosed with diabetes. Some families become more protective and controlling when an adolescent has diabetes (if you thought an early curfew was bad when you were young, imagine your parents wanting complete control over what you eat and drink). In such situations, families may get overtaxed and summon help from the extended family, neighbors, or the community. The neighborhood may be key, as a supportive community is proven to be advantageous (Waverijn et al., 2017). Statistics show that adolescents living in dangerous neighborhoods are more susceptible to engaging in risky behaviors, hence accelerating the course of their chronic illnesses (Obeidallah et al., 2001). Chronic Illness and Ethnicity Your cultural environment is important as well. The experience and outcomes of an illness are shaped by cultural factors that influence how it is perceived, labeled, and explained, and how the experience is valued (Broadbent, 2019). For example, African Americans with chronic illness have poorer outcomes than European Americans in the United States (Lederer et al., 2008). Therefore, we actually learn ways of being ill that depend on our cultural backgrounds (see Chapter 3). Someone coming from a self-reliant farm family may be taught to downplay illness and put on a brave face and keep on working. Someone else who grew up in a city may be more likely to follow the complete bed rest prescriptions of a doctor. Both the patients’ and the providers’ cultural approaches to the source of disease and illness affect patients’ care-seeking behavior and treatment opinions, choices, and compliance (Turner, 1996). In the past few years, practitioners have been sensitized to the role of cultural factors, especially acculturation, in rates of life-threatening chronic illnesses. For example, when treating migrants, practitioners are now more aware of casual factors of illnesses such as having to deal with changing diets and stress from their new environment, stressors that often lead to certain diseases like obesity and prostate cancer (Jasso et al., 2004). There is also extensive research on the role of linguistic competency (e.g., Ngo-Metzger et al., 2003) and ethnic match of patient and practitioner in coping with chronic illnesses (e.g., Tarn et al., 2005; see Chapter 9 for more on this topic). Some cultural groups react to chronic illnesses differently from others (Galanti, 2014). Many collectivist groups see chronic illnesses as something that an entire family or community, not just the individual, is responsible for and has to cope with. In the last section of Chapter 9, we discussed how the Hmong family rallied around the sick child with epilepsy and endured personal hardships to take care of her. There are similar cultural patterns across different religious and ethnic groups. For example, many church groups have organized programs to take care of chronically ill worshippers. Are there cultural differences in how ethnic groups cope with specific illnesses? Research in this area is growing. For example, Culver et al., (2004) tested for differences in coping responses in middle-class African American, Latinas, and European American women with early stage breast cancer. They found only two differences in coping (controlling for medical variables, education, and distress). Compared with European American women, the other two groups both reported using humor-based coping less and religion-based coping more. There was one difference in how coping related to distress: Venting related more strongly to elevated distress among Latinas than among non-Latinas (Culver et al., 2004). Religion (as seen in coping with breast cancer) plays a key role in understanding cultural differences in coping with chronic illnesses (Park & Carney, 2019). Spirituality in particular plays an especially strong role in the self-management of chronic illness among older women (Harvey, 2008). In a study directly testing the role of religion in coping with pain and psychological adaptation, Abraído--Lanza et al. (2004) found that Latinos with arthritis reported using high levels of religious coping. Further analysis indicated that religious coping was correlated with active but not passive coping and directly related to psychological well-being. Passive coping was associated with greater pain and worse adjustment. Findings such as this, together with similar work in other ethnic groups, such as African American (Holt et al., 2011), suggest that interventions and community-based outreach approaches should embrace an appreciation for expressions and experiences of spirituality for both patients and caregivers. The cultural group’s beliefs about health and illness are important as well (Arellano-Morales & Sosa, 2018). For many chronic medical problems, a patient’s coping behaviors and adherence to treatment will depend on the quality of the patient–practitioner interaction (see Chapter 8). Some earlier studies suggest that patients whose beliefs favor folk medicine are healthier when they seek treatment from folk healers rather than biomedical doctors (Kleinman et al., 1978; Mehl-Medrona, 1998; also see Chapter 3). This could be because of the corresponding belief systems as well as the relative closeness in social class between patient and practitioner. In other cases, it may be because the doctor’s own cultural identity may influence how they treat a patient from a similar culture (Gurung & Mehta, 2001). In many folk and traditional medical systems, a greater emphasis is also placed on communication, which can increase patient satisfaction and adherence to treatment. Prejudice and discrimination account for many negative outcomes for certain cultural groups. Lederer et al. (2008) conducted a retrospective cohort study of 280 non-Hispanic African American and 5,272 non-Hispanic European American adults age 40 years and older with chronic obstructive pulmonary disease (COPD). The patients were listed for lung transplantation in the United States between 1995 and 2004. After listing for lung transplantation, African American patients were less likely to undergo transplantation and more likely to die or to be removed from the list compared with the non-Hispanic European American patients. Unequal access to care may have contributed to these differences. African Americans in the study were more likely to have pulmonary hypertension, to be obese and diabetic, to lack private health insurance, and to live in poorer neighborhoods. Social Support Compared to individuals without disabilities, those with disabilities are more likely to experience social isolation and less likely to report obtaining enough social support (Bryson et al., 2019; Havercamp et al., 2004; Krahn et al., 2015). One strategy for enhancing support is through support groups and group psychotherapy, which have been shown to have positive effects on a range of psychosocial outcomes (Bardon et al., 2022). For those who have only recently been diagnosed with a disability or illness, support groups may be very helpful in fostering knowledge, coping mechanisms, and self-efficacy. Empirical studies and reviews show that people with more social support have more positive adjustment to chronic illnesses. Illnesses studied ranged from cancer (Rogers et al., 2017) to rheumatic diseases (Shim et al., 2017). Having a socially supportive environment often makes the patient more actively cope with the illness and less likely to disengage and get worse (Rosland et al., 2012). In the case of a chronic illness such as coronary heart disease, social network size and having a stressed partner can influence morbidity and mortality by influencing whether patients attend rehabilitation (Molloy et al., 2008). Social networks also help maintain quality of life and are particularly important for low SES individuals (Barden et al., 2016; Ruiz et al., 2019). Groups do not always work for everyone. Helgeson et al.(2000) determined the extent to which individual difference variables moderated the effects of an information-based educational group and how an emotion-focused peer discussion group helped women with breast cancer. Women who needed outside support (e.g., did not have strong personal connections) benefited the most from the educational group, and peer discussion groups were helpful for women who lacked support from their partners or doctors. Surprisingly, however, too much support can be detrimental. Helgeson et al. (2000) found that the discussion groups were harmful for women who already had high levels of personal support. These groups frequently use a medical model approach, taking place in clinics under the direction of professionals in the medical or mental health fields. However, there is also a need for less formal ways to connect and socialize with the disability community. Additional studies have focused on the experience of connecting with others with the same disability. Because it is unlikely that people with rare disorders will encounter others in their local community with the same disability, support conferences are sometimes offered by patient organizations as a way for people to connect. In a study of individuals with the rare condition Moebius syndrome, attending conferences more frequently was linked to higher social support and lower stigma (Bogart & Hemmesch, 2016). In surveys of people with physical disabilities, Silverman et al., (2017) found that nearly half of participants did not have any friends who also had physical disabilities. However, friendships between people with the same disabilities or friendships across different types of disabilities were linked to greater life satisfaction. There was additional evidence that friendships among people with disabilities reduced the negative effects of functional impairment on wellbeing, perhaps by learning adaptive strategies from each other. The cross-disability findings are particularly encouraging since people with disabilities, even rare ones, can benefit from relationships with persons who have any disability. '''Hospital Chapel'''. Many hospitals have chapels such as the one shown here. Caregivers can come and pray for their loved ones and even patients who are too sick to go to church can go say a prayer. There are some important cultural differences in how social support is used (Taylor et al., 2007; Wong & Lu, 2017). For example, a review of studies on culture and social support shows that Asians and Asian Americans are more reluctant to explicitly ask for support from others than are European Americans (Kim et al., 2008). This is likely due to their concern about the potentially negative relational consequences of such behaviors. Asians and Asian Americans are more likely to use and benefit from forms of support that do not involve explicit disclosure of personal stressful events and feelings of distress. A number of reviews provide important insights into interventions to help people cope with chronic illnesses. In one recent review, Martire and Helgeson (2017) suggest family members are the most important aid in children’s and adults’ illness management. Evidence suggests a dyadic approach to chronic illness management that targets the influence of close relationships may be the most helpful and sustainable method to affect patient behavior. Specifically, dyadic approaches aimed at helping patients and family members to find ways to set goals together may best benefit family members who are ill or are at risk because of poor health behaviors. Ghosh and Deb (2017) systematically reviewed positive psychology interventions in chronic physical illness and found writing is the most commonly used method for administration (see emotional expression in Chapter 6). Positive psychology interventions are considered feasible and acceptable by patients, but findings about their usefulness are inconclusive. == 11.4 - Coping With Terminal Illness and Death == === Raw Textbook Page === lc7osmeijppn60fnmpmmu9s1aqfoj03 2693569 2693463 2024-12-27T02:06:55Z Atcovi 276019 /* Raw Textbook Page */ 2693569 wikitext text/x-wiki == 11.1 - What is Disability and Chronic Illness? == * Disability affects 15% of the world's population. '''Disability Models''' This passage explores disability through three main models: moral, medical, and social. The '''moral model''' associates disability with divine intervention or punishment, emphasizing cultural and religious beliefs, and remains prevalent globally, though less so in Western cultures. The '''medical model''', dominant in the West, views disability as a biological dysfunction to be treated or cured, focusing responsibility on medical professionals and individuals. In contrast, the '''social model''' attributes disability to societal structures and attitudes, urging systemic changes to dismantle ableism and emphasizing a possible "cure" and "return to normalcy" for disabled people. The '''biopsychosocial model''', adopted by the World Health Organization (WHO) in the International Classification of Functioning (ICF), integrates aspects of all three models, highlighting how physical, personal, and societal factors interact to shape disability experiences. Disability dimensions, including visibility, functionality, and progression, influence individual adaptation and societal responses. Chronic illnesses like cardiovascular disease and diabetes are prevalent today due to increased life expectancy and medical advancements. The discussion also underscores the impact of cultural norms, health equity, and chronic conditions on life expectancy and societal roles in health and disability. === Raw Textbook Page === Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? What are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. == 11.2 - Coping With Disability and Chronic Illness == '''Goals of Treatment''' * Adaptation/coping is key. Can be defined as the affective, mental, and behavior chances that allow a disabled person to embrace their life despite their disability. The five major areas are success in performing daily tasks, reducing psych. disorders, reduce negative affect and improve positive affect, maintain a satisfactory functional status, and experience happiness in other areas of life. '''Quality of Life''' * '''QOL''' measures how well someone copes with chronic illness, considering physical, psychological, and social factors. * Initially assessed by physicians, QOL is now better evaluated by patients themselves based on their experiences with pain, emotional well-being, and functional status. * Tools like PROMIS, PROQOLID, OLGA, and Optum provide assessments for QOL. '''Biopsychosocial components of adaptation:''' # '''Biological Issues''': #* Chronic illnesses like cancer, coronary heart disease (CHD), and diabetes impact physical functioning and often involve pain, necessitating physical rehabilitation. #* Such conditions also influence psychological outlooks. # '''Psychological Issues''': #* Adaptation involves cognitive, emotional, and interpersonal adjustments, with key factors including: #** '''Appraisals''': Seeing illness as a challenge and perceiving social support improves QOL. #** '''Personality''': Traits like optimism and positive affect aid in coping, while depression and anxiety hinder it. #** '''Comparison''': Social comparisons (upward or downward) can impact self-esteem. #** '''Meaning''': Finding meaning in illness can enhance or detract from well-being, depending on context. '''Research Highlights''' * '''Optimism''': Strong predictor of better adaptation, linked to active coping and less distress across illnesses. * '''Gratitude''': Reduces depression and promotes better outcomes. * '''Social Support''': Critical for managing emotional and practical challenges. * '''Cultural Variations''': Different coping mechanisms observed across cultures (e.g., Chinese women with breast cancer). '''Challenges and Strategies:''' * Psychological reactions vary by illness and individual differences (e.g., premorbid mental health, cultural beliefs). * Interventions that foster optimism, gratitude, and supportive relationships are effective in improving QOL. === Raw Textbook Page === Improving one’s diet, refraining from smoking, and consuming minimal alcoholic beverages (see Chapter 7) may help prevent chronic illnesses, but do not guarantee avoiding these illnesses. Goals of Treatment Before we discuss how one can cope with having a chronic illness, it is important to consider some goals for treatment. Science has made many advances in the treatment of cancer and HIV infection, and some research suggests that illnesses such as CHD and diabetes can be reversed (e.g., Campbell & Campbell, 2016; Ornish et al., 1998); however, we still cannot cure these illnesses. Therefore, helping people cope with having these illnesses becomes very important. Adaptation to disability can be defined as a dynamic process of affective, cognitive, and behavioral changes that gradually approach an optimal state of well-being (Livneh & Antonak, 1997). Five major forms of adaptation are the successful performance of daily tasks, the minimizing psychological disorders, low levels of negative affect and high levels of positive affect, good functional status, and the experience of satisfaction in different areas of life (Stanton et al., 2001). Of all of these, the most common psychological outcome studied is the quality of life (Morgan & McGee, 2016). Quality of Life The most commonly used measure of how someone is coping with a chronic illness is a measure of their quality of life (QOL). Sometimes called health-related quality of life (HRQOL) or discussed as well-being, the past 40 years has seen an upsurge in research on QOL (Morgan & McGee, 2016). QOL features prominently in the study of how patients cope with diseases and is important for planning further treatment (Brodsky et al., 2017). QOL was originally a measure made by the physician, purely by whether the disease was present or absent. If the disease presence was strong, it was assumed that QOL would be low. It is now clear that patients are the best judges of their own QOL. Asking patients how much pain they are experiencing and how they feel (e.g., assessing depression and anxiety) is a valuable way to determine how well they are coping (Morrow et al., 2012). Assess your own QOL with one of the most common measures of QOL (Table 11.2). *See Procedures Manual, pages 13–15. Quality of life includes several components. Similar to measures of adaptation, QOL includes a measure of physical status and functioning, psychological status, social functioning, and the presence of the disease- or treatment-related symptoms. A wide array of measures assesses QOL. Figure 11.4 lists some of the major assessment tools, ranging from the generic to the specific, and separated by disease- or patient-specific types. In the age of the internet, can you Google for some? Yes indeed. The good news is you can find a number of credible collections of measures with accompanying psychometric information (Morgan & McGee, 2016). These include the Patient-Reported Outcomes Measurement System (www.promishealth.org), Patient Reported Outcomes and Quality of Life Instruments database (PROQOLID; Emery et al., 2005), the On-Line Guide to Quality-of-Life Assessment database (OLGA; http://www.olga-qol.com), and Optum (http://www.optum.com). A table depicting major measures of quality of life with column heads type, focus, and examples of assessment tools. Source: Reprinted with permission from Assessment in Health Psychology by Y. Benyamini, M. Johnston, & E. C. Karademas, ISBN 978-0-88937-452-2 ©2016 Hogrefe. www.hogrefe.com. Let’s take a look at the different biological, psychological, and sociocultural factors that can influence QOL and adaptation. Biopsychosocial Components of Adaptation Most of us will experience a disability or chronic illness at some point in our lives. However, it is clear that changing health behaviors can greatly reduce the chance of contracting some chronic illnesses (LaCaille & Hooker, 2019; Mermelstein & Brikmanis, 2019). Furthermore, psychological strategies can help one cope with chronic illness. For example, in a longitudinal study of patients with inflammatory bowel disease and arthritis, patients who displayed gratitude were less depressed later in the study (Sirois & Wood, 2017). In fact, feeling grateful was a significant predictor of lowered depression even after controlling for other psychological variables such as illness cognitions discussed in Chapter 10. Similarly, two other psychological variables, optimism and hope, are strong aids to helping patients cope with chronic diseases (Schiavon et al., 2017). Adaptation to chronic illnesses has many different components. Patients need to cope with not only their own affect, behaviors, and cognitions concerning the illness but also with revising their lifestyles to accommodate the treatment and coping with how others in their social networks respond to them because of their illness (Day, 2019; Hoyt & Stanton, 2019). They may experience many different feelings including anxiety, depression, and frustration, and may not be able to perform activities they used to, such as going to work or shopping for groceries. Daily tasks, changing symptoms, and fluctuating emotions can be overwhelming (Emery, 2019). There are numerous challenges to the process of integration; successful self-management with psychosocial, vocational, and existential support is critical. Next, I will discuss some of the different components of adaptation using the major approaches in health psychology. Biological Issues Biologically, different chronic illnesses will have different courses. For example, coronary heart disease (CHD) and cancer, the two leading causes of death for Americans, inflict significant changes in the body. Cancer causes cells to grow uncontrollably, harming surrounding tissue and limiting normal function. In CHD, the blood vessels around the heart are clogged with plaque and fat, changing blood flow and possibly leading to a heart attack. Other chronic illnesses such as diabetes and asthma similarly have physiological correlates, such as changes in insulin sensitivity and the blocking of breathing channels (Kalyva et al., 2016). The slow physiological changes limit functioning in many areas and are often accompanied by pain (Hoyt & Stanton, 2012). Consequently, physical rehabilitation is a big component of any treatment of chronic illnesses. The loss of function and increase in pain also have major consequences for how the patient views the world, and psychological issues need to be considered as well. Psychological Issues There has been growing interest in the role of psychological factors in adaptation to chronic illnesses and disability (Samson & Siam, 2008). In a review of both theoretical and empirical literature on adjusting to chronic illnesses, Stanton et al. (2001) identified two key multidimensional psychological aspects. First, the individual has to go through adaptations, which include cognitive aspects such as intrusive thoughts and changing views of the self, emotional aspects such as depression and anxiety, and behavioral and physical aspects such as dealing with pain or not being able to perform daily activities. Second, the person must make interpersonal adaptations, negotiating personal relationships with friends and family as well as professional relationships with health-care providers. Positive adaptation includes the mastery of illness-related tasks, the minimizing psychological disorder and negative feelings, perceptions of high quality of life, and the maintenance of adequate functional status and social roles (Hoyt & Stanton, 2019). Perhaps one of the most effective psychological resources that a person with a chronic illness or disability has is their mental approach to the situation and appraisals (see Chapter 5). Patients’ primary and secondary appraisals of the illness can correspondingly influence how they fare. If the illness is seen as a challenge (primary appraisal) and they believe they have a lot of social support to cope with it (secondary appraisal), they will probably have a higher QOL (Gatchel & Oordt, 2003b). For example, in a study of colorectal cancer-specific concerns in a population-based sample of colorectal cancer survivors, patients’ threat appraisals significantly predicted their quality of life up to 2 years after treatment (Steginga et al., 2009). A number of health psychologists have modified cognitive appraisal theory from its original context (i.e., stress) and have adapted it to help explain coping with chronic illnesses such as arthritis, breast cancer, prostate cancer, and AIDS (Merz et al., 2011; Schwartz & Rapkin, 2012). In fact, new work teases apart appraisals from personality factors. The Quality of Life Appraisal Profile–Version 2 (QOLAPv2) helps assess individual differences and is useful in explaining why people experiencing very different health states may report the same QOL (Rapkin et al., 2017). First developed with 4,173 respondents, the QOLAPv2 is useful across populations and provides better predictions of QOL than measures of personality alone. A caring healthcare nurse conducting physical therapy exercises with a senior adult patient at home. The nurse is assisting the patient who is holding dumbbells with her hands raised. Physical Therapy. Physical therapy is an important part of coping with the biological aspects of chronic illnesses, but mobility can influence the psychosocial aspects as well. Another common psychological reaction to a positive test result or even experiencing symptoms of a chronic illness is anxiety. Anxiety interferes with healthy functioning, causing a person to cope poorly and to delay the recognition and reporting of symptoms. Anxiety is often high when the patient is waiting for test results, receiving a diagnosis, and awaiting invasive medical procedures. Not knowing about the course of the illness or not having enough information about what the illness entails is especially anxiety provoking. Such lack of information-induced anxiety is more pronounced in populations of lower SES and in some ethnic groups. The most common negative reaction to a chronic illness is depression (Giardini et al., 2017). Depression can be either biological or psychological in nature and often goes undiagnosed because its symptoms are shadowed by the symptoms of the chronic illness. Unlike anxiety, depression tends to be a long-term reaction and increases as pain increases. When patients get depressed, they are less motivated to cope actively with the illness, and tend to interpret any bodily change negatively. Interestingly, the most important predictor of one’s mental health after acquiring a disability or chronic illness is their premorbid mental health. The form of psychological reaction varies also depending on the illness and varies considerably across individuals with the same illness. Personality factors, the amount of social support one receives or perceives to have, and cultural beliefs surrounding the illness can all influence coping with the illness and can alleviate depression and anxiety. Chapter 6 included details about the ways different personalities influence coping. The same relationships that link stress and coping link chronic illnesses and coping. The Big Five personality variables (conscientiousness, agreeableness, neuroticism, openness to experience, and extroversion; see Chapter 6) have been linked to coping in general (Smith, 2019) and coping with chronic illnesses in particular (Sirois, 2015). Similarly, being high in positive affect is also a good thing for those with chronic illnesses. Positive affect was significantly associated with having a lower risk of dying from any cause (i.e., all-cause mortality) in people with diabetes (Moskowitz et al., 2008). In a study of the role of religious involvement, spirituality, and physical or emotional functioning in a sample of African American men and women with cancer, positive affect was a key factor in predicting better adaptation (Holt et al., 2011). Optimism is another powerful personality characteristic in coping with chronic illnesses (Giardini et al., 2017). Carver et al. (1993) first demonstrated convincingly the role of optimism in women coping with breast cancer. When measured before surgery, the optimistic women were those using more active coping and facing the disease, and those with less distress. This pattern held for three further assessments at 3, 6, and 12 months after surgery. Optimism is also helpful in coping with diabetes mellitus, rheumatoid arthritis, and multiple sclerosis (Fournier et al., 2002a), breast cancer (Sohl et al., 2012), coronary bypass surgery (Tindle et al., 2012), and HIV infection (Peterson et al., 2012). Building optimism can go a long way. For example, falling, common in older chronically ill adults, predicts poorer physical health and greater negative emotions among the group (Ruthig et al., 2007). Falling also causes drops in optimism, which mediates the effects of falling on health and well-being. Recovery from falling can be enhanced by bolstering optimism (Ruthig et al., 2007). In general, different personality characteristics can greatly help coping (Smith, 2019). Another important component of psychological coping is related to how patients compare themselves with others with the disease and how much meaning they derive from the illness. For example, studies on upward and downward social comparison show that people can sometimes compare themselves with those better off than they are (“Boy, my coworker has the same problem, and he is doing so much better than I am.”) or worse off than themselves (“Oh, at least I am doing better than my neighbor who has the same illness.”). Women who cope better with breast cancer make comparisons with people who are faring less well than they are to enhance their own self-esteem (Wood et al., 1985). Chinese women facing breast cancer were also found to make the best of it. The essences of Chinese women’s experiences were that they faced the reality of the cancer diagnosis, took an active part in the cancer treatment, sustained an optimistic spirit, maintained physical activity, reflected, and then moved on (Fu et al., 2008). Finding meaning in your illness can often be beneficial, leading to lower mortality and morbidity (Hooker et al., 2018), but in some cases it can be detrimental to well-being as well. Originally, research documented that finding meaning in your experience can lead to positive well-being and better adaptation to the disease (Taylor, 1983). There are some important qualifications to this early finding. Tomich and Helgeson (2004) examined the consequences of finding meaning (they called it benefit finding) on QOL in 364 women diagnosed with stage I, II, or III breast cancer. Benefit finding and QOL were measured 4 months post diagnosis (Tl), 3 months after Tl (T2), and 6 months after T2 (T3). Women with lower socioeconomic status, minority women, and those with more severe levels of the disease perceived more benefits at baseline. Benefit finding was associated with more negative affect at baseline and also interacted with the stage of disease, such that negative relationships to QOL across time were limited to those with more severe disease. Findings suggest that there are qualifiers as to whether finding something good in the bad is, in itself, good or bad (Yanez et al., 2011). We discuss this further in Chapter 13. == 11.3 - Culture, Community, Chronic Illness, and Disability == The text explores how chronic illness and disability intersect with various social and cultural factors, influencing how individuals cope and adapt. Marginalized groups, such as those who are older, low-income, or from ethnic minorities, face unique challenges that are compounded by social discrimination. The environment, including family dynamics, neighborhood safety, and social support, significantly affects the management of chronic illnesses. Cultural beliefs and practices also play a role, with different groups relying on religion, community support, or traditional medicine for coping. Social support, both formal and informal, is crucial but must be tailored to individual needs, as excessive or mismatched support can sometimes be counterproductive. The importance of close relationships, such as family and community, is highlighted as a sustainable approach to managing chronic illnesses. Additionally, interventions like support groups and positive psychology strategies offer varying degrees of effectiveness, underscoring the need for personalized approaches to care. === Raw Textbook Page === Disability intersects with all other social groups, and in fact, is more likely to occur within other marginalized groups. Disabled people are more likely to also be older, female, Black, Hispanic, or indigenous, LGBTQ+, and low income (Okoro et al., 2018; Varadaraj et al., 2021). A person’s intersectionality has many implications for how they cope with chronic illnesses or disability. Jose, who lives with a large extended Mexican American family, is going to cope with a diagnosis of cancer differently from how Joshua copes, who lives alone and far away from his European American family. Jessica, a devout Catholic, may face breast cancer very differently from Carmel, who is agnostic. Friends, family, and society can make a big difference in how one copes. If you get a chronic illness that is stigmatized in your society, you are likely to be discriminated against for having the disease, and this discrimination can negatively influence your ability to cope with it. Family and Neighborhoods The environment in which you live can accentuate a disease or help control it (Gurung et al., 2004). Stressful events influence anxiety levels, thereby influencing adaptation to the disease (Lepore & Evans, 1996). In a major review of the ways that sociocultural factors can affect a patient, Taylor et al., (1997) traced the different ways that unhealthy environments—stressful work or family situations, living in a neighborhood with a high crime rate, being unemployed, or having multiple chronic burdens—can reduce social support and hurt adaptation to illness. As shown in Figure 11.5 each of these different elements plays a role in influencing perceptions and the availability of coping resources. The importance of social factors such as the family and community structures increases when the person with the chronic illness is a child (Lyon et al., 2011). For example, the family dynamics can change significantly when a child is diagnosed with diabetes. Some families become more protective and controlling when an adolescent has diabetes (if you thought an early curfew was bad when you were young, imagine your parents wanting complete control over what you eat and drink). In such situations, families may get overtaxed and summon help from the extended family, neighbors, or the community. The neighborhood may be key, as a supportive community is proven to be advantageous (Waverijn et al., 2017). Statistics show that adolescents living in dangerous neighborhoods are more susceptible to engaging in risky behaviors, hence accelerating the course of their chronic illnesses (Obeidallah et al., 2001). Chronic Illness and Ethnicity Your cultural environment is important as well. The experience and outcomes of an illness are shaped by cultural factors that influence how it is perceived, labeled, and explained, and how the experience is valued (Broadbent, 2019). For example, African Americans with chronic illness have poorer outcomes than European Americans in the United States (Lederer et al., 2008). Therefore, we actually learn ways of being ill that depend on our cultural backgrounds (see Chapter 3). Someone coming from a self-reliant farm family may be taught to downplay illness and put on a brave face and keep on working. Someone else who grew up in a city may be more likely to follow the complete bed rest prescriptions of a doctor. Both the patients’ and the providers’ cultural approaches to the source of disease and illness affect patients’ care-seeking behavior and treatment opinions, choices, and compliance (Turner, 1996). In the past few years, practitioners have been sensitized to the role of cultural factors, especially acculturation, in rates of life-threatening chronic illnesses. For example, when treating migrants, practitioners are now more aware of casual factors of illnesses such as having to deal with changing diets and stress from their new environment, stressors that often lead to certain diseases like obesity and prostate cancer (Jasso et al., 2004). There is also extensive research on the role of linguistic competency (e.g., Ngo-Metzger et al., 2003) and ethnic match of patient and practitioner in coping with chronic illnesses (e.g., Tarn et al., 2005; see Chapter 9 for more on this topic). Some cultural groups react to chronic illnesses differently from others (Galanti, 2014). Many collectivist groups see chronic illnesses as something that an entire family or community, not just the individual, is responsible for and has to cope with. In the last section of Chapter 9, we discussed how the Hmong family rallied around the sick child with epilepsy and endured personal hardships to take care of her. There are similar cultural patterns across different religious and ethnic groups. For example, many church groups have organized programs to take care of chronically ill worshippers. Are there cultural differences in how ethnic groups cope with specific illnesses? Research in this area is growing. For example, Culver et al., (2004) tested for differences in coping responses in middle-class African American, Latinas, and European American women with early stage breast cancer. They found only two differences in coping (controlling for medical variables, education, and distress). Compared with European American women, the other two groups both reported using humor-based coping less and religion-based coping more. There was one difference in how coping related to distress: Venting related more strongly to elevated distress among Latinas than among non-Latinas (Culver et al., 2004). Religion (as seen in coping with breast cancer) plays a key role in understanding cultural differences in coping with chronic illnesses (Park & Carney, 2019). Spirituality in particular plays an especially strong role in the self-management of chronic illness among older women (Harvey, 2008). In a study directly testing the role of religion in coping with pain and psychological adaptation, Abraído--Lanza et al. (2004) found that Latinos with arthritis reported using high levels of religious coping. Further analysis indicated that religious coping was correlated with active but not passive coping and directly related to psychological well-being. Passive coping was associated with greater pain and worse adjustment. Findings such as this, together with similar work in other ethnic groups, such as African American (Holt et al., 2011), suggest that interventions and community-based outreach approaches should embrace an appreciation for expressions and experiences of spirituality for both patients and caregivers. The cultural group’s beliefs about health and illness are important as well (Arellano-Morales & Sosa, 2018). For many chronic medical problems, a patient’s coping behaviors and adherence to treatment will depend on the quality of the patient–practitioner interaction (see Chapter 8). Some earlier studies suggest that patients whose beliefs favor folk medicine are healthier when they seek treatment from folk healers rather than biomedical doctors (Kleinman et al., 1978; Mehl-Medrona, 1998; also see Chapter 3). This could be because of the corresponding belief systems as well as the relative closeness in social class between patient and practitioner. In other cases, it may be because the doctor’s own cultural identity may influence how they treat a patient from a similar culture (Gurung & Mehta, 2001). In many folk and traditional medical systems, a greater emphasis is also placed on communication, which can increase patient satisfaction and adherence to treatment. Prejudice and discrimination account for many negative outcomes for certain cultural groups. Lederer et al. (2008) conducted a retrospective cohort study of 280 non-Hispanic African American and 5,272 non-Hispanic European American adults age 40 years and older with chronic obstructive pulmonary disease (COPD). The patients were listed for lung transplantation in the United States between 1995 and 2004. After listing for lung transplantation, African American patients were less likely to undergo transplantation and more likely to die or to be removed from the list compared with the non-Hispanic European American patients. Unequal access to care may have contributed to these differences. African Americans in the study were more likely to have pulmonary hypertension, to be obese and diabetic, to lack private health insurance, and to live in poorer neighborhoods. Social Support Compared to individuals without disabilities, those with disabilities are more likely to experience social isolation and less likely to report obtaining enough social support (Bryson et al., 2019; Havercamp et al., 2004; Krahn et al., 2015). One strategy for enhancing support is through support groups and group psychotherapy, which have been shown to have positive effects on a range of psychosocial outcomes (Bardon et al., 2022). For those who have only recently been diagnosed with a disability or illness, support groups may be very helpful in fostering knowledge, coping mechanisms, and self-efficacy. Empirical studies and reviews show that people with more social support have more positive adjustment to chronic illnesses. Illnesses studied ranged from cancer (Rogers et al., 2017) to rheumatic diseases (Shim et al., 2017). Having a socially supportive environment often makes the patient more actively cope with the illness and less likely to disengage and get worse (Rosland et al., 2012). In the case of a chronic illness such as coronary heart disease, social network size and having a stressed partner can influence morbidity and mortality by influencing whether patients attend rehabilitation (Molloy et al., 2008). Social networks also help maintain quality of life and are particularly important for low SES individuals (Barden et al., 2016; Ruiz et al., 2019). Groups do not always work for everyone. Helgeson et al.(2000) determined the extent to which individual difference variables moderated the effects of an information-based educational group and how an emotion-focused peer discussion group helped women with breast cancer. Women who needed outside support (e.g., did not have strong personal connections) benefited the most from the educational group, and peer discussion groups were helpful for women who lacked support from their partners or doctors. Surprisingly, however, too much support can be detrimental. Helgeson et al. (2000) found that the discussion groups were harmful for women who already had high levels of personal support. These groups frequently use a medical model approach, taking place in clinics under the direction of professionals in the medical or mental health fields. However, there is also a need for less formal ways to connect and socialize with the disability community. Additional studies have focused on the experience of connecting with others with the same disability. Because it is unlikely that people with rare disorders will encounter others in their local community with the same disability, support conferences are sometimes offered by patient organizations as a way for people to connect. In a study of individuals with the rare condition Moebius syndrome, attending conferences more frequently was linked to higher social support and lower stigma (Bogart & Hemmesch, 2016). In surveys of people with physical disabilities, Silverman et al., (2017) found that nearly half of participants did not have any friends who also had physical disabilities. However, friendships between people with the same disabilities or friendships across different types of disabilities were linked to greater life satisfaction. There was additional evidence that friendships among people with disabilities reduced the negative effects of functional impairment on wellbeing, perhaps by learning adaptive strategies from each other. The cross-disability findings are particularly encouraging since people with disabilities, even rare ones, can benefit from relationships with persons who have any disability. '''Hospital Chapel'''. Many hospitals have chapels such as the one shown here. Caregivers can come and pray for their loved ones and even patients who are too sick to go to church can go say a prayer. There are some important cultural differences in how social support is used (Taylor et al., 2007; Wong & Lu, 2017). For example, a review of studies on culture and social support shows that Asians and Asian Americans are more reluctant to explicitly ask for support from others than are European Americans (Kim et al., 2008). This is likely due to their concern about the potentially negative relational consequences of such behaviors. Asians and Asian Americans are more likely to use and benefit from forms of support that do not involve explicit disclosure of personal stressful events and feelings of distress. A number of reviews provide important insights into interventions to help people cope with chronic illnesses. In one recent review, Martire and Helgeson (2017) suggest family members are the most important aid in children’s and adults’ illness management. Evidence suggests a dyadic approach to chronic illness management that targets the influence of close relationships may be the most helpful and sustainable method to affect patient behavior. Specifically, dyadic approaches aimed at helping patients and family members to find ways to set goals together may best benefit family members who are ill or are at risk because of poor health behaviors. Ghosh and Deb (2017) systematically reviewed positive psychology interventions in chronic physical illness and found writing is the most commonly used method for administration (see emotional expression in Chapter 6). Positive psychology interventions are considered feasible and acceptable by patients, but findings about their usefulness are inconclusive. == 11.4 - Coping With Terminal Illness and Death == === Raw Textbook Page === Discuss the role of culture and religion in grief and death practices. Many chronic illnesses are treatable but not curable and some can be fatal. Some chronic illnesses, including certain types of cancer, are often referred to as terminal illnesses because people with these diseases often die after a relatively short time (although this time can range from months to a few years). Facing the reality of approaching death is an even bigger psychological challenge and can lead to many changes in outlook (Danel et al., 2017). What can be done to make the end of life easier? Problems with communication and visiting hours and too many administrative details can be trying to the patient and family members. Care is highly fragmented in a hospital and multiple caregivers pass through a patient’s room daily and nightly, monitoring the patient and performing different tests. In general, care should be taken to counter the effects of hospitalization (Buzgova et al., 2016). Patients and their families arrive with anxiety, and emotions are high if death is imminent. Health-care practitioners need to be explicitly prepared to address these issues. In particular, informed consent procedures should be closely followed whereby patients are told of their condition and the treatments available, if any. Patients should also be helped to accept their situation and prepare for death. This normally means helping patients to use their remaining time well. Psychological counseling should be made available for both the patient and their family. The patient may need help in facing death and in making sense of life. The family may need help to cope both with their grief and with the strain of caregiving. Both the patient and their family also may need help communicating with each other, in saying goodbye, and in dealing with sometimes conflicting needs (the patient fearing death and the family unable to imagine living without the patient). Religion has a long history of being included in studies of health (Koenig, 2015; Levin & Schiller, 1987). Research supports the link between religion and health (Park & Carney, 2019; Von Dras, 2017) although the bulk of the studies are correlational in nature. One of the most salient aspects of culture, religion is intrinsically tied to the other major elements of culture such as race and ethnicity. It is important to keep in mind the diversity of religious beliefs between and within groups of people. For instance, people from different races and ethnicities often have different religious beliefs. Also, even though North America is primarily Christian, there are still a significant number of North Americans who have non-Christian beliefs. Regardless of beliefs, turning to one’s spirituality can be a form of coping and can even help with pain management. However, this link is also an example of the differences between a correlational study and an experiment. Simply because people who are religious are also usually people who cope better with illness, does not allow us to conclude that religion causes better health. Nonetheless, what is important is that religion can help, and in the context of chronic and terminal illnesses, health psychologists should use any tool that can make a difference. One of the growing number of studies on different ethnic groups illustrates this point well (Tovar, 2017). Abraído-Lanza et al. (2004) tested for the link among religious, passive, and active coping, pain, and psychological adjustment in a sample of 200 Latinos with arthritis. The participants reported using high levels of religious coping that was correlated with active but not with passive coping. Religious coping was directly related to psychological well-being. Passive coping was associated with greater pain and worse adjustment (Abraído-Lanza et al., 2004). Traditional Latinos tend to be very religious, practicing Catholicism, curanderismo, or, more often, a blend of both (as discussed in Chapter 3). With the growing number of North Americans who are Latino, the findings of Abraído-Lanza et al. (2004) and other such studies suggest a greater focus on the role of ethnicity and religion on coping. A person’s religious beliefs often become more important as the end of life nears. Religious beliefs vary regarding the role of pain and of the role and significance of death. Different religions even have different ways to treat death and distinct ways to treat the lifeless human body. The deeper your faith, the more likely you are to turn to it if you or someone you love is dying (Boucher, 2017). The way death is treated within a culture can influence how well the patient copes (Corr & Corr, 2007; Doughty, 2017). For example, for devout Catholics, suffering is related to original sin and a Catholic has to face suffering like Christ did. Death is the freeing of the soul to the father who is in heaven. A righteous, well-lived life serves as preparation for death. As death draws near, the family and the terminally ill patient can draw solace from the visitation of a priest who will help the patient finalize their earthly affairs (Büssing & Poier, 2017). There is a final confession of sins, receiving of Holy Communion (a piece of bread or wafer that represents the body of Christ), an anointing, and a last blessing, known as the last rites. This scripted ritual goes a long way to help the terminally ill patient and family come to terms with the impending loss. Muslims, or followers of Islam, see death as the termination of the soul’s attachment to the body (Amer, 2017). Death is a blessing and a gift for the believer. To prepare, the person must do penance and be careful to not be under obligation to any other human being—the patient should make sure that they pay any dues or debts owed. Cleanliness at the time of death is more important than at any other time. Especially important is the edict that the seriously ill must die at home. To Hindus, suffering is part of maya, or illusion (Agnihotri & Agnihotri, 2017). The only way to transcend suffering is to be free from the cycle of birth and death and rebirth. The Hindu tries to work off bad karma from an early point in life and the place of your karmic cycle is indicated by your status in the world; for example, if you sinned in your past lives you will be reborn as an animal or, even worse, as a worm. This belief in predetermined fate helps reduce the anxiety of death. Other Eastern religions share some of these beliefs. Buddhists speak of and contemplate death often, in stark contrast to many non-Buddhists (Colgan et al., 2017). Pain is unavoidable, but attitudes and behavior influence suffering. According to Buddhists, the only way to avoid suffering is to free the self from desire, which is the cause of suffering. The Buddhist believes that as long as there is fear of death, life is not being lived to its fullest. Contemplating death can free us from fear, change the way we live and our attitude toward life, and help us face death healthily. The sense that death can be joyous is also reflected in different religious traditions (Doughty, 2017). The Irish wake is often a rousing celebration of the recently departed’s life and is accompanied by drinking and dancing. To the Sikhs of India, death is seen as a great opportunity to do something we put off all our lives. It is a chance to cleanse the soul of psychic fantasy, and life is an opportunity to practice dying, until one dies a death that will not have to be repeated. Death is not sad; friends chant and sing hymns near the dying to set a peaceful vibration and inspire the dying person to be in the best frame of mind. Hence, many religions downplay the sadness of death and emphasize the happiness coming from freedom and the unification with the creator. Death Across the Life Span Different cultural groups face death differently, whether it is one’s own death or the death of a close friend or family member (Corr & Corr, 2007; Noppe & Noppe, 2008). In addition, it matters how and when death arrives; the major causes of death vary across the life span. How and when death occurs automatically influences how survivors cope. Focusing on how we develop is important in understanding why mortality and morbidity varies across age groups. The main causes of death are not the same at each point of the life cycle (see Table 11.3). Congenital anomalies are the leading cause of infant deaths in the United States. Other major causes include complications from low birth weight (LBW), sudden infant death syndrome, and problems from labor, delivery, and other maternal complications (National Center for Health Statistics, 2017). The trauma of losing a child and the way parents cope can vary with the ethnic group and the expectations they have, and many other cultural variables can play a role. Many Mexican American mothers tend to have higher levels of stress during pregnancy. The idea that pregnancy is stressful is actually rooted in the culture: the Spanish word for labor is dolor, which also is the root for the words sorrow and pain. Other cultural variables beyond ethnicity can be important too. Many poor families do not have the health insurance or facilities to receive good prenatal care. Injury and accidents are the leading causes of death for children, adolescents, and young adults (National Center for Health Statistics, 2017). As we age, we succumb to more diseases that are related to unhealthy behaviors. The top three killers of adults (age 25 to 55) and older adults (age 55 and older) are coronary heart disease, cancer, and stroke (National Center for Health Statistics, 2017), each exacerbated by eating badly, smoking, and overindulging in alcohol. Aging should not always be associated with illness and sickness either. In one particularly impressive longitudinal study, researchers (clearly not the same ones) observed 1,500 Californians over 80 years. The study showed that eating well, getting physical activity, and giving and receiving a lot of social support are some of the factors that can provide many happy years of life for older adults (Friedman & Martin, 2011). There were also some surprises: Some of the people who worked the hardest, lived the longest, getting and staying married was not directly associated with longer lives for women, and the traits most associated with thriving were persistence and being prudent. The physical deterioration of cells associated with aging is related to specific diseases in adulthood. For example, many older adults experience marked problems in thinking and remembering or dementia. The most common problem is one that you will have heard about: Alzheimer’s disease, a degenerative disease of the brain that leads to dementia, makes everyday tasks like grocery shopping difficult. Other major causes of dementia include Parkinson’s disease and stroke. These differences in causes of mortality indicate how different areas of health psychological research will be applied at different times of the life cycle. Prevention of injuries should be a major goal when intervening with children, a decrease in unhealthy behaviors is a pertinent goal for children and young adults and help in coping with chronic and terminal disease is a critical goal for older persons. The Path to Death As the moment of death approaches, some explicit physiological changes accompany the different psychological stages. Dying patients often experience incontinence, losing control of their bladder and other bodily functions. In particular, patients may be unable to control their salivation and will not be able to feed themselves or even eat solid food as their digestive systems reduce functioning. With cancer or CHD, there is often an increase in pain and medical practitioners prescribe high doses of morphine, even putting the delivery of morphine under the patient’s own control. In this way, patients can self-administer medication to alleviate their pain and suffering. Patients may experience severe memory problems or have problems concentrating. Interactions with caregivers and hospital staff can become difficult, which can lead to misunderstandings and miscommunication. Friends and family often have trouble facing the patient in this state, and the patient may not want to be seen by anyone. Talking about death is taboo among many North Americans, so the last few days of a patient’s life can be very difficult as visitors and even medical personnel do not always know how to approach the topic. Consequently, death education is an important area of research. Given the number of tragedies on college campuses, there is a special emphasis on helping students cope with grief (Cox et al., 2015; Cupit et al., 2016). Facilitating Death Should life be terminated if a patient is in tremendous pain and discomfort or is comatose? This is one of the most controversial ethical issues regarding health care. Euthanasia, physician-assisted suicide, and the withdrawing of life-sustaining treatment (sometimes called passive euthanasia) are some of the most difficult moral and ethical dilemmas we face today. Euthanasia is the termination of life by the injection of a lethal drug. Assisted suicide involves a physician supplying a lethal drug while not actually administering it themself. When life-sustaining treatment is withdrawn, the underlying disease takes its own course. All are subjects of intense national debate (Bulmer et al., 2017). One of the most famous scenarios involves Terri Schiavo. By 2005, Schiavo, a Florida woman, had been in a persistent vegetative state for 15 years. Her husband, Michael Schiavo, battled her parents over whether his wife should be allowed to die. He argued that because she was brain dead it would not be fair to keep her alive. Terri Schiavo suffered heart failure from a potassium imbalance in 1990. Her husband said his wife told him that she would not want to be kept alive artificially. Doctors who testified on behalf of Michael Schiavo said that his wife had no hope for recovery. She was fed through a tube but breathed on her own. Terri Schiavo’s parents, Bob and Mary Schindler, maintained that their daughter could be helped with therapy. After years of litigation and appeals, Terri Schiavo’s feeding tube was removed in October 2004, only to be reinserted 6 days later after the Florida legislature, in emergency session, passed a law that affected only Terri Schiavo. The legislation gave Governor Jeb Bush the power to intervene in the case, and he ordered the feeding tube reinserted. In early 2005, the tube was removed again and Terri Schiavo died on March 31, 2005, of starvation and dehydration. What should have been done here? Should she have been kept alive? Was Terri conscious of the world around her? Did she experience psychological pain by being kept alive? Her unresponsive state made it difficult to answer any of these questions. Ripped from the headlines: “Nerve implant ‘restores consciousness’ to man in persistent vegetative state.” Not ripped, the actual headline. And I’m not making it up. This is exactly what The Guardian reported in September 2017. Physicians fitted a man in a coma for 15 years (yes, same as Terri Schiavo when she died) with a neural implant that stimulated the vagus nerve. He started to track objects with his eyes and began to stay awake (Devlin, 2017). You can bet all future cases are going to be even tougher to decide. Apart from this now classic case, the most publicized event that put these procedures into public consciousness was Dr. Jack Kevorkian’s assistance in 130 suicides since 1990. This Michigan doctor helped patients end their own lives even in the face of threats to his own life. Three juries refused to convict him despite a Michigan statute established for that purpose. The uproar surrounding his case led to intense political movements with advocates on both sides of the issue, and Kevorkian was imprisoned. He served 8 years of a 10- to 25-year sentence for second-degree murder, was released in 2007 and even ran for Congress, but lost. He died of liver disease in 2011 without assistance. In 1994, Oregon became the first state to legalize forms of assisted suicide. Be warned; how surveys are worded can influence attitudes toward assisting death (Magelssen et al., 2016). While some are against assisted suicide and euthanasia because of religious reasons, Not Dead Yet (notdeadyet.org), an organization of disability activists, opposes euthanasia, assisted suicide, and passive euthanasia, because they argue these are deadly forms of discrimination against disabled and older people. In a society full of suicide prevention efforts, disability is the one community these efforts may not include. This sends a message that disabled lives are not worth living. Assisted suicide, more often than not, may end up being a de facto alternative to suffering caused by delays or denials in health care and palliative care, Not Dead Yet argues. Society should shift focus toward ensuring that people with disabilities have access to resources and supports that allow them to have a higher quality of life. For example, a simple and low-cost intervention, such as having a volunteer visit, can extend and improve the life of terminally ill patients (Herbst-Damm & Kulik, 2005). One of the most important ethical principles in medicine is that the patient has autonomy (Angell, 1997). Terminally ill patients may spend months experiencing excruciating pain and discomfort in the process of dying. The extent of pain felt and the amount of cognition present are criteria that can be used to argue for allowing a person to end their own life. It is still very hard to draw a line. Even if a person is in extreme pain, palliative care (a form of treatment aimed at alleviating symptoms without necessarily affecting the cause) could be used (Tanuseputro et al., 2017). If a person is in a coma and has no measurable cognitive functioning, there is still no guarantee that cognition will not return or that the person is not thinking or feeling. Sometimes the decision to cut off life support is made easier by the patient having filled out an advance directive or living will in which they clearly specify the conditions under which life support should be terminated. Families also play a major role in this issue (Mayer et al., 2017). However, research shows that surrogate preferences can inaccurately reflect patients’ treatment wishes (Haley et al., 2002). Families often provide the majority of care for individuals with chronic illness for many reasons, including a sense of attachment, cultural expectations, and preferences for avoiding institutional care. Although it is optimal for the families, patients, and health-care providers to have ongoing discussions about goals of care, it is often only when the patient’s condition worsens that decisions regarding end-of-life care take place. Research suggests that family members are often key decision makers for end-of-life issues regardless of patients’ prior preferences concerning end-of-life care. Doctors tend to consult with family members even in the presence of written advance directives from their patients (Mowery, 2007). A woman touching the cheeks of Terri Schiavo, who is lying in a bed. Terri Schiavo. Terri Schiavo of Florida had been in a coma for 15 years. Her husband wanted to turn off life support but her family wanted to keep her alive. She died in March 2005. What information can help decide what should have been done? Are There Stages? Many in the lay population have heard of the concept of stages of death and that people who are dying experience a series of emotions. Spoiler alert: It’s not what you think. This somewhat inaccurate belief stems from work published by Elisabeth Kübler-Ross (1969). Kübler-Ross interviewed more than 200 dying patients and concluded that the process of dying involves five stages that vary in their emotional content and intensity. First comes denial, an initial reaction to the thought of death. This stage lasts around 2 days, can be a form of emotional coping (see Chapter 6), and can mask anxiety without necessarily removing it. Next comes anger, a stage in which patients are upset that death is happening to them. In many ways, the fact that they are dying violates a sense of the world being just. Most people believe that they do not deserve to die because they have been good or at least have not been bad enough to be punished with death. There is often misplaced resentment and a lot of irritability. Bargaining comes next. Patients try to restore their belief in a just world and may promise to be good or live life better (e.g., give a lot to charity) in exchange for life. This trading for life then gives way to depression. The patient feels a lack of control and now grieves in expectation of their death, a process known as anticipatory grief. The depression is often driven by a realization that a person will be losing their past and will also be losing all that was possible in a future. Finally, the patient may reach a stage of acceptance in which they fully acknowledge that death cannot be avoided. At this point, the patient is often very weak and faces death with a peaceful calm. Although these stages sound appropriate, and you probably can nod your head and see how a dying person could go through them, there is little empirical evidence for these stages (Jurecic, 2017). There is research suggesting each of these emotions are common (e.g., acceptance, Davis et al., 2017), but not to prove stages exist. Kübler-Ross used only cross-sectional research in a single culture and did not follow patients as they got closer to death. The fact is that people may experience these stages but not necessarily in the order just described. It is a dynamic process and people may go back and forth to different “stages.” One constant feature in Kübler-Ross’s stages of death is that most people will experience some depression just before death. How someone experiences death varies based on their culture, how much social support they have, the physiological progression of their disease, and other factors. Consequently, other researchers have attempted to explain the experience of dying (e.g., Pattison, 1977; Shneidman, 1980). When the time of death draws closer and it is clear that little can be done for the dying patient, it is important to help the person and their family prepare for death. We have discussed already some of the traditional ways this has been done, such as psychological counseling. There is an important additional dimension to consider when you look beyond the biology and psychology surrounding dying: culture. From a psychological perspective, fear, depression, and even denial of death may be common for patients and their families, but the exact experiences vary significantly across cultural groups (Galanti, 2014; Irish et al., 1993). European American health practitioners are often unintentionally ethnocentric, and this ethnocentricity makes it difficult for them to fully comprehend the experiences of people from other cultures. With an emotionally charged situation such as dying, this issue becomes even more important. Cultural differences become more evident from even a basic level of definition of key terms. It may seem clear what being dead is, but do not take death for granted. In many cultures, people are considered officially dead when Western biomedicine would consider them still alive and vice versa (Rosenblatt, 1993). There are correspondingly some significant differences in the expression and experience of emotions such as grief and loss. In some cultures, it is normal for people to cut themselves or otherwise hurt themselves to express their loss. Some East Indians, for example, fast for weeks as a sign of grieving. There are also cultural differences in the fear of death. African Americans report higher levels of death anxiety than European Americans (Depaola et al., 2003). Some of these cultural variations are seen in the rituals that accompany death. Making sure that the adequate ritual is conducted for the bereaved person is often critical to the health and coping of those left behind. Although they are not given attention in the health psychological literature to date, cultural differences in dealing with the dead may have important implications for psychological adjustment. Many cultural beliefs may clash with the beliefs of Western biomedicine, and hospital policies may prohibit certain practices, but they are important practices nonetheless. For example, in the American Indian culture, the burning of sage and other herbs is part of many religious ceremonies and is also used to prepare the soul of the dying person for the afterlife. Hospitals have nonsmoking policies and lighting a fire may seem clearly out of the question. But if sage is not burned, it could jeopardize the happiness of the dying patient’s soul and greatly hurt their family. Health-care professionals have to be aware of such cultural practices and negotiate a way to satisfy all concerned. For Muslims, there are also clear-cut practices that have to be followed at the time of death. As soon as a relative sees that the person is dead, they must turn the body to face Mecca (the site in the Middle East of the Kaaba, the Muslim’s holiest space). They also have someone sitting close by read the Koran (the word of God as channeled through the prophet Mohammed), close the body’s mouth and eyes and cover the face, and quickly bathe the body and cover it with white cotton (Gilanshah, 1993). Cultural Rituals for Death and Dying. Different cultures have very different rituals for death and dying. These caskets are part of the burial rituals of the people of Ghana who use different shapes of coffins for different individuals. Health practitioners are more effective in helping individuals during the difficult time of coping with death when they are sensitive to different cultural traditions. Many of the specific considerations needed are difficult for members of different ethnic groups to mention themselves. In the middle of coping with loss, it may be too much to expect a member of a different cultural group to explain exactly what is needed. It is likewise difficult for health-care workers to know all the different cultural idiosyncrasies surrounding death, but both groups need to work toward ensuring an adaptive experience for all concerned. For some groups, this sharing and explaining of cultural routines may be especially difficult. For example, given the negative points in the history of African Americans and American Indians in North America (Washington, 2006), both these ethnic groups have particularly strained relationships with European Americans and the health-care institution in particular (Barrett, 1998). Consequently, the development of separate models to understand how different cultural groups understand death and dying has increased (Corr & Corr, 2007). Barrett (1995, 1998), for example, has derived a list of special considerations for caregivers working with African Americans who experience loss (Table 11.5). Although devised for African Americans, these models serve as good reminders for health professionals working with any cultural group. #Understand the sociocultural influences from both Western and African traditions that combine to influence the attitudes, beliefs, and values. #Acknowledge and appreciate the uniqueness of the subgroups of African Americans (when in doubt, ask). #Be sensitive to basic differences in quality of life and differences in death rates and causes for African Americans versus European Americans. #Understand the impact of collective losses that African Americans often grieve for. #Include a consideration of SES as well as religion and spirituality. #Acknowledge the role of cultural mistrust regarding health. #Be sensitive to the value placed by African Americans on expressions of condolence. #Understand the role played by and expectations for the clergy and spiritual leaders (often higher expectations than for the medical community). Source: Barrett (1998). Reprinted with permission. Sex, Gender, and Death In the context of culture, it is also important to look at sex differences relating to the experience of death. Scholarship in death studies suggests that the different perceptions and experiences of men and women must also (in addition to cultural differences) be taken into consideration to best help those dying as well as those caring for the dead and grieving (Noppe, 2004). Martin and Doka (2000) remarked that the benchmark for grieving is normally set as how women handle loss. Women tend to show emotion, seek social support, talk about loss, and allow time to grieve openly, things not normally done by men (Cook, 1988). In a major review of gender differences in adjustment to bereavement, Stroebe et al., (2001) reported that women express their emotions more than men, although they found little evidence for the hypothesis that working through grief helps them recover faster. Particularly interesting is the fact that men suffer relatively greater health consequences when grieving than women, possibly because widowers get less support than widows (Stroebe & Stroebe, 1983). Specifically, widowers are significantly more distressed and depressed than widows and also have a higher incidence of mental illnesses. Widows have been found to suffer from fewer physical health problems and illnesses than widowers and are less likely to die during the period of acute grief after the loss of a spouse (Stroebe et al., 2001). Keeping these ethnic and sex differences in mind is clearly important in understanding how different subgroups of people experience the certainty of death. dw20xzto33chdsozoeg8yhc9mc1qo3s 2693570 2693569 2024-12-27T02:16:53Z Atcovi 276019 /* 11.4 - Coping With Terminal Illness and Death */ 2693570 wikitext text/x-wiki == 11.1 - What is Disability and Chronic Illness? == * Disability affects 15% of the world's population. '''Disability Models''' This passage explores disability through three main models: moral, medical, and social. The '''moral model''' associates disability with divine intervention or punishment, emphasizing cultural and religious beliefs, and remains prevalent globally, though less so in Western cultures. The '''medical model''', dominant in the West, views disability as a biological dysfunction to be treated or cured, focusing responsibility on medical professionals and individuals. In contrast, the '''social model''' attributes disability to societal structures and attitudes, urging systemic changes to dismantle ableism and emphasizing a possible "cure" and "return to normalcy" for disabled people. The '''biopsychosocial model''', adopted by the World Health Organization (WHO) in the International Classification of Functioning (ICF), integrates aspects of all three models, highlighting how physical, personal, and societal factors interact to shape disability experiences. Disability dimensions, including visibility, functionality, and progression, influence individual adaptation and societal responses. Chronic illnesses like cardiovascular disease and diabetes are prevalent today due to increased life expectancy and medical advancements. The discussion also underscores the impact of cultural norms, health equity, and chronic conditions on life expectancy and societal roles in health and disability. === Raw Textbook Page === Disability can be considered the world’s largest minority group, affecting 15% of the global population (World Health Organization, 2011). Disability is more common than you think!. The reason people tend to underestimate the number of people with disabilities is that many cultures have stereotypical representations of what disability looks like. Most people picture a wheelchair user, but disability is much broader! In fact, many of the health conditions in this textbook—such as cancer, HIV/AIDS, and diabetes—can be considered disabilities in certain contexts. Additionally, the majority of disabilities are invisible, meaning that you can’t tell someone is disabled by looking at them. Disability is one of the only social categories that you can be born into or join at any time. Chances are, you or a loved one will experience disability at some point in your lives, so it is vital that we learn about this minority group. Disability Models If you are thinking this all depends on how disability is defined and the cultural context, you are thinking like a good health psychologist! Let’s discuss three of the main models of disability—or ways of thinking about the cause of disability and what should be done about it. People who ascribe to a moral model of disability believe that disability is a representation of divine intervention or punishment for sin. There are a variety of cultural and religious beliefs about what should be done about disability, including atoning for sins or involving religious healers. While this model is not dominant in Western cultures, it is actually the most prevalent model of disability worldwide, and can be found in certain African and Asian cultures. For example, in rural Botswana, people with disabilities and their families have been historically hidden away to avoid bringing shame to the family or community (Jost et al., 2022). However, we still see relics of the moral model in Western media. For example, in many Disney movies and James Bond movies, the villain has a disability or disfigurement, perpetuating the stereotypical connection between disability and evil. The medical model of disability is the default way of thinking about disability in many Western cultures. If you are from one of these cultures, chances are you have been influenced by the medical model. Under this model, disability is a dysfunction, pathology, or limitation in functioning compared to the average person. The goal is to cure the disability, or to make someone as normal as possible. This model places the responsibility of managing disability into medical professionals, individuals with disabilities, and their families. While treatment advances under the medical model have made great strides in reducing people’s pain and extending lives, our understanding of disability is not complete until we consider the roles of society and culture. That’s where the social model of disability comes in, which states that the “problem” of disability lies primarily in society and is the responsibility of society at large. This model highlights that disability, like many other social identities including race and gender, is socially constructed. Like the “isms” affecting members of other minoritized social groups, people with disabilities are often targets of ableism. Ableism is stereotyping, prejudice, discrimination, and social oppression toward disabled people (Bogart & Dunn, 2019). Unlike the other disability models discussed, the social model places the responsibility on society to ameliorate disability through policy, accommodations, and dismantling ableism. As an example of social model thinking, a thought exercise might involve imagining a city in which everyone had the same impairment, say paraplegia (Geoff Adams-Spink, 2011). Using a wheelchair would not be abnormal or stigmatized. Buildings might have shorter ceilings and doors. Tables at restaurants would not have chairs, because residents would roll in on their own! A person who does not have paraplegia who comes to town would be the disabled one, experiencing stigma for their difference, bumping their heads on door frames and stooping in rooms; restaurants would not be accessible to them. This example shows the power of society in creating or ameliorating disability. There is cultural value and meaning to be found in all of these models. In 2001, the World Health Organization brought together a committee including disabled people, advocates, and healthcare experts to develop a framework for understanding disability. This resulted in the International Classification of Functioning, Disability and Health (ICF; World Health Organization, 2001; Figure 11.1), which is based on the biopsychosocial model. This framework is helpful when thinking about disability around the world because it combines ideas from the medical model and social model, while recognizing the role of cultural beliefs around morality and religion in constructing disability. Figure 11.1 International Classification of Functioning, Disability, and Health A diagram depicting the framework for understanding and measuring functioning, disability, and health. Source: World Health Organization. (2002). ICF Beginner’s Guide: Towards a Common Language for Functioning, Disability and Health. https://www.who.int/publications/m/item/icf-beginner-s-guide-towards-a-common-language-for-functioning-disability-and-health. Figure 11.1 Description Under this model, a health condition is disabling when body functions or structures are affected, daily activities are limited, and social participation is restricted. These factors may become disabling based on interactions with contextual factors such as the built environment, cultural norms, and personal factors, such as an individual’s financial resources and resilience. This means that a health condition is not necessarily disabling if appropriate personal and environmental factors are present. For example, a wheelchair user living in a place with accessible public transit, who has sufficient financial resources, social support, and coping skills would be much less disabled than someone with the same health condition in a less suitable context. According to the ICF and the Americans with Disabilities Act, a condition reaches the level of a disability when it substantially limits one or more major life activity, such as caring for oneself, walking, seeing, hearing, communicating, and working. Although there are many different underlying conditions that can cause disability, there are common dimensions that cross-cut disability and affect people’s experiences (Figure 11.2). Disability types include mobility, communication, intellectual, cognitive, chronic health, sensory, and mental health, to name a few. Some conditions, such mobility, sensory, and intellectual disabilities, are frequently and even stereotypically linked to the publics’ concept of disability. Other categories, such as invisible disabilities, chronic health issues, rare disorders, and mental health disabilities, are less frequently acknowledged by the general population as disabilities. These non-stereotypic disabilities are less likely to be acknowledged and supported, which means those who have them may experience greater stigma and a lack of resources. Figure 11.2 Disability Dimensions A table depicting Disability Dimensions with column heads Disability Dimensions Relevant to Health and Description. The time of disability onset can be at birth (congenital) or it can be acquired at any time in someone’s life. These who are born with their disability or who develop disability in childhood may have an adaptive advantage compared to those who acquire it later (Bogart, 2020). There is no functional loss in this situation; rather, infants with the disability proceed through their early development while learning to function in their physical and social environment. Similarly, people with congenital or early-onset impairments could develop a more consistent identity. People who develop their disability later, however, need to relearn how to function. People with acquired disabilities frequently express feeling grief about this change in their identity, function, and how other people view them (Adler et al., 2021). There is evidence that those with congenital or early-onset disabilities have stronger disability identity and disability self-efficacy, which in turn are associated with higher life satisfaction (Bogart, 2014). Whether a disability is visible or invisible plays an important role in the type of ableism they may experience. The majority of disabilities, including many chronic illnesses described in this book, are invisible. A certain amount of overt ableism may be avoided for those with invisible disabilities because they frequently “pass” for nondisabled. However, because their symptoms are not immediately visible, even medical professionals may dismiss or invalidate them when they seek a diagnosis or treatment (Munro et al., 2022). On the other hand, those with visible disabilities—such as those who have amputated limbs, have facial differences, or utilize assistive devices—are frequently noticed in social settings and draw comments, questions, and stares (Bogart et al., 2012). As a result, ableism affects both those with invisible and obvious disabilities, albeit in different ways (subtle vs. overt). Functional impairment, pain, and fatigue are additional key disability dimensions. According to the ICF, both the physical and social surroundings have a significant impact on functional impairment—one’s ability to perform activities of daily living. Myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS) is a condition well known for causing fatigue. A defining symptom of ME is postexertional malaise, which is a worsening of fatigue and other symptoms following energy expenditure. This worsens with activity but does not go away with sleep (Centers for Disease Control and Prevention (CDC), 2021). Because of their invisible nature and subjective symptoms (rather than objectively measurable biomarkers), diseases like ME are among the most historically contested and challenging to treat disabilities. The course of a disability describes the extent to which it changes over time. Disabilities can be temporary or acute, such as a broken bone, or chronic, lasting or expected to last more than 6 months. Chronic disabilities may become more severe over time (progressive), be life-limiting (terminal), relapse and remit (episodic), or remain stable. Conditions with a stable course, like an amputated limb) are unlikely to get worse or better, and this predictability facilitates adaptation. On the other hand, the unpredictability involved in progressive, terminal, and episodic conditions requires dynamic adaptation to frequently changing symptoms. Multiple sclerosis is an example of an episodic condition. These can be especially challenging to adapt to, because it may be difficult to plan activities since the person may not know when a flare will come on, requiring them to cancel. Additionally, repeated cancelling of plans with friends, or friends and family who do not accommodate needs when planning activities, can strain relationships. Finally, people with these conditions can experience increased ableism. A person with an episodic condition may have a flare that requires them to use a wheelchair one day, while the next week, they may be able to walk. Due to public ignorance, these individuals are sometimes labeled as malingerers or “fakers.” Some chronic illnesses and disabilities may even be terminal, such as, in some cases, cancer, diabetes, cardiovascular disease (CVD), and coronary heart disease (CHD). Most adults are affected by chronic illnesses and 70% ultimately die as a result of one (CDC, 2018). Like episodic and progress disabilities, terminal disabilities involve uncertainty and they present the extra challenge of reckoning with impending death and grief. Ponder This What do you think are the most common reasons people die? In what ways can your family and neighborhood influence the development of a chronic illness? How do you think religious beliefs and health interact? What are the most common chronic illnesses? Historically, some evidence suggests most people died relatively young. Archaeological evidence suggests the main causes of death were predation by animals and other hostile humans. There were few, if any, chronic illnesses. Most illnesses resulting from viruses or bacteria were short-lived simply because there were few cures for them—if you got sick, you died. During the Roman Empire (around a.d. 100), life expectancy was between 22 and 25 years. Recent estimates, shown in Table 11.1, suggest that Western women born in 2010 will live about 81 years and men will live about 76 years (Murphy et al., 2012). This is a big change even when compared to only 100 years ago: Women born in 1900 lived on average 48.3 years and men lived 46.3 years. This change in life expectancy is largely due to the immense improvements in medicine that can postpone death. However, we do not all have the same life expectancy: Table 11.1 shows dramatic ethnic differences in life expectancies both by sex and by ethnicity throughout the years. African American and European American men’s and women’s life expectancies changed over time, and both groups have different life expectancies today. There is also a significant sex difference—women live on average 5 years longer than men. Science has yet to explain this fact. The reasons may be that women give and receive more social support, may be biologically fitter, and engage in fewer risky behaviors. 1. Includes races other than White and Black. 2. Race categories are consistent with 1977 Office of Management and Budget (OMB) standards. Multiple-race data were reported for deaths by 37 states and the District of Columbia in 2010 and by 34 states and the District of Columbia in 2009, and were reported for births (used as the denominator in computing infant mortality rates by 38 states and the District of Columbia in 2010 and by 33 states and the District of Columbia in 2009. Multiple-race data for these reporting areas were bridged to single-race categories of the 1977 OMB standards for comparability with other reporting areas. 3. Rates for 2009 are revised and may differ from rates previously published. 4. Per 100,000 U.S. standard population, based on the year 2000 standard. 5. Life expectancies for 2009 have been updated and may differ from those previously published. 6. Deaths under age 1 year per 1,000 live births in specified group. Today, the major causes of death are heart disease, cancer, COVID-19, accidents, and stroke (CDC, 2021). There are surprising statistics: more than 83 million Americans have a CVD (the total population of the United States is 327 million; U.S. Census, n.d.); 76 million Americans have high blood pressure; nearly 7 million Americans have had strokes (American Heart Association, 2023), and 12 million men and women have some type of cancer (Jemal et al., 2017). Diabetes, an illness that can hasten the onset of CVDs, is a common chronic illness with more than 18 million Americans estimated to have either type 1 or type 2 diabetes (American Heart Association, 2018). In fact, heart disease and stroke account for approximately 65% of deaths due to diabetes (CDC, 2018). Figure 11.3 shows the projected levels of major chronic diseases by the year 2023. == 11.2 - Coping With Disability and Chronic Illness == '''Goals of Treatment''' * Adaptation/coping is key. Can be defined as the affective, mental, and behavior chances that allow a disabled person to embrace their life despite their disability. The five major areas are success in performing daily tasks, reducing psych. disorders, reduce negative affect and improve positive affect, maintain a satisfactory functional status, and experience happiness in other areas of life. '''Quality of Life''' * '''QOL''' measures how well someone copes with chronic illness, considering physical, psychological, and social factors. * Initially assessed by physicians, QOL is now better evaluated by patients themselves based on their experiences with pain, emotional well-being, and functional status. * Tools like PROMIS, PROQOLID, OLGA, and Optum provide assessments for QOL. '''Biopsychosocial components of adaptation:''' # '''Biological Issues''': #* Chronic illnesses like cancer, coronary heart disease (CHD), and diabetes impact physical functioning and often involve pain, necessitating physical rehabilitation. #* Such conditions also influence psychological outlooks. # '''Psychological Issues''': #* Adaptation involves cognitive, emotional, and interpersonal adjustments, with key factors including: #** '''Appraisals''': Seeing illness as a challenge and perceiving social support improves QOL. #** '''Personality''': Traits like optimism and positive affect aid in coping, while depression and anxiety hinder it. #** '''Comparison''': Social comparisons (upward or downward) can impact self-esteem. #** '''Meaning''': Finding meaning in illness can enhance or detract from well-being, depending on context. '''Research Highlights''' * '''Optimism''': Strong predictor of better adaptation, linked to active coping and less distress across illnesses. * '''Gratitude''': Reduces depression and promotes better outcomes. * '''Social Support''': Critical for managing emotional and practical challenges. * '''Cultural Variations''': Different coping mechanisms observed across cultures (e.g., Chinese women with breast cancer). '''Challenges and Strategies:''' * Psychological reactions vary by illness and individual differences (e.g., premorbid mental health, cultural beliefs). * Interventions that foster optimism, gratitude, and supportive relationships are effective in improving QOL. === Raw Textbook Page === Improving one’s diet, refraining from smoking, and consuming minimal alcoholic beverages (see Chapter 7) may help prevent chronic illnesses, but do not guarantee avoiding these illnesses. Goals of Treatment Before we discuss how one can cope with having a chronic illness, it is important to consider some goals for treatment. Science has made many advances in the treatment of cancer and HIV infection, and some research suggests that illnesses such as CHD and diabetes can be reversed (e.g., Campbell & Campbell, 2016; Ornish et al., 1998); however, we still cannot cure these illnesses. Therefore, helping people cope with having these illnesses becomes very important. Adaptation to disability can be defined as a dynamic process of affective, cognitive, and behavioral changes that gradually approach an optimal state of well-being (Livneh & Antonak, 1997). Five major forms of adaptation are the successful performance of daily tasks, the minimizing psychological disorders, low levels of negative affect and high levels of positive affect, good functional status, and the experience of satisfaction in different areas of life (Stanton et al., 2001). Of all of these, the most common psychological outcome studied is the quality of life (Morgan & McGee, 2016). Quality of Life The most commonly used measure of how someone is coping with a chronic illness is a measure of their quality of life (QOL). Sometimes called health-related quality of life (HRQOL) or discussed as well-being, the past 40 years has seen an upsurge in research on QOL (Morgan & McGee, 2016). QOL features prominently in the study of how patients cope with diseases and is important for planning further treatment (Brodsky et al., 2017). QOL was originally a measure made by the physician, purely by whether the disease was present or absent. If the disease presence was strong, it was assumed that QOL would be low. It is now clear that patients are the best judges of their own QOL. Asking patients how much pain they are experiencing and how they feel (e.g., assessing depression and anxiety) is a valuable way to determine how well they are coping (Morrow et al., 2012). Assess your own QOL with one of the most common measures of QOL (Table 11.2). *See Procedures Manual, pages 13–15. Quality of life includes several components. Similar to measures of adaptation, QOL includes a measure of physical status and functioning, psychological status, social functioning, and the presence of the disease- or treatment-related symptoms. A wide array of measures assesses QOL. Figure 11.4 lists some of the major assessment tools, ranging from the generic to the specific, and separated by disease- or patient-specific types. In the age of the internet, can you Google for some? Yes indeed. The good news is you can find a number of credible collections of measures with accompanying psychometric information (Morgan & McGee, 2016). These include the Patient-Reported Outcomes Measurement System (www.promishealth.org), Patient Reported Outcomes and Quality of Life Instruments database (PROQOLID; Emery et al., 2005), the On-Line Guide to Quality-of-Life Assessment database (OLGA; http://www.olga-qol.com), and Optum (http://www.optum.com). A table depicting major measures of quality of life with column heads type, focus, and examples of assessment tools. Source: Reprinted with permission from Assessment in Health Psychology by Y. Benyamini, M. Johnston, & E. C. Karademas, ISBN 978-0-88937-452-2 ©2016 Hogrefe. www.hogrefe.com. Let’s take a look at the different biological, psychological, and sociocultural factors that can influence QOL and adaptation. Biopsychosocial Components of Adaptation Most of us will experience a disability or chronic illness at some point in our lives. However, it is clear that changing health behaviors can greatly reduce the chance of contracting some chronic illnesses (LaCaille & Hooker, 2019; Mermelstein & Brikmanis, 2019). Furthermore, psychological strategies can help one cope with chronic illness. For example, in a longitudinal study of patients with inflammatory bowel disease and arthritis, patients who displayed gratitude were less depressed later in the study (Sirois & Wood, 2017). In fact, feeling grateful was a significant predictor of lowered depression even after controlling for other psychological variables such as illness cognitions discussed in Chapter 10. Similarly, two other psychological variables, optimism and hope, are strong aids to helping patients cope with chronic diseases (Schiavon et al., 2017). Adaptation to chronic illnesses has many different components. Patients need to cope with not only their own affect, behaviors, and cognitions concerning the illness but also with revising their lifestyles to accommodate the treatment and coping with how others in their social networks respond to them because of their illness (Day, 2019; Hoyt & Stanton, 2019). They may experience many different feelings including anxiety, depression, and frustration, and may not be able to perform activities they used to, such as going to work or shopping for groceries. Daily tasks, changing symptoms, and fluctuating emotions can be overwhelming (Emery, 2019). There are numerous challenges to the process of integration; successful self-management with psychosocial, vocational, and existential support is critical. Next, I will discuss some of the different components of adaptation using the major approaches in health psychology. Biological Issues Biologically, different chronic illnesses will have different courses. For example, coronary heart disease (CHD) and cancer, the two leading causes of death for Americans, inflict significant changes in the body. Cancer causes cells to grow uncontrollably, harming surrounding tissue and limiting normal function. In CHD, the blood vessels around the heart are clogged with plaque and fat, changing blood flow and possibly leading to a heart attack. Other chronic illnesses such as diabetes and asthma similarly have physiological correlates, such as changes in insulin sensitivity and the blocking of breathing channels (Kalyva et al., 2016). The slow physiological changes limit functioning in many areas and are often accompanied by pain (Hoyt & Stanton, 2012). Consequently, physical rehabilitation is a big component of any treatment of chronic illnesses. The loss of function and increase in pain also have major consequences for how the patient views the world, and psychological issues need to be considered as well. Psychological Issues There has been growing interest in the role of psychological factors in adaptation to chronic illnesses and disability (Samson & Siam, 2008). In a review of both theoretical and empirical literature on adjusting to chronic illnesses, Stanton et al. (2001) identified two key multidimensional psychological aspects. First, the individual has to go through adaptations, which include cognitive aspects such as intrusive thoughts and changing views of the self, emotional aspects such as depression and anxiety, and behavioral and physical aspects such as dealing with pain or not being able to perform daily activities. Second, the person must make interpersonal adaptations, negotiating personal relationships with friends and family as well as professional relationships with health-care providers. Positive adaptation includes the mastery of illness-related tasks, the minimizing psychological disorder and negative feelings, perceptions of high quality of life, and the maintenance of adequate functional status and social roles (Hoyt & Stanton, 2019). Perhaps one of the most effective psychological resources that a person with a chronic illness or disability has is their mental approach to the situation and appraisals (see Chapter 5). Patients’ primary and secondary appraisals of the illness can correspondingly influence how they fare. If the illness is seen as a challenge (primary appraisal) and they believe they have a lot of social support to cope with it (secondary appraisal), they will probably have a higher QOL (Gatchel & Oordt, 2003b). For example, in a study of colorectal cancer-specific concerns in a population-based sample of colorectal cancer survivors, patients’ threat appraisals significantly predicted their quality of life up to 2 years after treatment (Steginga et al., 2009). A number of health psychologists have modified cognitive appraisal theory from its original context (i.e., stress) and have adapted it to help explain coping with chronic illnesses such as arthritis, breast cancer, prostate cancer, and AIDS (Merz et al., 2011; Schwartz & Rapkin, 2012). In fact, new work teases apart appraisals from personality factors. The Quality of Life Appraisal Profile–Version 2 (QOLAPv2) helps assess individual differences and is useful in explaining why people experiencing very different health states may report the same QOL (Rapkin et al., 2017). First developed with 4,173 respondents, the QOLAPv2 is useful across populations and provides better predictions of QOL than measures of personality alone. A caring healthcare nurse conducting physical therapy exercises with a senior adult patient at home. The nurse is assisting the patient who is holding dumbbells with her hands raised. Physical Therapy. Physical therapy is an important part of coping with the biological aspects of chronic illnesses, but mobility can influence the psychosocial aspects as well. Another common psychological reaction to a positive test result or even experiencing symptoms of a chronic illness is anxiety. Anxiety interferes with healthy functioning, causing a person to cope poorly and to delay the recognition and reporting of symptoms. Anxiety is often high when the patient is waiting for test results, receiving a diagnosis, and awaiting invasive medical procedures. Not knowing about the course of the illness or not having enough information about what the illness entails is especially anxiety provoking. Such lack of information-induced anxiety is more pronounced in populations of lower SES and in some ethnic groups. The most common negative reaction to a chronic illness is depression (Giardini et al., 2017). Depression can be either biological or psychological in nature and often goes undiagnosed because its symptoms are shadowed by the symptoms of the chronic illness. Unlike anxiety, depression tends to be a long-term reaction and increases as pain increases. When patients get depressed, they are less motivated to cope actively with the illness, and tend to interpret any bodily change negatively. Interestingly, the most important predictor of one’s mental health after acquiring a disability or chronic illness is their premorbid mental health. The form of psychological reaction varies also depending on the illness and varies considerably across individuals with the same illness. Personality factors, the amount of social support one receives or perceives to have, and cultural beliefs surrounding the illness can all influence coping with the illness and can alleviate depression and anxiety. Chapter 6 included details about the ways different personalities influence coping. The same relationships that link stress and coping link chronic illnesses and coping. The Big Five personality variables (conscientiousness, agreeableness, neuroticism, openness to experience, and extroversion; see Chapter 6) have been linked to coping in general (Smith, 2019) and coping with chronic illnesses in particular (Sirois, 2015). Similarly, being high in positive affect is also a good thing for those with chronic illnesses. Positive affect was significantly associated with having a lower risk of dying from any cause (i.e., all-cause mortality) in people with diabetes (Moskowitz et al., 2008). In a study of the role of religious involvement, spirituality, and physical or emotional functioning in a sample of African American men and women with cancer, positive affect was a key factor in predicting better adaptation (Holt et al., 2011). Optimism is another powerful personality characteristic in coping with chronic illnesses (Giardini et al., 2017). Carver et al. (1993) first demonstrated convincingly the role of optimism in women coping with breast cancer. When measured before surgery, the optimistic women were those using more active coping and facing the disease, and those with less distress. This pattern held for three further assessments at 3, 6, and 12 months after surgery. Optimism is also helpful in coping with diabetes mellitus, rheumatoid arthritis, and multiple sclerosis (Fournier et al., 2002a), breast cancer (Sohl et al., 2012), coronary bypass surgery (Tindle et al., 2012), and HIV infection (Peterson et al., 2012). Building optimism can go a long way. For example, falling, common in older chronically ill adults, predicts poorer physical health and greater negative emotions among the group (Ruthig et al., 2007). Falling also causes drops in optimism, which mediates the effects of falling on health and well-being. Recovery from falling can be enhanced by bolstering optimism (Ruthig et al., 2007). In general, different personality characteristics can greatly help coping (Smith, 2019). Another important component of psychological coping is related to how patients compare themselves with others with the disease and how much meaning they derive from the illness. For example, studies on upward and downward social comparison show that people can sometimes compare themselves with those better off than they are (“Boy, my coworker has the same problem, and he is doing so much better than I am.”) or worse off than themselves (“Oh, at least I am doing better than my neighbor who has the same illness.”). Women who cope better with breast cancer make comparisons with people who are faring less well than they are to enhance their own self-esteem (Wood et al., 1985). Chinese women facing breast cancer were also found to make the best of it. The essences of Chinese women’s experiences were that they faced the reality of the cancer diagnosis, took an active part in the cancer treatment, sustained an optimistic spirit, maintained physical activity, reflected, and then moved on (Fu et al., 2008). Finding meaning in your illness can often be beneficial, leading to lower mortality and morbidity (Hooker et al., 2018), but in some cases it can be detrimental to well-being as well. Originally, research documented that finding meaning in your experience can lead to positive well-being and better adaptation to the disease (Taylor, 1983). There are some important qualifications to this early finding. Tomich and Helgeson (2004) examined the consequences of finding meaning (they called it benefit finding) on QOL in 364 women diagnosed with stage I, II, or III breast cancer. Benefit finding and QOL were measured 4 months post diagnosis (Tl), 3 months after Tl (T2), and 6 months after T2 (T3). Women with lower socioeconomic status, minority women, and those with more severe levels of the disease perceived more benefits at baseline. Benefit finding was associated with more negative affect at baseline and also interacted with the stage of disease, such that negative relationships to QOL across time were limited to those with more severe disease. Findings suggest that there are qualifiers as to whether finding something good in the bad is, in itself, good or bad (Yanez et al., 2011). We discuss this further in Chapter 13. == 11.3 - Culture, Community, Chronic Illness, and Disability == The text explores how chronic illness and disability intersect with various social and cultural factors, influencing how individuals cope and adapt. Marginalized groups, such as those who are older, low-income, or from ethnic minorities, face unique challenges that are compounded by social discrimination. The environment, including family dynamics, neighborhood safety, and social support, significantly affects the management of chronic illnesses. Cultural beliefs and practices also play a role, with different groups relying on religion, community support, or traditional medicine for coping. Social support, both formal and informal, is crucial but must be tailored to individual needs, as excessive or mismatched support can sometimes be counterproductive. The importance of close relationships, such as family and community, is highlighted as a sustainable approach to managing chronic illnesses. Additionally, interventions like support groups and positive psychology strategies offer varying degrees of effectiveness, underscoring the need for personalized approaches to care. === Raw Textbook Page === Disability intersects with all other social groups, and in fact, is more likely to occur within other marginalized groups. Disabled people are more likely to also be older, female, Black, Hispanic, or indigenous, LGBTQ+, and low income (Okoro et al., 2018; Varadaraj et al., 2021). A person’s intersectionality has many implications for how they cope with chronic illnesses or disability. Jose, who lives with a large extended Mexican American family, is going to cope with a diagnosis of cancer differently from how Joshua copes, who lives alone and far away from his European American family. Jessica, a devout Catholic, may face breast cancer very differently from Carmel, who is agnostic. Friends, family, and society can make a big difference in how one copes. If you get a chronic illness that is stigmatized in your society, you are likely to be discriminated against for having the disease, and this discrimination can negatively influence your ability to cope with it. Family and Neighborhoods The environment in which you live can accentuate a disease or help control it (Gurung et al., 2004). Stressful events influence anxiety levels, thereby influencing adaptation to the disease (Lepore & Evans, 1996). In a major review of the ways that sociocultural factors can affect a patient, Taylor et al., (1997) traced the different ways that unhealthy environments—stressful work or family situations, living in a neighborhood with a high crime rate, being unemployed, or having multiple chronic burdens—can reduce social support and hurt adaptation to illness. As shown in Figure 11.5 each of these different elements plays a role in influencing perceptions and the availability of coping resources. The importance of social factors such as the family and community structures increases when the person with the chronic illness is a child (Lyon et al., 2011). For example, the family dynamics can change significantly when a child is diagnosed with diabetes. Some families become more protective and controlling when an adolescent has diabetes (if you thought an early curfew was bad when you were young, imagine your parents wanting complete control over what you eat and drink). In such situations, families may get overtaxed and summon help from the extended family, neighbors, or the community. The neighborhood may be key, as a supportive community is proven to be advantageous (Waverijn et al., 2017). Statistics show that adolescents living in dangerous neighborhoods are more susceptible to engaging in risky behaviors, hence accelerating the course of their chronic illnesses (Obeidallah et al., 2001). Chronic Illness and Ethnicity Your cultural environment is important as well. The experience and outcomes of an illness are shaped by cultural factors that influence how it is perceived, labeled, and explained, and how the experience is valued (Broadbent, 2019). For example, African Americans with chronic illness have poorer outcomes than European Americans in the United States (Lederer et al., 2008). Therefore, we actually learn ways of being ill that depend on our cultural backgrounds (see Chapter 3). Someone coming from a self-reliant farm family may be taught to downplay illness and put on a brave face and keep on working. Someone else who grew up in a city may be more likely to follow the complete bed rest prescriptions of a doctor. Both the patients’ and the providers’ cultural approaches to the source of disease and illness affect patients’ care-seeking behavior and treatment opinions, choices, and compliance (Turner, 1996). In the past few years, practitioners have been sensitized to the role of cultural factors, especially acculturation, in rates of life-threatening chronic illnesses. For example, when treating migrants, practitioners are now more aware of casual factors of illnesses such as having to deal with changing diets and stress from their new environment, stressors that often lead to certain diseases like obesity and prostate cancer (Jasso et al., 2004). There is also extensive research on the role of linguistic competency (e.g., Ngo-Metzger et al., 2003) and ethnic match of patient and practitioner in coping with chronic illnesses (e.g., Tarn et al., 2005; see Chapter 9 for more on this topic). Some cultural groups react to chronic illnesses differently from others (Galanti, 2014). Many collectivist groups see chronic illnesses as something that an entire family or community, not just the individual, is responsible for and has to cope with. In the last section of Chapter 9, we discussed how the Hmong family rallied around the sick child with epilepsy and endured personal hardships to take care of her. There are similar cultural patterns across different religious and ethnic groups. For example, many church groups have organized programs to take care of chronically ill worshippers. Are there cultural differences in how ethnic groups cope with specific illnesses? Research in this area is growing. For example, Culver et al., (2004) tested for differences in coping responses in middle-class African American, Latinas, and European American women with early stage breast cancer. They found only two differences in coping (controlling for medical variables, education, and distress). Compared with European American women, the other two groups both reported using humor-based coping less and religion-based coping more. There was one difference in how coping related to distress: Venting related more strongly to elevated distress among Latinas than among non-Latinas (Culver et al., 2004). Religion (as seen in coping with breast cancer) plays a key role in understanding cultural differences in coping with chronic illnesses (Park & Carney, 2019). Spirituality in particular plays an especially strong role in the self-management of chronic illness among older women (Harvey, 2008). In a study directly testing the role of religion in coping with pain and psychological adaptation, Abraído--Lanza et al. (2004) found that Latinos with arthritis reported using high levels of religious coping. Further analysis indicated that religious coping was correlated with active but not passive coping and directly related to psychological well-being. Passive coping was associated with greater pain and worse adjustment. Findings such as this, together with similar work in other ethnic groups, such as African American (Holt et al., 2011), suggest that interventions and community-based outreach approaches should embrace an appreciation for expressions and experiences of spirituality for both patients and caregivers. The cultural group’s beliefs about health and illness are important as well (Arellano-Morales & Sosa, 2018). For many chronic medical problems, a patient’s coping behaviors and adherence to treatment will depend on the quality of the patient–practitioner interaction (see Chapter 8). Some earlier studies suggest that patients whose beliefs favor folk medicine are healthier when they seek treatment from folk healers rather than biomedical doctors (Kleinman et al., 1978; Mehl-Medrona, 1998; also see Chapter 3). This could be because of the corresponding belief systems as well as the relative closeness in social class between patient and practitioner. In other cases, it may be because the doctor’s own cultural identity may influence how they treat a patient from a similar culture (Gurung & Mehta, 2001). In many folk and traditional medical systems, a greater emphasis is also placed on communication, which can increase patient satisfaction and adherence to treatment. Prejudice and discrimination account for many negative outcomes for certain cultural groups. Lederer et al. (2008) conducted a retrospective cohort study of 280 non-Hispanic African American and 5,272 non-Hispanic European American adults age 40 years and older with chronic obstructive pulmonary disease (COPD). The patients were listed for lung transplantation in the United States between 1995 and 2004. After listing for lung transplantation, African American patients were less likely to undergo transplantation and more likely to die or to be removed from the list compared with the non-Hispanic European American patients. Unequal access to care may have contributed to these differences. African Americans in the study were more likely to have pulmonary hypertension, to be obese and diabetic, to lack private health insurance, and to live in poorer neighborhoods. Social Support Compared to individuals without disabilities, those with disabilities are more likely to experience social isolation and less likely to report obtaining enough social support (Bryson et al., 2019; Havercamp et al., 2004; Krahn et al., 2015). One strategy for enhancing support is through support groups and group psychotherapy, which have been shown to have positive effects on a range of psychosocial outcomes (Bardon et al., 2022). For those who have only recently been diagnosed with a disability or illness, support groups may be very helpful in fostering knowledge, coping mechanisms, and self-efficacy. Empirical studies and reviews show that people with more social support have more positive adjustment to chronic illnesses. Illnesses studied ranged from cancer (Rogers et al., 2017) to rheumatic diseases (Shim et al., 2017). Having a socially supportive environment often makes the patient more actively cope with the illness and less likely to disengage and get worse (Rosland et al., 2012). In the case of a chronic illness such as coronary heart disease, social network size and having a stressed partner can influence morbidity and mortality by influencing whether patients attend rehabilitation (Molloy et al., 2008). Social networks also help maintain quality of life and are particularly important for low SES individuals (Barden et al., 2016; Ruiz et al., 2019). Groups do not always work for everyone. Helgeson et al.(2000) determined the extent to which individual difference variables moderated the effects of an information-based educational group and how an emotion-focused peer discussion group helped women with breast cancer. Women who needed outside support (e.g., did not have strong personal connections) benefited the most from the educational group, and peer discussion groups were helpful for women who lacked support from their partners or doctors. Surprisingly, however, too much support can be detrimental. Helgeson et al. (2000) found that the discussion groups were harmful for women who already had high levels of personal support. These groups frequently use a medical model approach, taking place in clinics under the direction of professionals in the medical or mental health fields. However, there is also a need for less formal ways to connect and socialize with the disability community. Additional studies have focused on the experience of connecting with others with the same disability. Because it is unlikely that people with rare disorders will encounter others in their local community with the same disability, support conferences are sometimes offered by patient organizations as a way for people to connect. In a study of individuals with the rare condition Moebius syndrome, attending conferences more frequently was linked to higher social support and lower stigma (Bogart & Hemmesch, 2016). In surveys of people with physical disabilities, Silverman et al., (2017) found that nearly half of participants did not have any friends who also had physical disabilities. However, friendships between people with the same disabilities or friendships across different types of disabilities were linked to greater life satisfaction. There was additional evidence that friendships among people with disabilities reduced the negative effects of functional impairment on wellbeing, perhaps by learning adaptive strategies from each other. The cross-disability findings are particularly encouraging since people with disabilities, even rare ones, can benefit from relationships with persons who have any disability. '''Hospital Chapel'''. Many hospitals have chapels such as the one shown here. Caregivers can come and pray for their loved ones and even patients who are too sick to go to church can go say a prayer. There are some important cultural differences in how social support is used (Taylor et al., 2007; Wong & Lu, 2017). For example, a review of studies on culture and social support shows that Asians and Asian Americans are more reluctant to explicitly ask for support from others than are European Americans (Kim et al., 2008). This is likely due to their concern about the potentially negative relational consequences of such behaviors. Asians and Asian Americans are more likely to use and benefit from forms of support that do not involve explicit disclosure of personal stressful events and feelings of distress. A number of reviews provide important insights into interventions to help people cope with chronic illnesses. In one recent review, Martire and Helgeson (2017) suggest family members are the most important aid in children’s and adults’ illness management. Evidence suggests a dyadic approach to chronic illness management that targets the influence of close relationships may be the most helpful and sustainable method to affect patient behavior. Specifically, dyadic approaches aimed at helping patients and family members to find ways to set goals together may best benefit family members who are ill or are at risk because of poor health behaviors. Ghosh and Deb (2017) systematically reviewed positive psychology interventions in chronic physical illness and found writing is the most commonly used method for administration (see emotional expression in Chapter 6). Positive psychology interventions are considered feasible and acceptable by patients, but findings about their usefulness are inconclusive. == 11.4 - Coping With Terminal Illness and Death == '''Introduction''' '''Chronic and Terminal Illnesses:''' * Chronic illnesses are treatable but often not curable, with some being fatal (e.g., certain cancers, also called terminal illnesses). * Facing terminal illnesses presents significant psychological challenges, requiring patients to confront their mortality and adjust their outlook on life. '''End-of-Life Care:''' * Effective end-of-life care addresses emotional, psychological, and logistical challenges faced by patients and their families. ** '''Hospital Challenges:''' Issues like fragmented care, strict visiting hours, and excessive administrative processes can be distressing. ** '''Support Measures:''' Clear communication, informed consent, psychological counseling, and preparation for death are critical. ** '''Family Support:''' Families need help with grief, caregiving stress, and emotional communication, such as saying goodbye. '''Role of Religion and Spirituality:''' * Religion, an integral aspect of culture, often becomes more important at the end of life and serves as a coping mechanism. ** Religious practices and beliefs can aid in pain management, emotional resilience, and acceptance of death. ** Studies show correlations between religious coping (e.g., prayer, rituals) and psychological well-being in patients, though causation is not established. ** Cultural and ethnic diversity in religious beliefs highlights the need for tailored spiritual care. '''Perspectives Across Religions:''' * '''Christianity (Catholicism):''' Suffering is tied to original sin; death is seen as liberation of the soul. Rituals like last rites provide solace. * '''Islam:''' Death is the soul’s detachment from the body, viewed as a blessing. Preparing includes settling debts, penance, and cleanliness, with a preference for dying at home. * '''Hinduism:''' Death and suffering are part of the karmic cycle; liberation comes from transcending this cycle. Karma’s role reduces anxiety about death. * '''Buddhism:''' Death is inevitable; fear of death reflects unfulfilled life. Contemplating mortality helps reduce fear and fosters healthier attitudes toward life and death. * '''Other Traditions:''' Irish wakes celebrate life with joy, while Sikhs see death as an opportunity for soul cleansing and reunification with the creator. '''Broader Implications:''' Religious and cultural practices help frame death positively, emphasizing liberation, spiritual growth, and joy. This understanding should guide health psychologists and caregivers to use holistic and culturally sensitive approaches to end-of-life care. '''Death Across the Life Span''' ==== Physiological and Psychological Changes in Dying Patients ==== * Near the end of life, patients often experience physiological changes, such as incontinence, reduced digestive function, increased pain, memory problems, and difficulty concentrating. * High doses of morphine, often self-administered, are used to manage pain. * Emotional and social difficulties arise as family and friends may struggle to interact with the patient in this vulnerable state, compounded by societal taboos around discussing death. * Death education, particularly for younger populations, is critical in helping people cope with loss. ---- ==== Ethical Dilemmas in End-of-Life Care ==== * '''Euthanasia:''' Active termination of life via a lethal drug. * '''Assisted Suicide:''' Physician provides a lethal drug but does not administer it. * '''Passive Euthanasia:''' Withdrawing life-sustaining treatment and allowing the disease to progress naturally. These practices are controversial, with ethical, moral, religious, and societal implications. ---- ==== Notable Cases and Debates ==== # '''Terri Schiavo Case (2005):''' #* Persistent vegetative state for 15 years. #* Conflict between her husband (advocating for life support removal) and her parents (wanting to continue life support). #* Highlights challenges in determining consciousness, psychological pain, and ethical considerations. # '''Jack Kevorkian:''' #* Assisted in 130 suicides, sparking national debates and legal reforms. #* His actions influenced legislation, such as Oregon's 1994 legalization of physician-assisted suicide. # '''Medical Advances:''' #* A vagus nerve implant in 2017 partially restored consciousness to a patient in a persistent vegetative state, complicating future decision-making. ---- ==== Perspectives and Advocacy ==== * '''Religious and Disability Groups:''' ** Many religious groups oppose euthanasia on moral grounds. ** Organizations like ''Not Dead Yet'' argue that euthanasia and assisted suicide discriminate against disabled and elderly individuals, emphasizing the need for better palliative care and support systems. * '''Patient Autonomy:''' ** Respect for the patient’s wishes, often expressed through advance directives or living wills, is a cornerstone of ethical medical care. ** Family members play a significant role but may misinterpret or override patient preferences. ---- ==== Palliative Care and Ethical Considerations ==== * Palliative care focuses on alleviating suffering without directly treating the cause. * Decisions to withdraw life support are easier when advance directives are in place, yet families and healthcare providers may still face challenges aligning their actions with the patient’s wishes. ---- ==== Key Questions for Decision-Making ==== * Is the patient’s condition reversible? * Does the patient have an advance directive or living will? * What is the role of family in interpreting the patient’s wishes? * What societal, religious, or cultural factors influence the decision? * Can palliative care adequately address pain and suffering? The interplay of these considerations highlights the complexity and sensitivity of end-of-life decisions. === Simplified Summary of the Stages of Death and Cultural Perspectives on Death and Dying === '''Stages of Death by Kübler-Ross:''' * Elisabeth Kübler-Ross proposed five stages of dying: '''denial''', '''anger''', '''bargaining''', '''depression''', and '''acceptance'''. * These stages are not universally sequential or empirically supported; people may experience emotions dynamically and not in this order. * Factors such as culture, social support, and disease progression influence individual experiences of death. '''Cultural Perspectives on Death:''' * Cultural practices surrounding death differ significantly: ** '''American Indians''': Use sage-burning rituals for spiritual preparation, often conflicting with hospital policies. ** '''Muslims''': Follow strict protocols like facing the body toward Mecca and reciting the Qur'an. ** '''Ghana''': Unique coffin shapes as part of burial traditions. * '''Health Practitioners''': Need to respect cultural practices for effective support, despite challenges due to ethnocentric biases or institutional limitations. '''Gender Differences in Grief:''' * '''Women''': Tend to openly express emotions and seek social support. * '''Men''': Less likely to express emotions, leading to greater health consequences during grief, including higher rates of depression and illness. * Widowers face more challenges than widows, both mentally and physically, during acute grief periods. '''Health Practitioner Guidelines:''' * Be culturally sensitive and informed about the unique needs of different groups. * Understand that cultural mistrust may affect relationships, especially with groups like African Americans and American Indians. * Tailor support approaches to include socioeconomic status, religion, and spirituality. Understanding death as a universal yet deeply individual and cultural experience is crucial for providing compassionate care. === Raw Textbook Page === Discuss the role of culture and religion in grief and death practices. Many chronic illnesses are treatable but not curable and some can be fatal. Some chronic illnesses, including certain types of cancer, are often referred to as terminal illnesses because people with these diseases often die after a relatively short time (although this time can range from months to a few years). Facing the reality of approaching death is an even bigger psychological challenge and can lead to many changes in outlook (Danel et al., 2017). What can be done to make the end of life easier? Problems with communication and visiting hours and too many administrative details can be trying to the patient and family members. Care is highly fragmented in a hospital and multiple caregivers pass through a patient’s room daily and nightly, monitoring the patient and performing different tests. In general, care should be taken to counter the effects of hospitalization (Buzgova et al., 2016). Patients and their families arrive with anxiety, and emotions are high if death is imminent. Health-care practitioners need to be explicitly prepared to address these issues. In particular, informed consent procedures should be closely followed whereby patients are told of their condition and the treatments available, if any. Patients should also be helped to accept their situation and prepare for death. This normally means helping patients to use their remaining time well. Psychological counseling should be made available for both the patient and their family. The patient may need help in facing death and in making sense of life. The family may need help to cope both with their grief and with the strain of caregiving. Both the patient and their family also may need help communicating with each other, in saying goodbye, and in dealing with sometimes conflicting needs (the patient fearing death and the family unable to imagine living without the patient). Religion has a long history of being included in studies of health (Koenig, 2015; Levin & Schiller, 1987). Research supports the link between religion and health (Park & Carney, 2019; Von Dras, 2017) although the bulk of the studies are correlational in nature. One of the most salient aspects of culture, religion is intrinsically tied to the other major elements of culture such as race and ethnicity. It is important to keep in mind the diversity of religious beliefs between and within groups of people. For instance, people from different races and ethnicities often have different religious beliefs. Also, even though North America is primarily Christian, there are still a significant number of North Americans who have non-Christian beliefs. Regardless of beliefs, turning to one’s spirituality can be a form of coping and can even help with pain management. However, this link is also an example of the differences between a correlational study and an experiment. Simply because people who are religious are also usually people who cope better with illness, does not allow us to conclude that religion causes better health. Nonetheless, what is important is that religion can help, and in the context of chronic and terminal illnesses, health psychologists should use any tool that can make a difference. One of the growing number of studies on different ethnic groups illustrates this point well (Tovar, 2017). Abraído-Lanza et al. (2004) tested for the link among religious, passive, and active coping, pain, and psychological adjustment in a sample of 200 Latinos with arthritis. The participants reported using high levels of religious coping that was correlated with active but not with passive coping. Religious coping was directly related to psychological well-being. Passive coping was associated with greater pain and worse adjustment (Abraído-Lanza et al., 2004). Traditional Latinos tend to be very religious, practicing Catholicism, curanderismo, or, more often, a blend of both (as discussed in Chapter 3). With the growing number of North Americans who are Latino, the findings of Abraído-Lanza et al. (2004) and other such studies suggest a greater focus on the role of ethnicity and religion on coping. A person’s religious beliefs often become more important as the end of life nears. Religious beliefs vary regarding the role of pain and of the role and significance of death. Different religions even have different ways to treat death and distinct ways to treat the lifeless human body. The deeper your faith, the more likely you are to turn to it if you or someone you love is dying (Boucher, 2017). The way death is treated within a culture can influence how well the patient copes (Corr & Corr, 2007; Doughty, 2017). For example, for devout Catholics, suffering is related to original sin and a Catholic has to face suffering like Christ did. Death is the freeing of the soul to the father who is in heaven. A righteous, well-lived life serves as preparation for death. As death draws near, the family and the terminally ill patient can draw solace from the visitation of a priest who will help the patient finalize their earthly affairs (Büssing & Poier, 2017). There is a final confession of sins, receiving of Holy Communion (a piece of bread or wafer that represents the body of Christ), an anointing, and a last blessing, known as the last rites. This scripted ritual goes a long way to help the terminally ill patient and family come to terms with the impending loss. Muslims, or followers of Islam, see death as the termination of the soul’s attachment to the body (Amer, 2017). Death is a blessing and a gift for the believer. To prepare, the person must do penance and be careful to not be under obligation to any other human being—the patient should make sure that they pay any dues or debts owed. Cleanliness at the time of death is more important than at any other time. Especially important is the edict that the seriously ill must die at home. To Hindus, suffering is part of maya, or illusion (Agnihotri & Agnihotri, 2017). The only way to transcend suffering is to be free from the cycle of birth and death and rebirth. The Hindu tries to work off bad karma from an early point in life and the place of your karmic cycle is indicated by your status in the world; for example, if you sinned in your past lives you will be reborn as an animal or, even worse, as a worm. This belief in predetermined fate helps reduce the anxiety of death. Other Eastern religions share some of these beliefs. Buddhists speak of and contemplate death often, in stark contrast to many non-Buddhists (Colgan et al., 2017). Pain is unavoidable, but attitudes and behavior influence suffering. According to Buddhists, the only way to avoid suffering is to free the self from desire, which is the cause of suffering. The Buddhist believes that as long as there is fear of death, life is not being lived to its fullest. Contemplating death can free us from fear, change the way we live and our attitude toward life, and help us face death healthily. The sense that death can be joyous is also reflected in different religious traditions (Doughty, 2017). The Irish wake is often a rousing celebration of the recently departed’s life and is accompanied by drinking and dancing. To the Sikhs of India, death is seen as a great opportunity to do something we put off all our lives. It is a chance to cleanse the soul of psychic fantasy, and life is an opportunity to practice dying, until one dies a death that will not have to be repeated. Death is not sad; friends chant and sing hymns near the dying to set a peaceful vibration and inspire the dying person to be in the best frame of mind. Hence, many religions downplay the sadness of death and emphasize the happiness coming from freedom and the unification with the creator. Death Across the Life Span Different cultural groups face death differently, whether it is one’s own death or the death of a close friend or family member (Corr & Corr, 2007; Noppe & Noppe, 2008). In addition, it matters how and when death arrives; the major causes of death vary across the life span. How and when death occurs automatically influences how survivors cope. Focusing on how we develop is important in understanding why mortality and morbidity varies across age groups. The main causes of death are not the same at each point of the life cycle (see Table 11.3). Congenital anomalies are the leading cause of infant deaths in the United States. Other major causes include complications from low birth weight (LBW), sudden infant death syndrome, and problems from labor, delivery, and other maternal complications (National Center for Health Statistics, 2017). The trauma of losing a child and the way parents cope can vary with the ethnic group and the expectations they have, and many other cultural variables can play a role. Many Mexican American mothers tend to have higher levels of stress during pregnancy. The idea that pregnancy is stressful is actually rooted in the culture: the Spanish word for labor is dolor, which also is the root for the words sorrow and pain. Other cultural variables beyond ethnicity can be important too. Many poor families do not have the health insurance or facilities to receive good prenatal care. Injury and accidents are the leading causes of death for children, adolescents, and young adults (National Center for Health Statistics, 2017). As we age, we succumb to more diseases that are related to unhealthy behaviors. The top three killers of adults (age 25 to 55) and older adults (age 55 and older) are coronary heart disease, cancer, and stroke (National Center for Health Statistics, 2017), each exacerbated by eating badly, smoking, and overindulging in alcohol. Aging should not always be associated with illness and sickness either. In one particularly impressive longitudinal study, researchers (clearly not the same ones) observed 1,500 Californians over 80 years. The study showed that eating well, getting physical activity, and giving and receiving a lot of social support are some of the factors that can provide many happy years of life for older adults (Friedman & Martin, 2011). There were also some surprises: Some of the people who worked the hardest, lived the longest, getting and staying married was not directly associated with longer lives for women, and the traits most associated with thriving were persistence and being prudent. The physical deterioration of cells associated with aging is related to specific diseases in adulthood. For example, many older adults experience marked problems in thinking and remembering or dementia. The most common problem is one that you will have heard about: Alzheimer’s disease, a degenerative disease of the brain that leads to dementia, makes everyday tasks like grocery shopping difficult. Other major causes of dementia include Parkinson’s disease and stroke. These differences in causes of mortality indicate how different areas of health psychological research will be applied at different times of the life cycle. Prevention of injuries should be a major goal when intervening with children, a decrease in unhealthy behaviors is a pertinent goal for children and young adults and help in coping with chronic and terminal disease is a critical goal for older persons. The Path to Death As the moment of death approaches, some explicit physiological changes accompany the different psychological stages. Dying patients often experience incontinence, losing control of their bladder and other bodily functions. In particular, patients may be unable to control their salivation and will not be able to feed themselves or even eat solid food as their digestive systems reduce functioning. With cancer or CHD, there is often an increase in pain and medical practitioners prescribe high doses of morphine, even putting the delivery of morphine under the patient’s own control. In this way, patients can self-administer medication to alleviate their pain and suffering. Patients may experience severe memory problems or have problems concentrating. Interactions with caregivers and hospital staff can become difficult, which can lead to misunderstandings and miscommunication. Friends and family often have trouble facing the patient in this state, and the patient may not want to be seen by anyone. Talking about death is taboo among many North Americans, so the last few days of a patient’s life can be very difficult as visitors and even medical personnel do not always know how to approach the topic. Consequently, death education is an important area of research. Given the number of tragedies on college campuses, there is a special emphasis on helping students cope with grief (Cox et al., 2015; Cupit et al., 2016). Facilitating Death Should life be terminated if a patient is in tremendous pain and discomfort or is comatose? This is one of the most controversial ethical issues regarding health care. Euthanasia, physician-assisted suicide, and the withdrawing of life-sustaining treatment (sometimes called passive euthanasia) are some of the most difficult moral and ethical dilemmas we face today. Euthanasia is the termination of life by the injection of a lethal drug. Assisted suicide involves a physician supplying a lethal drug while not actually administering it themself. When life-sustaining treatment is withdrawn, the underlying disease takes its own course. All are subjects of intense national debate (Bulmer et al., 2017). One of the most famous scenarios involves Terri Schiavo. By 2005, Schiavo, a Florida woman, had been in a persistent vegetative state for 15 years. Her husband, Michael Schiavo, battled her parents over whether his wife should be allowed to die. He argued that because she was brain dead it would not be fair to keep her alive. Terri Schiavo suffered heart failure from a potassium imbalance in 1990. Her husband said his wife told him that she would not want to be kept alive artificially. Doctors who testified on behalf of Michael Schiavo said that his wife had no hope for recovery. She was fed through a tube but breathed on her own. Terri Schiavo’s parents, Bob and Mary Schindler, maintained that their daughter could be helped with therapy. After years of litigation and appeals, Terri Schiavo’s feeding tube was removed in October 2004, only to be reinserted 6 days later after the Florida legislature, in emergency session, passed a law that affected only Terri Schiavo. The legislation gave Governor Jeb Bush the power to intervene in the case, and he ordered the feeding tube reinserted. In early 2005, the tube was removed again and Terri Schiavo died on March 31, 2005, of starvation and dehydration. What should have been done here? Should she have been kept alive? Was Terri conscious of the world around her? Did she experience psychological pain by being kept alive? Her unresponsive state made it difficult to answer any of these questions. Ripped from the headlines: “Nerve implant ‘restores consciousness’ to man in persistent vegetative state.” Not ripped, the actual headline. And I’m not making it up. This is exactly what The Guardian reported in September 2017. Physicians fitted a man in a coma for 15 years (yes, same as Terri Schiavo when she died) with a neural implant that stimulated the vagus nerve. He started to track objects with his eyes and began to stay awake (Devlin, 2017). You can bet all future cases are going to be even tougher to decide. Apart from this now classic case, the most publicized event that put these procedures into public consciousness was Dr. Jack Kevorkian’s assistance in 130 suicides since 1990. This Michigan doctor helped patients end their own lives even in the face of threats to his own life. Three juries refused to convict him despite a Michigan statute established for that purpose. The uproar surrounding his case led to intense political movements with advocates on both sides of the issue, and Kevorkian was imprisoned. He served 8 years of a 10- to 25-year sentence for second-degree murder, was released in 2007 and even ran for Congress, but lost. He died of liver disease in 2011 without assistance. In 1994, Oregon became the first state to legalize forms of assisted suicide. Be warned; how surveys are worded can influence attitudes toward assisting death (Magelssen et al., 2016). While some are against assisted suicide and euthanasia because of religious reasons, Not Dead Yet (notdeadyet.org), an organization of disability activists, opposes euthanasia, assisted suicide, and passive euthanasia, because they argue these are deadly forms of discrimination against disabled and older people. In a society full of suicide prevention efforts, disability is the one community these efforts may not include. This sends a message that disabled lives are not worth living. Assisted suicide, more often than not, may end up being a de facto alternative to suffering caused by delays or denials in health care and palliative care, Not Dead Yet argues. Society should shift focus toward ensuring that people with disabilities have access to resources and supports that allow them to have a higher quality of life. For example, a simple and low-cost intervention, such as having a volunteer visit, can extend and improve the life of terminally ill patients (Herbst-Damm & Kulik, 2005). One of the most important ethical principles in medicine is that the patient has autonomy (Angell, 1997). Terminally ill patients may spend months experiencing excruciating pain and discomfort in the process of dying. The extent of pain felt and the amount of cognition present are criteria that can be used to argue for allowing a person to end their own life. It is still very hard to draw a line. Even if a person is in extreme pain, palliative care (a form of treatment aimed at alleviating symptoms without necessarily affecting the cause) could be used (Tanuseputro et al., 2017). If a person is in a coma and has no measurable cognitive functioning, there is still no guarantee that cognition will not return or that the person is not thinking or feeling. Sometimes the decision to cut off life support is made easier by the patient having filled out an advance directive or living will in which they clearly specify the conditions under which life support should be terminated. Families also play a major role in this issue (Mayer et al., 2017). However, research shows that surrogate preferences can inaccurately reflect patients’ treatment wishes (Haley et al., 2002). Families often provide the majority of care for individuals with chronic illness for many reasons, including a sense of attachment, cultural expectations, and preferences for avoiding institutional care. Although it is optimal for the families, patients, and health-care providers to have ongoing discussions about goals of care, it is often only when the patient’s condition worsens that decisions regarding end-of-life care take place. Research suggests that family members are often key decision makers for end-of-life issues regardless of patients’ prior preferences concerning end-of-life care. Doctors tend to consult with family members even in the presence of written advance directives from their patients (Mowery, 2007). A woman touching the cheeks of Terri Schiavo, who is lying in a bed. Terri Schiavo. Terri Schiavo of Florida had been in a coma for 15 years. Her husband wanted to turn off life support but her family wanted to keep her alive. She died in March 2005. What information can help decide what should have been done? Are There Stages? Many in the lay population have heard of the concept of stages of death and that people who are dying experience a series of emotions. Spoiler alert: It’s not what you think. This somewhat inaccurate belief stems from work published by Elisabeth Kübler-Ross (1969). Kübler-Ross interviewed more than 200 dying patients and concluded that the process of dying involves five stages that vary in their emotional content and intensity. First comes denial, an initial reaction to the thought of death. This stage lasts around 2 days, can be a form of emotional coping (see Chapter 6), and can mask anxiety without necessarily removing it. Next comes anger, a stage in which patients are upset that death is happening to them. In many ways, the fact that they are dying violates a sense of the world being just. Most people believe that they do not deserve to die because they have been good or at least have not been bad enough to be punished with death. There is often misplaced resentment and a lot of irritability. Bargaining comes next. Patients try to restore their belief in a just world and may promise to be good or live life better (e.g., give a lot to charity) in exchange for life. This trading for life then gives way to depression. The patient feels a lack of control and now grieves in expectation of their death, a process known as anticipatory grief. The depression is often driven by a realization that a person will be losing their past and will also be losing all that was possible in a future. Finally, the patient may reach a stage of acceptance in which they fully acknowledge that death cannot be avoided. At this point, the patient is often very weak and faces death with a peaceful calm. Although these stages sound appropriate, and you probably can nod your head and see how a dying person could go through them, there is little empirical evidence for these stages (Jurecic, 2017). There is research suggesting each of these emotions are common (e.g., acceptance, Davis et al., 2017), but not to prove stages exist. Kübler-Ross used only cross-sectional research in a single culture and did not follow patients as they got closer to death. The fact is that people may experience these stages but not necessarily in the order just described. It is a dynamic process and people may go back and forth to different “stages.” One constant feature in Kübler-Ross’s stages of death is that most people will experience some depression just before death. How someone experiences death varies based on their culture, how much social support they have, the physiological progression of their disease, and other factors. Consequently, other researchers have attempted to explain the experience of dying (e.g., Pattison, 1977; Shneidman, 1980). When the time of death draws closer and it is clear that little can be done for the dying patient, it is important to help the person and their family prepare for death. We have discussed already some of the traditional ways this has been done, such as psychological counseling. There is an important additional dimension to consider when you look beyond the biology and psychology surrounding dying: culture. From a psychological perspective, fear, depression, and even denial of death may be common for patients and their families, but the exact experiences vary significantly across cultural groups (Galanti, 2014; Irish et al., 1993). European American health practitioners are often unintentionally ethnocentric, and this ethnocentricity makes it difficult for them to fully comprehend the experiences of people from other cultures. With an emotionally charged situation such as dying, this issue becomes even more important. Cultural differences become more evident from even a basic level of definition of key terms. It may seem clear what being dead is, but do not take death for granted. In many cultures, people are considered officially dead when Western biomedicine would consider them still alive and vice versa (Rosenblatt, 1993). There are correspondingly some significant differences in the expression and experience of emotions such as grief and loss. In some cultures, it is normal for people to cut themselves or otherwise hurt themselves to express their loss. Some East Indians, for example, fast for weeks as a sign of grieving. There are also cultural differences in the fear of death. African Americans report higher levels of death anxiety than European Americans (Depaola et al., 2003). Some of these cultural variations are seen in the rituals that accompany death. Making sure that the adequate ritual is conducted for the bereaved person is often critical to the health and coping of those left behind. Although they are not given attention in the health psychological literature to date, cultural differences in dealing with the dead may have important implications for psychological adjustment. Many cultural beliefs may clash with the beliefs of Western biomedicine, and hospital policies may prohibit certain practices, but they are important practices nonetheless. For example, in the American Indian culture, the burning of sage and other herbs is part of many religious ceremonies and is also used to prepare the soul of the dying person for the afterlife. Hospitals have nonsmoking policies and lighting a fire may seem clearly out of the question. But if sage is not burned, it could jeopardize the happiness of the dying patient’s soul and greatly hurt their family. Health-care professionals have to be aware of such cultural practices and negotiate a way to satisfy all concerned. For Muslims, there are also clear-cut practices that have to be followed at the time of death. As soon as a relative sees that the person is dead, they must turn the body to face Mecca (the site in the Middle East of the Kaaba, the Muslim’s holiest space). They also have someone sitting close by read the Koran (the word of God as channeled through the prophet Mohammed), close the body’s mouth and eyes and cover the face, and quickly bathe the body and cover it with white cotton (Gilanshah, 1993). Cultural Rituals for Death and Dying. Different cultures have very different rituals for death and dying. These caskets are part of the burial rituals of the people of Ghana who use different shapes of coffins for different individuals. Health practitioners are more effective in helping individuals during the difficult time of coping with death when they are sensitive to different cultural traditions. Many of the specific considerations needed are difficult for members of different ethnic groups to mention themselves. In the middle of coping with loss, it may be too much to expect a member of a different cultural group to explain exactly what is needed. It is likewise difficult for health-care workers to know all the different cultural idiosyncrasies surrounding death, but both groups need to work toward ensuring an adaptive experience for all concerned. For some groups, this sharing and explaining of cultural routines may be especially difficult. For example, given the negative points in the history of African Americans and American Indians in North America (Washington, 2006), both these ethnic groups have particularly strained relationships with European Americans and the health-care institution in particular (Barrett, 1998). Consequently, the development of separate models to understand how different cultural groups understand death and dying has increased (Corr & Corr, 2007). Barrett (1995, 1998), for example, has derived a list of special considerations for caregivers working with African Americans who experience loss (Table 11.5). Although devised for African Americans, these models serve as good reminders for health professionals working with any cultural group. #Understand the sociocultural influences from both Western and African traditions that combine to influence the attitudes, beliefs, and values. #Acknowledge and appreciate the uniqueness of the subgroups of African Americans (when in doubt, ask). #Be sensitive to basic differences in quality of life and differences in death rates and causes for African Americans versus European Americans. #Understand the impact of collective losses that African Americans often grieve for. #Include a consideration of SES as well as religion and spirituality. #Acknowledge the role of cultural mistrust regarding health. #Be sensitive to the value placed by African Americans on expressions of condolence. #Understand the role played by and expectations for the clergy and spiritual leaders (often higher expectations than for the medical community). Source: Barrett (1998). Reprinted with permission. Sex, Gender, and Death In the context of culture, it is also important to look at sex differences relating to the experience of death. Scholarship in death studies suggests that the different perceptions and experiences of men and women must also (in addition to cultural differences) be taken into consideration to best help those dying as well as those caring for the dead and grieving (Noppe, 2004). Martin and Doka (2000) remarked that the benchmark for grieving is normally set as how women handle loss. Women tend to show emotion, seek social support, talk about loss, and allow time to grieve openly, things not normally done by men (Cook, 1988). In a major review of gender differences in adjustment to bereavement, Stroebe et al., (2001) reported that women express their emotions more than men, although they found little evidence for the hypothesis that working through grief helps them recover faster. Particularly interesting is the fact that men suffer relatively greater health consequences when grieving than women, possibly because widowers get less support than widows (Stroebe & Stroebe, 1983). Specifically, widowers are significantly more distressed and depressed than widows and also have a higher incidence of mental illnesses. Widows have been found to suffer from fewer physical health problems and illnesses than widowers and are less likely to die during the period of acute grief after the loss of a spouse (Stroebe et al., 2001). Keeping these ethnic and sex differences in mind is clearly important in understanding how different subgroups of people experience the certainty of death. 3q6s84ul1432bvl8647cvv8accl6ytx 2 317545 2693461 2024-12-26T23:52:34Z Tule-hog 2984180 adapt from [[:b:User:1234qwer1234qwer4/BookCat.js]] 2693461 javascript text/javascript // based on https://en.wikipedia.org/wiki/User:DannyS712/Draft_no_cat.js // Set `var bookCatAJAX = true;` to add BookCat without reloading the page $(() => { const bookCat = {}; window.bookCat = bookCat; bookCat.config = { name: '[[User:Tule-hog/BookCat.js|BookCat.js]]', debug: false }; bookCat.summary = "Added {{[[Template:BookCat|BookCat]]}} using " + bookCat.config.name; bookCat.run = function () { var editSummary = bookCat.summary; if ( bookCat.config.debug ) { console.log ( editSummary ); } var api = new mw.Api(); api.get( { action: 'query', titles: mw.config.get( 'wgPageName' ), prop: 'revisions', rvprop: 'content', rvslots: 'main', formatversion: 2 } ).done( function ( response ) { console.log( response ); var text = response.query.pages[0].revisions[0].slots.main.content; if(text.match(/\{\{\s*Bookcat\s*\}\}/i)){ return mw.notify("BookCat already present."); } text += '\n\n{{BookCat}}'; api.postWithEditToken( { action: 'edit', minor: true, title: mw.config.get( 'wgPageName' ), text: text, summary: editSummary } ).done( function() { if(bookCatAJAX === undefined || bookCatAJAX == false){ location.reload(); } else mw.notify("BookCat successfully added."); } ); } ); }; }); $( document ).ready( () => { if ( mw.config.get( 'wgNamespaceNumber' ) === 0 && mw.config.get('wgAction') === 'view' && mw.config.get('wgCategories').every(e => !e.includes("Book:")) ) { mw.loader.using( [ 'mediawiki.util' ], function () { var link = mw.util.addPortletLink( 'p-cactions', '#', 'BookCat', 'ca-bookcat', 'Add BookCat'); $( link ).click( function ( event ) { event.preventDefault(); mw.loader.using( 'mediawiki.api', window.bookCat.run ); } ); } ); } } ); jaism3uwp9ud9augd7vzm53yrlu55ko 10 317546 2693481 2024-12-27T00:15:13Z Watchduck 137431 New resource with "{{Collapsible START|sharp families by quaestor weight|collapsed light}} {{multiple image | align = left | total_width = 420 | image1 = 3-ary Boolean functions; families with quaestor weight 1.svg | image2 = 3-ary Boolean functions; families with quaestor weight 1 (indices).svg | footer = quaestor weight 1 }} {{multiple image | align = left | total_width = 420 | image1 = 3-ary Boolean functions; families with quaestor weight 3.svg | image2 = 3-ary Boolean functio..." 2693481 wikitext text/x-wiki {{Collapsible START|sharp families by quaestor weight|collapsed light}} {{multiple image | align = left | total_width = 420 | image1 = 3-ary Boolean functions; families with quaestor weight 1.svg | image2 = 3-ary Boolean functions; families with quaestor weight 1 (indices).svg | footer = quaestor weight 1 }} {{multiple image | align = left | total_width = 420 | image1 = 3-ary Boolean functions; families with quaestor weight 3.svg | image2 = 3-ary Boolean functions; families with quaestor weight 3 (indices).svg | footer = quaestor weight 3 }}<noinclude> [[Category:Families of Boolean functions]] </noinclude> 02kl8cyxjt4bgmkhzdm0ydfw68giu5s 10 317547 2693482 2024-12-27T00:18:16Z Watchduck 137431 New resource with "{| class="wikitable collapsible collapsed" style="text-align: center;" !colspan="8"| arity to integer |- !rowspan="2" colspan="2"| sequence !colspan="6"| arity |- ! 0 !! 1 !! 2 !! 3 !! 4 !! 5 |- | {{oeislink|A000231}} | families | 2 || 3 || 7 || 46 || 4336 || 134281216 |- | | '''super'''-families | 1 || 2 || 5 || 30 || 1973 || 57805981 |- | | '''balanced''' families <small>(central values of {{oeislink|A054724}})</small> | || 1 || 3 || 14 || 870 || 18796230 |- | {{oe..." 2693482 wikitext text/x-wiki {| class="wikitable collapsible collapsed" style="text-align: center;" !colspan="8"| arity to integer |- !rowspan="2" colspan="2"| sequence !colspan="6"| arity |- ! 0 !! 1 !! 2 !! 3 !! 4 !! 5 |- | {{oeislink|A000231}} | families | 2 || 3 || 7 || 46 || 4336 || 134281216 |- | | '''super'''-families | 1 || 2 || 5 || 30 || 1973 || 57805981 |- | | '''balanced''' families <small>(central values of {{oeislink|A054724}})</small> | || 1 || 3 || 14 || 870 || 18796230 |- | {{oeislink|A001320}} | '''self-complementary''' families | || 1 || 3 || 14 || 240 || 63488 |- | {{oeislink|A051502}} | number of families with '''maximal size''' <math>2^{arity}</math> <small>(right column of {{oeislink|A227725}})</small> | 2 || 1 || 2 || 23 || 3904 || 134156284 |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> hqk787tf5cwit9zb90luh0wknjsswwf 2693538 2693482 2024-12-27T00:55:10Z Watchduck 137431 2693538 wikitext text/x-wiki {| class="wikitable collapsible collapsed" style="text-align: center;" !colspan="8"| arity to integer |- !rowspan="2" colspan="2"| !colspan="6"| arity |- ! 0 !! 1 !! 2 !! 3 !! 4 !! 5 |- | {{oeislink|A000231}} | families | 2 || 3 || 7 || 46 || 4336 || 134281216 |- | | '''super'''-families | 1 || 2 || 5 || 30 || 1973 || 57805981 |- | | '''balanced''' families <small>(central values of {{oeislink|A054724}})</small> | || 1 || 3 || 14 || 870 || 18796230 |- | {{oeislink|A001320}} | '''self-complementary''' families | || 1 || 3 || 14 || 240 || 63488 |- | {{oeislink|A051502}} | number of families with '''maximal size''' <math>2^{arity}</math> <small>(right column of {{oeislink|A227725}})</small> | 2 || 1 || 2 || 23 || 3904 || 134156284 |}<noinclude> [[Category:Families of Boolean functions]] </noinclude> 68qnca5ivdswp8qq1a9avrk6kkad5yz 0 317548 2693487 2024-12-27T00:25:51Z Tule-hog 2984180 Tule-hog moved page [[Network+/Architecture/Media/Introduction]] to [[Network+/Old guides/Network media]]: alter parent 2693487 wikitext text/x-wiki #REDIRECT [[Network+/Old guides/Network media]] r5h8yqpwl6rgp4wjlr7y3h6i03vw54h 2693490 2693487 2024-12-27T00:26:10Z Tule-hog 2984180 nominate speedy 2693490 wikitext text/x-wiki {{speedy|C10}} #REDIRECT [[Network+/Old guides/Network media]] b3forecrwlfx2aqwla7q7rw5drhnbj9 1 317549 2693489 2024-12-27T00:25:51Z Tule-hog 2984180 Tule-hog moved page [[Talk:Network+/Architecture/Media/Introduction]] to [[Talk:Network+/Old guides/Network media]]: alter parent 2693489 wikitext text/x-wiki #REDIRECT [[Talk:Network+/Old guides/Network media]] grcc63dzk2jnvu6yxho4qw68fh6swee 0 317550 2693495 2024-12-27T00:31:41Z Tule-hog 2984180 Tule-hog moved page [[Network+/Architecture/Services/Reverse Proxy/Review list of available proxy servers]] to [[Network+/Activities/Review list of available proxy servers]]: alter parent 2693495 wikitext text/x-wiki #REDIRECT [[Network+/Activities/Review list of available proxy servers]] mmfofzidgvn891ut35xwpuq496anmlg 2693496 2693495 2024-12-27T00:32:02Z Tule-hog 2984180 nominate speedy 2693496 wikitext text/x-wiki {{speedy|C10}} #REDIRECT [[Network+/Activities/Review list of available proxy servers]] kzp2n17kcxju1u22giob33ayr3wkq65 0 317551 2693501 2024-12-27T00:35:34Z Tule-hog 2984180 Tule-hog moved page [[Network+/Standards/OSI Model/Introduction]] to [[Network+/Old guides/OSI Model]]: alter parent 2693501 wikitext text/x-wiki #REDIRECT [[Network+/Old guides/OSI Model]] j787gal055zpskyctpi7544h0ke91i7 2693502 2693501 2024-12-27T00:36:01Z Tule-hog 2984180 nominate speedy 2693502 wikitext text/x-wiki {{speedy|C10}} #REDIRECT [[Network+/Old guides/OSI Model]] 1pznzkirgcfde6u0rbwgkw26e90jxdj 0 317552 2693504 2024-12-27T00:38:27Z Tule-hog 2984180 Tule-hog moved page [[Network+/Standards/OSI Model/OSI Components]] to [[Network+/Old guides/OSI Model/OSI Components]]: alter parent 2693504 wikitext text/x-wiki #REDIRECT [[Network+/Old guides/OSI Model/OSI Components]] mbrosb3e0sk1s9o6vhlo49a6buqss21 2693508 2693504 2024-12-27T00:40:32Z Tule-hog 2984180 nominate speedy 2693508 wikitext text/x-wiki {{speedy|C10}} #REDIRECT [[Network+/Old guides/OSI Model/OSI Components]] 14lenpn413gex7aq5lot303yfdlvjng 2 317553 2693537 2024-12-27T00:55:09Z Mickie-Mickie 2230175 creating a personal page 2693537 wikitext text/x-wiki ''Welcome to the personal page of Mickie'' bu3szrsomsbz8m2h19vppws5e5doank 0 317554 2693540 2024-12-27T00:58:23Z Tule-hog 2984180 Tule-hog moved page [[Network+/Standards/TCP/IP Model/Introduction]] to [[Network+/Old guides/IP Model]]: alter parent 2693540 wikitext text/x-wiki #REDIRECT [[Network+/Old guides/IP Model]] efflpmprqg0t5b57tkf20bn5le1w7jy 2693546 2693540 2024-12-27T01:01:02Z Tule-hog 2984180 nominate speedy 2693546 wikitext text/x-wiki {{speedy|C10}} #REDIRECT [[Network+/Old guides/IP Model]] p8nyr3jhoqr2y7vclpc9hllfozg8inc 1 317555 2693542 2024-12-27T00:58:23Z Tule-hog 2984180 Tule-hog moved page [[Talk:Network+/Standards/TCP/IP Model/Introduction]] to [[Talk:Network+/Old guides/IP Model]]: alter parent 2693542 wikitext text/x-wiki #REDIRECT [[Talk:Network+/Old guides/IP Model]] nyx0vfkbkccos5nxmkk54immx3yqtjn 10 317558 2693580 2024-12-27T04:25:11Z Tule-hog 2984180 test conditional language 2693580 wikitext text/x-wiki {{ambox |type=speedy |text=This {{#if:{{NAMESPACE}}|{{lc:{{NAMESPACE}}}}|[[wv:mainspace|mainspace]]}} page may qualify for [[Wikiversity:Deletions|speedy deletion]] because: {{{reason|{{{1|''no [[Wikiversity:Deletions|reason]] given''}}}}}}<br/>{{#ifeq:{{{self}}} |yes |If you disagree, please remove this notice.<br /> |If you disagree or intend to fix it, and '''you have not contributed to it before''', you may remove this notice. If you have contributed before and disagree, please explain why on {{#ifeq:{{NAMESPACE}} |{{TALKSPACE}} |this discussion page |[[{{TALKPAGENAME}}|the discussion page]] }}, after adding <span style="font-family:'Lucida Console', monospace">{{Tl|hangon}}</span> to the top of the {{#if:{{NAMESPACE}}|{{lc:{{NAMESPACE}}}}|resource}}. This will alert [[Wikiversity:Support staff|curators and custodians]] to your intention, and may permit you the time to write your explanation. }} ---- <span class="plainlinks">''Before [{{fullurl:{{SUBJECTPAGENAME}}|action=delete}} deleting] check the [[{{TALKPAGENAME}}|discussion page]], [[Special:Whatlinkshere/{{SUBJECTPAGENAME}}|what links here]], [{{fullurl:{{SUBJECTPAGENAME}}|action=history}} history] ([{{fullurl:{{SUBJECTPAGENAME}}|diff=0}} last edit]), the [{{fullurl:Special:Log|page={{SUBJECTPAGENAMEE}}}} page log], and [[Wikiversity:Deletions]]. {{{notes|}}}''{{subpagesif}}</span> }}<includeonly> [[Category:Candidates for speedy deletion|{{PAGENAME}}]]</includeonly><noinclude> {{Documentation}} [[Category:Deletion templates|{{PAGENAME}}]] </noinclude> 3plbgf78aajv0ar0dsusdlowir8hx9r 6 317559 2693585 2024-12-27T05:54:32Z Young1lim 21186 {{Information |Description=LIB.2A: Shared Libraries (20241227 - 20241226) |Source={{own|Young1lim}} |Date=2024-12-27 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2693585 wikitext text/x-wiki == Summary == {{Information |Description=LIB.2A: Shared Libraries (20241227 - 20241226) |Source={{own|Young1lim}} |Date=2024-12-27 |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}} cs4dcxv5cv4aq25cxu83mecv9vdew9j 6 317560 2693592 2024-12-27T10:46:20Z Young1lim 21186 {{Information |Description=Borrows (20241224 - 20241223) |Source={{own|Young1lim}} |Date=2024-12-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2693592 wikitext text/x-wiki == Summary == {{Information |Description=Borrows (20241224 - 20241223) |Source={{own|Young1lim}} |Date=2024-12-24 |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}} 3oyfc3nlvlnbpdzhtrg5rbknx6q1j2p 2693594 2693592 2024-12-27T10:47:01Z Young1lim 21186 /* Summary */ 2693594 wikitext text/x-wiki == Summary == {{Information |Description=Borrows (20241224 - 20241223) |Source={{own|Young1lim}} |Date=2024-12-27 |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}} 3z3tmlwwx52dlm9hlzmzlhaabr1oyg0 6 317561 2693595 2024-12-27T10:47:21Z Young1lim 21186 {{Information |Description=Borrows (20241225 - 20241224) |Source={{own|Young1lim}} |Date=2024-12-27 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2693595 wikitext text/x-wiki == Summary == {{Information |Description=Borrows (20241225 - 20241224) |Source={{own|Young1lim}} |Date=2024-12-27 |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}} 5cq6g2ua83yd8ip6w4jjdnp6ydqj8xd 6 317562 2693597 2024-12-27T10:48:15Z Young1lim 21186 {{Information |Description=Borrows (20241226 - 20241225) |Source={{own|Young1lim}} |Date=2024-12-27 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2693597 wikitext text/x-wiki == Summary == {{Information |Description=Borrows (20241226 - 20241225) |Source={{own|Young1lim}} |Date=2024-12-27 |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}} laoelu7iqxii77i52qrwme0p06m2rpa 6 317563 2693599 2024-12-27T10:49:11Z Young1lim 21186 {{Information |Description=Borrows (20241227 - 20241226) |Source={{own|Young1lim}} |Date=2024-12-27 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2693599 wikitext text/x-wiki == Summary == {{Information |Description=Borrows (20241227 - 20241226) |Source={{own|Young1lim}} |Date=2024-12-27 |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}} e6j2krkpse7quukhbaytkq3ony92qrp